He is suspended permanent. for what? in three space on earth. It will align itself in such a way that one end points towards the north. and the other towards the south. This is the principle of our compass.

他被永久停职了。因为什么？在地球上的三个地方。它会以这样一种方式排列，一端指向北方。另一端指向南方。这就是我们指南针的原理。

We use the compass for navigation.

我们用指南针来导航。

There is a magnet, a magnetic bar inside of the compass. One end points towards the north and another end points towards the south. and you use this direction to navigate.

指南针里面有一根磁棒。一端指向北方，另一端指向南方。你利用这个方向来进行导航。

And one end is defined as the north pole, and the top point is defined as south pole.

其中一端被定义为北极，另一端被定义为南极。

So you have a north pole and a south pole. for magnet.

所以，你有北极和南极。对于磁铁来说。

So why this happens? Because the magnet has magnetic field. just like this picture shows. This dashed line are magnetic field lines. So the magnetic bar, the bar magnet, produce magnetic field lines.

那么，为什么会发生这种情况呢？因为磁铁具有磁场。就像这张图所展示的那样。这些虚线就是磁力线。所以，条形磁铁会产生磁力线。

and use a to represent the magnetic field.

我们用这个符号来表示磁场。

magnetic flux density.

这就是磁通密度。

And bar magnets always have to align one end. for example, and one end the north. It's because the magnetic field lines.

条形磁铁总是有一端会指向特定的方向。比如，一端是北极。这是因为磁力线的缘故。

try to align them with that of the earth.

它们会试图与地球的磁场方向保持一致。

The earth is actually is a very giant magnet.

实际上，地球就是一个巨大的磁铁。

So if you we have south pole. and north pole. on the earth, right? So the earth is also a magnet. If you have a small magnet,

所以，如果你知道它有南极，也有北极。在地球上，对吧？所以，地球也是一个磁铁。如果你有一个小磁铁，

and this small magnet will try to align direction. with the earth.

它会自动调整方向。与地球对齐。

And we have introduced the magnetic field lines, right? It's a dashed line. It's show in the last figure. Magnetic field line is from north to south. outside of the magnet.

我们之前介绍过磁感线，对吧？它是一条虚线。在最后一张图中有显示。磁感线是从北极指向南极的。这是在磁铁外部。

and can be traced using a small compass.

可以用一个小指南针来追踪。

We use compass for navigation.

我们用指南针来导航。

but the inside, the magnetic field lines travel from south to north. and finally you will get a closed loop, right? that's what the figure shows. For outside of the magnet, the magnetic field lines travel from north to south. but inside, it travels from south to north.

但在内部，磁感线是从南极指向北极的。最后，你会得到一个闭合的回路，对吧？这就是图中所展示的。对于磁铁外部来说，磁感线是从北极指向南极的。而在内部，则是从南极指向北极。

And the magnetic field lines always exist in loops. because outside it travels from north to south, and inside it travels from south to north. so it's a loop. There's no beginning or no end. You don't know where it starts. You don't know where it starts and where it ends. There's no beginning or end. And they never intersect each other. There are many magnetic field lines. But they never intersect.

而且，磁感线总是以闭合回路的形式存在。因为外部是从北向南，而内部是从南向北。所以，这就形成了一个闭合回路。它没有起点，也没有终点。你无法确定它从哪里开始。你也不知道它在哪里结束。没有开始，也没有结束。而且，它们永远不会相互交叉。磁感线有很多条。但它们从不相交。

Here's one magnetic field line. Probably there's another line. And they never intersect.

这里有一条磁感线。可能还有另一条。它们依然不会相交。

And they always try to shorten themselves. So how can we find the shortest way from north to south?

而且，它们总是试图缩短自己。那么，我们该如何找到从北到南的最短路径呢？

And always try to shorten themselves to find the shortest path. outside.

它们总是通过缩短自身来寻找最短的路径。这是在外部情况下的表现。

And magnetic field is generally represented in vector notation.

通常，磁场是用矢量符号来表示的。

I think all of you saw already these videos at high school, right? This is the symbol of B. We use the symbol B to represent the magnetic fields. It has also a direction. So it's a vector. The magnetic field also has direction.

我想大家在高中时应该都看过这些视频，对吧？这是B的符号。我们用符号B来表示磁场。它也有方向。所以它是一个矢量。磁场也有方向。

You know the equation F = I times L times B. So it's the formula. If you use the B x L, you will get the force. B is the magnetic field here. It has direction and also has magnitude. So it's a vector.

你们知道那个公式：F等于I乘以L乘以B。这就是公式。如果用B乘以L，就能得到力。这里的B就是磁场。它既有方向，也有大小。所以它是一个矢量。

And sometimes magnetic field also defined as magnetic flux density.

有时候，磁场也被定义为磁通量密度。

The magnetic flux density is a quantity to measure the magnetic field. at a given point. So how strong the magnetic field is.

磁通量密度是一个用来衡量磁场的物理量。在特定点上。也就是磁场的强弱。

It can use to be quantified.

它可以被量化。

or to measure the magnetic field.

用来测量磁场。

And the unit of a magnetic field or flux density, flux density or magnetic field is

磁场或磁通密度的单位是

tesla, used T.

特斯拉，用T表示。

T is the unit of magnetic fields. or we can also use ampere meter.

T是磁场的单位。或者我们也可以用安培米。

why can we also use newton divided by ampere meter? because you know the formula, right? force equals to B I A. I is the current. I is the ampere. A is the length. force has the unit newton. The newton divided by I and A will get B. force has the unit ampere. A is length. so force divided by current and length you will get the the magnetism. so it also has unit newton per ampere meter.

为什么我们也可以用牛顿除以安培米呢？因为你知道公式，对吧？力等于BIA。I是电流。单位是安培。A是长度。力的单位是牛顿。那么，牛顿除以I和A，得到的就是B。力的单位是安培。A是长度。所以，力除以电流和长度，就会得到磁感应强度。所以它也有单位，就是牛顿每安培。每米。

and the earth is also a very big magnet. but its magnetic field is quite small. the earth is also a magnet, but its magnetic field is very small.

而且，地球也是一个非常大的磁铁。不过它的磁场相当弱。地球也是磁铁，但它的磁场非常小。

and here's a typical conclusion.

这里有一个典型的结论。

a typical conclusion for magnet.

这是关于磁铁的一个典型结论。

So we know that the lost pole attract and this south pole

我们知道，异极相吸，也就是南极。

so this is known as a magnetic force. unlike force attract. Like force repel.

这被称为磁力。同极相斥。

so here you can see when we put the north pole closer,

在这里，大家可以看到，当我们把北极靠近时，

close the south pole, they will attract. attract with each other.

它会吸引南极。它们会相互吸引。也就是彼此吸引。

so the magnetic field lines also

所以，磁场的磁感线也是

travel from north to south. because there are two magnets. The figure below means the like poles, they both have the end. They also have the north pole. So they will pair each other.

从北极指向南极。因为这里有两个磁铁。下面的图表示同极相对，它们的两端都是相同的极性。而且都是北极。所以它们会相互排斥。

Here, the magnetic field lines travel from north to south outside. On inside, travel from south to north.

在这里，磁感线是从外部从北极指向南极的。而在内部，则是从南极指向北极。

And when they attract each other, the north pole of one magnet also has the magnetic field lines. travel to another south pole.

当它们相互吸引时，北极也会产生磁感线。并指向另一个磁铁的南极。

but if you have the same pole, they will repel each other. there's no magnetic field lines.

但是，如果两个磁极相同，它们就会相互排斥。此时就没有磁感线了。

travel from one magnet to another.

磁感线不会从一个磁铁延伸到另一个磁铁。

okay, magnetic induction. magnetic induction is a process by which a material is magnetized. magnetized by an external magnetic field.

好的，磁感应。磁感应是一个过程，通过这个过程，材料会被磁化。它是被外部磁场磁化的。

So here is an example. Here we have a U-shape magnet. If we put a material between the U-shape. the material will be magnetized.

这里有一个例子。这里有一个U形磁铁。如果我们把一种材料放在U形磁铁之间，这种材料就会被磁化。

Here is an example. If you place a soft iron,

这里有一个例子。如果你把一块软铁

see, H is the symbol for magnetic field. and we increase it again and again.

看，H是磁场的符号。我们会不断地增加它。

So in your H field, which means the magnetic field. He put a soft iron block.

所以，H场就是指磁场。他放了一块软铁块。

in a magnetic field. The flux will pass through the iron. and temporarily magnetize it.

放在磁场中。磁通量会穿过这块铁。并使其暂时磁化。

due to magnetic induction.

这是由于磁感应现象。

and here's a very important law. There's a line through the core. There's no line through the core, okay? The lines travel around the core. It never goes through.

这里有一个非常重要的定律。有一条线穿过铁芯。没有线穿过铁芯，明白吗？这些线是绕着铁芯走的。它永远不会穿过铁芯。

That means if there's an object, there's an object inside of this core.

这意味着，如果有一个物体，它就在这个铁芯内部。

If there's an object inside of this core, object will not be magnetized.

如果铁芯里面有物体，这个物体就不会被磁化。

Okay, here is the direction. the direction of the magnetic field.

好的，这里是方向。磁场的方向。

So, in magnetic flux, is a measure of the total magnetic field. the total magnetic field which pass through a given area.

那么，在磁通量中，它是总磁场强度的度量。即穿过给定面积的总磁场。

So if you define the area, and you can use the magnetic flux to measure. how many magnetic fields pass through this area. And here you get the simple equation to compute the magnetic flux. phi is the magnetic field strength, magnetic field. B.

所以，如果你定义了一个面积，就可以用磁通量来衡量。有多少磁场穿过这个面积。这里有一个简单的公式，可以用来计算磁通量。Phi代表磁场强度，也就是磁场。用B表示。

or flux density.

或者叫磁通密度。

And theta is the angle. It's the angle between the direction of the magnetic field.

而theta是角度。它是磁场方向之间的夹角。

and the area.

以及面积。

The area sometimes has an angle. with the magnetic field.

面积有时也会有一个角度。与磁场有关。

The unit of a magnetic flux is Weber. And based on this equation, we know the unit of magnetic field is tesla. and area is meter square. So weber can be expressed by tesla times meter square.

磁通量的单位是韦伯。根据这个公式，我们可以知道，磁场强度的单位是特斯拉。而面积的单位是平方米。所以，韦伯可以用特斯拉乘以平方米来表示。

or you change to other units, one tesla is one weber. weber divided by meter square.

或者换一种说法，1特斯拉等于1韦伯。也就是韦伯除以平方米。

Okay, now we talk about the relation between the magnetic field and electrical currents. Magnetic field can produce current, right? And current also can produce magnetic field. This is the famous Faraday's law. We will introduce them later.

好的，现在我们来讨论一下磁场和电流之间的关系。磁场可以产生电流，对吧？而电流也可以产生磁场。这就是著名的法拉第定律。我们稍后会详细介绍。

So if a wire with electrical current I,

那么，如果有一根通有电流I的导线，

it will produce a magnetic field. around. Its directions can be given by the right hand. right hand thumb rule.

它就会产生一个磁场。就在导线周围。其方向可以用右手定则来判断。也就是右手拇指法则。

And here's the right hand thumb rule.

这就是右手拇指法则的示意图。

because the current flow is into away the reviewer. If you use the right hand, your thumb is going inside. so the direction of the four fingers is the direction of the magnetic field lines.

因为电流是流向远离观察者的方向的。如果你用右手，你的拇指就会指向里面。所以，四指弯曲的方向就是磁场的方向。

So, electrical currents can produce magnetic fields. And we can use the right hand thumb rule to determine the direction. of the H field or magnetic field.

所以，电流可以产生磁场。我们可以用右手拇指法则来确定方向。即H场或磁场的方向。

And here's another example. If you have a current is going through this wire, and it will produce the magnetic field. around this wire.

这里还有一个例子。如果电流流过这根导线，它就会产生磁场。就在导线周围。

in circle, in circle this line.

这个磁场是环绕着导线的，呈圆形分布。

and if you have a coil, just like this example shows, if you have a coil, the relationship between the current and the H field. It's found using the right-hand rule. We have introduced this in the previous slide. you can use the right-hand thumb rule determine the direction of the magnetic field.

如果你有一个线圈，就像这个例子展示的，那么就是电流与H场之间的关系。这可以用右手定则来确定。我们在之前的幻灯片中已经介绍过了。你可以用右手拇指法则来确定磁场的方向。

Here is an example. If you have a coil like this, so if you use the right-hand rule, your four fingers work in the direction of the current. and your thumb is the direction of the magnetic flux line.

这里有一个例子。如果你有一个像这样的线圈，那么使用右手定则，四根手指的方向就是电流的方向。而拇指的方向就是磁感线的方向。

inside of the magnet. Okay, you just use your right-hand rule like this. and the thumb is the direction of the magnetic lines inside the magnet.

也就是磁铁内部的方向。好的，你就这样使用右手定则。拇指指向的就是磁铁内部的磁感线方向。

So you can determine this is a north. this is a S pole. Because the magnetic field lines travel inside of the magnet.

所以你可以判断，这是一个北极。这是南极。因为磁感线是在磁铁内部从南极指向北极的。

from south to north. That means you can use this method to determine which end is a north. which end part is south pole. Or if the south pole and north pole is given, you can also use this method to determine the direction of the current.

也就是从南到北。这意味着，你可以用这种方法来判断哪一端是北极。哪一端是南极。或者，如果已经给出了南极和北极，你也可以用这种方法，来确定电流的方向。

or the other way around.

或者反过来也可以。

if the current direction is given, you can use this method to determine where is the north pole. and where is the south pole.

如果已知电流方向，就可以用这个方法来判断北极的位置。以及南极的位置。

Okay, now we have this equation again. If you put a wire, in a magnetic field, with current I,

好的，现在我们又得到了这个公式。如果你放一根导线，放在磁场中，通以电流I，

and because of the magnetic field, it will produce a force. the magnitude of the force is the

那么由于磁场的作用，就会产生一个力。这个力的大小是

magnetic field times the current. So it's simple.

磁场强度乘以电流。所以，很简单。

and times the length of wire.

再乘以导线的长度。

and also about here's a wire, but if you just have one moving charge, you can use the q v b. to compute the force applying to the charge. q is the, we know q is the symbol for electric charge, right? v is the velocity. of the charge. How quick does it move? How quick does the charge move? b is the magnetic field.

另外，这里是一根导线，但如果你只有一个运动的电荷，你可以用qvB来计算。来计算作用在电荷上的力。这里的q，我们知道是电荷的符号，对吧？v是速度。也就是电荷的速度。它移动得多快？电荷移动的速度有多快？B是磁场。

and q b e is the force applied to the charge.

而q、B、E就是作用在电荷上的力。

so basically the equation, if you put a wire, this length is a. has current i.

所以，基本上这个公式就是，如果你有一根导线，它的长度是a。电流是I。

the magnetic field can produce the force. to the wire. this makes the wire move.

磁场会产生力。作用在这根导线上。这会让导线移动。

here's an example. You have two magnets, right? if you put a material or a wire, if there's no current,

这里有一个例子。你有两块磁铁，对吧？如果你把一种材料或导线放在中间，如果没有电流，

the material or the wire will not move. but if you close the switch, we close the switch, there will be current. the wire will move.

材料或导线也不会移动。但是如果你闭合开关，我们闭合开关，就会有电流。导线就会移动。

and the coil will move. This is the principle of electric motor.

线圈也会移动。这就是电动机的原理。

There are many, currently we have many electrical cars. on the public road. They use the electric motor. to drive the car. Electric motor exactly, if you give the current, it also a magnet.

现在有很多电动汽车。在公共道路上行驶。它们使用的是电动机。来驱动汽车。电动机就是这样的，如果你给它通电，它也会变成磁铁。

and the magnet will be rotated.

而且磁铁会旋转。

here we have two parallel wires. and for each wire, if we put them in a magnetic field, there will be a force for each wire.

这里我们有两根平行的导线。对于每一根导线，如果我们把它放在磁场中，就会受到一个力的作用。作用在每根导线上。

And if the current has the same direction, the wire will attract.

如果电流方向相同，这两根导线会相互吸引。

And if the wire has a different direction or opposite direction, the wire will repel each other. because for each wire there will be a force.

如果电流方向相反，那么这两根导线就会相互排斥。因为每根导线都会受到力的作用。

And based on, you can determine the direction of the force based on the right-hand rule. right-hand thumb rule.

你可以根据右手定则来判断力的方向。也就是右手定则。

If you know the direction of the magnetic field and you know the current direction, you can use your right hand. to determine the force for each wire. Once you determine the force for each wire, you can then find whether they are attracting each other or repel. And the conclusion is that if the current has the same direction, they will attract each other. if the current has different direction, they will repel.

如果你知道磁场的方向和电流的方向，就可以用你的右手。来确定每根导线所受的力。一旦确定了每根导线所受的力，你就能判断它们之间是相互吸引还是相互排斥。结论是，如果电流方向相同，它们就会相互吸引。如果电流方向相反，它们就会相互排斥。

Okay, for Ampere's law, the famous person that

好的，关于安培定律，那位著名的人物

the scientist Paul Ulrich.

就是科学家保罗·尤利希。

He was inspired towards science through reading.

他是通过阅读受到启发，从而投身科学的。

He has read a lot of books.

他读了很多书。

And also practical experiment and finally he found the current can produce magnetic field.

此外，他还进行了大量的实验，最终发现电流可以产生磁场。

And during the time of Oersted, the source of the commercial electricity was chemical cell batteries. And at that time, there is also source, voltage source or current source. and they are typically chemical.

在奥斯特那个时代，商业用电的来源是化学电池。当时还有另一种电源，那就是电压源或电流源。它们通常是化学形式的。

So Oersted's main contribution is that he have found the relation between the magnetic field and the electrical current.

因此，奥斯特的主要贡献在于，他发现了这种关系。即磁场与电流之间的关系。

And from the left figure shows the current can produce a magnetic field.

左边的图显示，电流可以产生磁场。

And the right figure shows the magnetic field can produce a current.

而右边的图显示，磁场可以产生电流。

And the both has achieved these results. Oersted has done a lot of work. a lot of experiments. And some of them fails. and maybe only one succeed.

两者都取得了这些成果。奥斯特为此做了大量的工作。进行了许多实验。其中一些失败了。也许只有一项成功了。

Those success is just the final results. but what happens? success is the final results behind one thousand failures.

这些成功只是最终的结果。但过程是怎样的呢？成功背后，是一千次失败的积累。

for i guess the first observation is that if we change the current, we can produce the magnetic fields. So a changing current can produce magnetic fields.

我认为，第一个观察结果是，如果我们改变电流，就能产生磁场。所以，变化的电流可以产生磁场。

and also the other way around. If we change the magnetic fields, we can produce current.

反之亦然。如果我们改变磁场，就能产生电流。

It's the final conclusion. A changing current can produce magnetic fields and a change magnetic field can produce current.

这就是最终的结论。变化的电流可以产生磁场，而变化的磁场可以产生电流。

But if the magnet is static,

但是，如果磁场是静止的，

you will have zero voltage output. and if you're moving, if you're moving the magnet, if you're moving the magnet into the coil,

那么输出电压将为零。如果你让它移动，如果你把磁铁移入线圈中，

you will find the voltage is not zero. So, and if you move it faster, it's reduce. and appears as voltage meter reading.

你会发现电压不为零。而且，如果你移动得更快，电压会更高。这就会在电压表上显示出来。

This observation that if we change the magnetic field, it can produce current.

这一观察结果表明，如果我们改变磁场，就能产生电流。

When the magnet is moving into the coil, the magnetic field changes. that's reason it can have current. and the voltmeter is not zero.

当磁铁进入线圈时，磁场就会发生变化。这就是为什么会产生电流的原因。而且电压表也不为零。

and if you move the magnet outside, out of the coil,

如果你把磁铁移出来，也就是移出线圈，

you can also receive voltage. but in the reverse direction.

同样也能检测到电压。不过方向是相反的。

And here's a brief summary. So a change magnetic flux, is a circuit. and electric voltage in a circuit. So if you change the magnetic field, you can produce current.

这里是一个简要的总结。所以，磁通量的变化，会在电路中产生感应电流。也就是电路中的电压。因此，如果你改变磁场，就能产生电流。

And the magnitude of the electric voltage is proportional to the rate of changes of the flux.

而电压的大小与磁通量的变化率成正比。

And here is the equation to express this. phi, we know phi is the magnetic flux, right? So the rate of changes of the flux d phi divided by dt.

这里有一个公式来表达这一点。这里的φ，我们知道就是磁通量，对吧？那么，磁通量的变化率就是dφ除以dt。

is the rate of change of the flux. If the magnetic field changes, I get a current.

这就是磁通量的变化率。如果磁场发生变化，就会产生电流。

So here is the voltage.

所以，这里就是电压。

When you combine these two sentences together, you can find a changing magnetic flux in one coil also produce voltage across that coil. A changing magnetic flux will produce voltage.

把这两句话结合起来，你就能发现一个线圈中的磁通量变化也会在另一个线圈中产生电压。磁通量的变化会产生电压。

And here's the equation to express this. this description. phi is the magnetic flux. D phi by dt is the rate of change.

这里有一个公式来表达这一点。这个描述。phi代表磁通量。d phi/dt表示变化的速率。

phi can also be produced by current, right? So magnetic field can also be produced by currents. And if you change the currents, the magnetic fields can also be changed.

而磁通量也可以由电流产生，对吧？所以，磁场也可以由电流产生。如果你改变电流，磁场也会随之改变。

And we can use the description below, it says that the direction of the induced electromotive force are always

我们可以用下面的描述来解释，它指出，感应电动势的方向总是

try to prevent the motion.

试图阻止这种运动。

so you can set up a current that opposes the motion. direction of the induced electromotive force will always tend to set up a current. and prevent the motion.

所以，你可以产生一个与运动方向相反的电流。感应电动势的方向总是倾向于产生一个电流。从而阻止运动。

So we can use this description to determine the direction of the current. because the induced current will always try to prevent the motion.

所以，我们可以用这个描述来确定电流的方向。因为感应电流总是试图阻止运动。

prevent the motion.

阻止运动。

application of Lenz's law. application of Lenz's law, electromagnetic induction.

楞次定律的应用。楞次定律的应用，就是电磁感应。

and here you have a voltmeter. If you change the magnetic field or the flux, you can see the pointer rotates. That means it produces voltage.

这里有一个电压表。如果你改变磁场或磁通量，你就会看到指针会转动。这意味着它产生了电压。

And you can use a voltmeter to measure the voltage.

你可以用电压表来测量电压。

And there's many other use cases. electrical generator. It produces electricity, right? If there's no electrical generator, it can't produce electricity. But how do they produce? They change the flux here. It will produce electricity.

还有许多其他的应用场景。比如发电机。它能产生电能，对吧？如果没有发电机，就无法产生电。那它们是怎么产生的呢？就是通过改变这里的磁通量。这样就能产生电了。

Then how can we change the flux? If this coil rotates, the flux will always

那么，我们该如何改变磁通量呢？如果这个线圈旋转，磁通量就会不断变化。

because we know that the flux is the magnetic field times the area. times cosine theta, right? theta is the angle between the magnetic field and the coil. So if the coil rotates, because of the different angles, it will produce different magnetic flux.

因为我们知道，磁通量等于磁场强度乘以面积。再乘以cosineθ，对吧？这里的θ是指磁场与线圈之间的夹角。所以，当线圈旋转时，由于角度不同，就会产生不同的磁通量。

and finally, you are getting the current.

最终，就会产生电流。

from the generator.

来自发电机。

okay, we have introduced the RC circuit. in our previous lecture. here is a RA circuit. R is the inductor, L is the inductor. And also have a typical time constant.

好的，我们已经介绍了RC电路。是在上一节课中。这里是一个RL电路。R是电阻，L是电感。它也有一个典型的时间常数。

And here, of course, it will derive the time constant.

当然，这里会推导出时间常数。

The time constant is L divided by R. And for the capacitor, it's the R times the capacitance. for the RC.

时间常数等于L除以R。而对于电容电路，则是R乘以电容。也就是RC电路。

for the RC circuit, the time constant is R times the capacitance. But now the time constant is L divided by R. L is the inductance.

对于RC电路，时间常数是R乘以电容。但现在的时间常数却是L除以R。L是电感量。

Let's see how the curve changes. So here is a switch. We want to close the switch. Before we close the switch, or before t1, the circuit is open. so there is no current. and it won't change.

我们来看看曲线是如何变化的。这里有一个开关。我们想要闭合这个开关。在闭合开关之前，也就是t=1之前，电路是断开的。所以没有电流。而且也不会发生变化。

So y-axis is the current and x-axis is the time. And then we close the switch.

所以，y轴是电流，x轴是时间。然后我们闭合开关。

the current will increase slowly from zero. to obtain a stable value.

电流会从零开始缓慢增加。最终达到一个稳定的值。

The stable value is the voltage divided by resistance. So this is the final current. Because the inductance, inductor,

这个稳定值就是电压除以电阻。所以，这就是最终的电流。因为电感器，

can be finally regarded as a wire. So it's a wire, there's no resistance here. So the final stable current is just the voltage divided by the resistance. And the current increase slowly until the stable. And for the voltage, when the switch is closed,

最终可以被视为一根导线。所以它是一根导线，这里没有电阻。因此，最终的稳定电流就是电压除以电阻。电流会缓慢上升，直到达到稳定状态。那么对于电压来说，当开关闭合时，

and the inductance are not the voltage of the source

电感的影响并不明显。电源的电压

and it increase slowly. and it increase slowly until zero. because finally it's a wire, right? so it's like a short circuit. so there's no voltage here.

也会缓慢上升。一直增加到零。因为最终它只是一根导线，对吧？所以这就像是一个短路。所以这里没有电压。

And the voltage can be expressed as the inductance times the the rate change.

而电压可以表示为电感乘以电流的变化率。

And here's a

这里有一个

so if you just have one turn, for the coil.

假设你只有一圈线圈，也就是一个线圈。

or you just consider one turn. And the contribution of this one turn that I've given here plus is Z1.

或者你只考虑一圈。那么这一圈的贡献就是我这里给出的，就是Z1。

And the contribution of this one turn to the magnetic flux is phi1. phi is the proportionality for current. flows through the coil.

这一圈对磁通的贡献是Φ1。Φ是电流的比例系数。它流过线圈。

So if the current is larger, it will produce more magnetic flux.

所以，如果电流更大，它就会产生更多的磁通。

And k, so coefficient, is a constant value. but if you have n turn, so the total flux phi will be n times k. times i. i is the current.

而k，也就是这个系数，是一个常数。但是，如果你有n圈，那么总磁通Φ就等于n乘以k。再乘以电流i。i就是电流。

And based on the Faraday's law, we know the voltage is minus n. and it is the number of the turns times the rate change of the flux. And the flux can also be expressed by n k i.

根据法拉第定律，我们知道电压等于负的n。也就是线圈的匝数乘以磁通量的变化率。而磁通量也可以表示为n、k和i的乘积。

Because n is a constant value, k is a constant value, so we just put it outside. of the derivation. So finally you will get the k minus k n square. times the rate change of current.

因为n和k都是常数，所以我们可以把它们提到外面去。在求导的时候。所以最后得到的结果是k减去k n的平方。再乘以电流的变化率。

and k n square is a constant value. it's defined as the inductance. defined as the inductance of the coil. And we use L here. So finally you will get the voltage is the inductance times the rate change of the current. And don't forget the sign is minus. because as we mentioned before.

而k n的平方是一个常数。它被定义为电感。也就是线圈的电感。这里我们用L来表示。所以最后得到，电压等于电感乘以电流的变化率。别忘了，前面还有一个负号。因为正如我们之前提到的。

that produced voltage always try to prevent the change of the flux.

产生的电压总是试图阻止磁通量的变化。

It's the change of flux. And the produced voltage try to prevent the change of the flux. That's the reason we have a minus here.

就是磁通量的变化。而产生的电压试图阻止这种变化。这就是为什么这里有一个负号的原因。

And L is the inductance, it's a unit Henry. We use a symbol L.

而L代表电感，单位是亨利。我们用字母L来表示。

And inductance Henry can also be expressed by the voltage, one voltage times second. divided by ampere or ampere. So voltage times seconds per ampere.

电感（亨利）也可以用电压乘以时间来表示。再除以安培。也就是伏特·秒每安培。

So based on this equation, we can find the voltage across an inductor. It's proportional. It's proportional the time rate change of current. It's proportional because of the L. L is a constant value. and di divided by dt is the time rate of the current. So the voltage is proportional to the time rate

那么，根据这个公式，我们可以求出电感两端的电压。它是成正比的。与电流随时间的变化率成正比。之所以成正比，是因为L。L是一个常数。而di/dt就是电流的变化率。所以，电压与电流的变化率成正比。

and if we draw the equation using this figure, and x-axis is the time rate change of the while y-axis is the voltage. So you can get a linear line. The slope is the inductance.

如果我们用这张图来表示这个公式，横轴就是电流的变化率。纵轴则是电压。这样就能得到一条直线。线。斜率就是电感值。

And the inductance is the property of the inductor.

而电感是电感器本身的特性。

and it's always try to prevent the change of the current.

它总是试图阻止电流的变化。

and it also determines how much electric voltage is generated. per unit change of current.

它同时也决定了会产生多大的感应电压。也就是每单位电流变化所对应的电压。

Here's an example. It has an inductance of one henry. So its current changes. it's one ampere. per second. Based on the equation, so we can guess the produced electrical voltage is also one volt. just opposite the change of the current.

这里有一个例子。它的电感值为1亨利。那么，当它的电流发生变化时，变化量是1安培。每秒。根据公式，我们可以推断出产生的感应电压是1伏特。它的大小与电流的变化相反。

The produced electrical voltage tries to prevent this change.

产生的感应电压试图阻止这种变化。

And you know the voltage is the derivation of the current. right? So if you integrate, you can get the current. using the voltage.

大家都知道，电压是电流的导数。对吧？所以，如果你进行积分，就能得到电流。利用电压。

and the inductance is the k s square. and we change the equation a little bit.

而电感是k s的平方。我们对公式做了一点修改。

and we can finally get five times divided by i.

最终，我们可以得到5除以i。

five times n is also defined as a flux linkage.

5乘以n也被定义为磁链。

so this is the total flux linking the coil. total flux linking the coil times the number of the turns. number of turns is n. and the total flux linking the coil is phi. so five times n is the flux linkage.

所以，这就是与线圈相连的总磁链。总磁链乘以匝数。这里的匝数就是n。而与线圈相连的总磁链是φ。所以，5乘以n就是磁链。

so the inductance can also be described as the flux linkage. per unit current.

因此，电感也可以被描述为磁链。每单位电流。

so here is per unit current. and very similar to capacitors. in a capacitor, we know capacitor is a charge divided by voltage, right? so it's the charge q per unit voltage. so the inductance can also be described as the flux linkage per unit current. and here we have inductance is the flux linkage per unit current.

所以这里是每单位电流。这与电容非常相似。我们知道，电容是电荷除以电压，对吧？所以它是单位电压下的电荷q。那么，电感也可以被描述为单位电流下的磁链。这里我们看到，电感就是磁链与单位电流的比值。

c is q divided by v. and here we have phi divided by i.

而电容C等于电荷q除以电压v。而这里我们看到的是磁链除以电流I。

so inductance.

这就是电感。

and here again, the equation again, how can we calculate the inductance? you know the inductance is the flux linkage. phi times n divided by i. and in between the following bits,

这里再次给出了公式，我们如何计算电感呢？大家知道，电感就是磁链。也就是磁链乘以匝数，再除以电流。在接下来的内容中，

you can get n squared. times phi.

你可以得到n的平方。再乘以磁链。

and here we introduce another new variable, which is defined as reluctance. reluctance represent the opposition to magnetic opposition to magnetic flux. It depends on the geometry and material. So reluctance is defined as n and i.

在这里，我们引入另一个新的变量，它被定义为磁阻。磁阻代表了对磁场的阻碍。也就是对磁通的阻碍。它取决于几何形状和材料。因此，磁阻的定义是磁链除以电流。

divided by the flux phi.

再除以磁通。

The unit is ampere turns, ampere turns per we.

单位是安匝，即每韦伯的安匝数。

because here mmf is m i, right? mmf is equivalent to m i. i is the current. so the unit is ampere turns. so the reluctance has unit ampere turns per we.

因为这里的磁动势是NI，对吧？磁动势等同于NI。其中I代表电流。所以单位是安匝。因此，磁阻的单位是安匝每韦伯。

flux has unit we.

而磁通的单位是韦伯。

mmf is just used here to replace the m i.

在这里，我们只是用磁动势来代替NI。

and here we change the formula a little bit. we know the inductance is n square. divided by the flux phi.

这里我们对公式做了一点改动。我们知道，电感等于匝数的平方。再除以磁通。

so the inductance is equal to n square divided by phi.

所以，电感等于N的平方除以磁通。

and the phi can be removed. so finally you will get

而磁通可以被约掉。最后，你会得到

n square divided by reluctance. and reluctance can also be expressed by the mu zero mu r a and l. finally you will get a form to calculate the remux. which is very similar to capacitance. Instead of using epsilon zero and epsilon r, here we use the mu zero and mu r.

N的平方除以磁阻。而磁阻也可以表示为μ0、μr、A和l。最后，你会得到一个计算磁阻的公式。它和电容非常相似。这里不是用ε0和εr，而是用μ0和μr。

and a is the same. and l is also the same. and for the inductance, we have n square. because here we have n. We have n turns, right? For the capacitor, there's no turns. For the inductance, we have n turns. So here we have n square. mu zero, one is mu zero.

而A是一样的。l也是一样的。对于电感，我们有n的平方。因为这里有一个n。我们有n匝线圈，对吧？而对于电容，没有匝数。对于电感，有n匝，所以这里是n的平方。μ0，一个是μ0。

mu zero is also a constant. it's a constant value. It's the permeability. permeability of free space.

μ0也是一个常数。它是一个常数值，就是磁导率。即真空磁导率。

If you remember epsilon zero.

如果你还记得ε0的话。

epsilon zero is the

ε0是

permittivity. permittivity of free space, right? and mu zero, mu zero here is permeability of free space. and mu r is relative permeability.

介电常数。也就是真空介电常数，对吧？而这里的μ0，就是真空磁导率。而μr是相对磁导率。

and if you have, so this equation is applicable to the coil at the left side. if you have a coil at the right side, it's a circle. And for this case, the equation is still applicable. but here we have to pay attention to the area. The area is the cross-sectional, circular cross-sectional.

那么，这个公式适用于左边的线圈。如果右边也有一个线圈，它是一个圆。在这种情况下，这个公式仍然适用。但是这里，我们必须注意面积。这个面积是指横截面积，也就是圆形的横截面积。

a circular cross section through the coil.

就是穿过线圈的圆形横截面。

here is the a, okay? a is the cross section of the circle.

这里就是A，好吗？A就是圆的横截面。

and a is the average length.

而a是平均长度。

It's the mean or average circumference.

也就是平均周长。

You can get the circumference if you consider it as a circle.

如果你把它看作一个圆，就可以算出周长。

So it's not the cross-sectional area of the wire. This should be pay attention.

所以，这并不是导线的横截面积。这一点需要注意。

It's not the cross-sectional area of the wire, it's the cross-sectional of the circle.

它不是导线的横截面积，而是圆的横截面积。

And this equation is very similar to what we have introduced before. in a capacitor. Thanks, John. Only difference is that we replace the epsilon zero and a epsilon r. It's mu zero and a mu r.

这个公式和我们之前介绍过的非常相似。就是在电容器中。谢谢，约翰。唯一的区别是，我们把真空介电常数ε0替换成了相对介电常数εr。同样，也是用真空磁导率μ0和相对磁导率μr来替换。

mu is permeability of free space, mu zero. It's also a constant value, similar to epsilon zero. And mu r is relative permeability.

μ代表真空磁导率，也就是μ0。它也是一个常数，类似于ε0。而μr是相对磁导率。

And here we analyze the currents in your earth.

在这里，我们分析地球中的电流。

in power-based circuits.

即基于功率的电路。

And we know the coils try to prevent change, right? The coil's opposite. And a circuit is a coil. They are trying to oppose. And thus delay any changes. be it voltage or current.

我们都知道，线圈会试图阻止电流的变化，对吧？线圈的作用正好相反。而电路本身就是一个线圈。它们都在试图抵抗变化。从而延缓任何变化。无论是电压还是电流。

So try to oppose and thus delay any change in its voltage or current. Here is an example. If you consider a circuit with a resistor R, in parallel with a coil. of inductance L and resistance R L.

所以，它试图抵抗并延缓电压或电流的变化。这里有一个例子。如果我们考虑一个包含电阻R的电路，并且与一个线圈并联。该线圈具有电感L和电阻RL。

And RL is very small, is smaller than R.

而且RL非常小，比R还要小。

and if it's close to switch,

如果在闭合开关时，

current i1, jumps

电流I1，会突然跳变。

almost instantaneously. to its steady state value. Its steady state value is voltage divided by R. We know the voltage, right? And we know the resistance R. So the current i1 jumps instantaneously to its steady state value.

几乎是瞬间发生的。达到其稳态值。其稳态值等于电压除以R。我们知道电压，对吧？我们也知道电阻R。所以，电流I1会瞬间跳变到其稳态值。

But for i2, it takes much longer. It takes much longer to reach its steady state.

但是对于I2，则需要更长的时间。达到其稳态状态。

According to the next figure, it's increase slowly, slowly, slowly, and then finally reach its steady state. It takes much longer. Because the coil try to oppose, try to oppose.

根据下一张图，它是缓慢、缓慢、缓慢地增加，然后最终达到稳态。这需要更长的时间。因为线圈试图阻碍，试图抵抗。

the change. So it's anti-delay the change.

这种变化。所以，它会延缓这种变化。

So that means if you want to change the current, the coil wants the change to be slower. so it reaches the steady state much more slower.

这意味着，如果你想改变电流，线圈会让这种变化变得更慢。所以，达到稳态的速度会慢得多。

And finally, it will also reach the steady state, which is the voltage divided by the inductance to resistance. inductance resistance R, L.

最后，它也会达到稳态，此时的电流等于电压除以电感与电阻的比值。也就是电感电阻R、L。

So in this case, it also stores the energy. when it reaches the steady state, it has a steady current. But when we open the switch, you can see the curve like this. the current decrease slowly. And in this case, the inductor is considered as the source. and provide energy. to the circuit.

所以在这种情况下，它也会储存能量。当达到稳态时，电流是稳定的。但是，当我们打开开关时，你可以看到，曲线是这样的。电流会缓慢下降。在这种情况下，电感被视为一个电源。它会提供能量。给电路。

if the switch is open, the inductor in this case is the source and provide current to the circuit.

如果开关断开，那么在这种情况下，电感就是电源，会提供电流，给电路。

and the current decrease slowly. Finally, reach zero. If all the energy is released.

电流会缓慢衰减。最终会降到零。这意味着所有的能量都已经释放了。

And here you can see a similar figure here. For the current here, current increase slowly. it's the storage process. You can also define as a charging process, right? So the energy is stored in the inductor.

在这里，你可以看到类似的图表。这里的电流是缓慢上升的。这是一个储能过程。你也可以把它称为充电过程，对吧？所以，能量被存储在电感中。

And finally, when we open the switch, the inductor will release its energy. and you can see the decay of current. from steady state to zero.

最后，当我们断开开关时，电感会释放它的能量。你可以看到电流的衰减。从稳定状态降到零。

And at the steady state, for direct current, the inductance can be considered as a wire. So it's short-circuit.

而在稳态下，对于直流电来说，电感可以被视为一根导线。所以，这就是短路。

At the steady state, the inductor is a wire, so it can be considered as a short-circuit. And as I mentioned before, you know, capacitor. for steady state, it's open. So for capacitors, it's open circuit.

在稳态时，电感相当于一根导线，所以可以把它看作短路。正如我之前提到的，电容器。在稳态下，它是断开的。所以对于电容器来说，就是开路。

And as we have explained before, the current through the inductor cannot be changed suddenly. because the current always try to prevent the change. it must be delayed. So the current in the inductor cannot be changed suddenly.

正如我们之前解释过的，流过电感的电流不能突变，不能突然改变。因为电流总是试图阻止这种变化。所以必须有一个延迟。因此，电感中的电流不能突然改变。

So the coils are used to provide high current protection. to semiconductor devices.

所以，线圈用于提供过流保护。用于半导体器件。

Because the coil can always try to prevent the change. and thus delay the changing. So if in your circuit there's spark, and you can use this inductor, prevent the spark.

因为线圈总是试图阻止电流的变化。从而延缓这种变化。所以，如果你的电路中出现火花，你可以使用这个电感，来防止产生火花。

and do the protection.

起到保护作用。

We don't want to spark. So you can use this inductor to prevent the spark.

我们不希望出现火花。所以，你可以用这个电感来防止火花。

Great, and final part. The inductor is also used to store energy. How much energy? How much energy is actually stored in the inductor? And here is a typical R-L circuit. and connected with a voltage source. So if the switch is closed, which means the inductor is being charged. So energy is stored. But how much energy? And based on the Ohm's law, we can guess we have resistance voltage. and we have the inductor voltage, right? And if we add them together, we might get the voltage source.

很好，最后一点。电感还可以用来储存能量。能储存多少能量呢？实际上，电感中储存了多少能量？这是一个典型的RL电路。它连接了一个电压源。那么，如果开关闭合，这意味着电感是在充电的。所以，能量就被储存起来了。但是，具体是多少能量呢？根据欧姆定律，我们可以推测，这里有电阻和电压。还有电感的电压，对吧？如果我们把它们加起来，可能会得到电源电压。

And the voltage of the resistor can be expressed by the Ohm's law. It's the resistance times the current.

而电阻的电压可以用欧姆定律来表示，就是电阻乘以电流。

And the voltage of the inductor is the inductance times the rate change of the current.

而电感的电压则是电感乘以电流的变化率。

And for this equation, if you multiply i dt for both left and right side, and do the integration, for the first term, on the left side, it's v times i dt. and for each term on the right side, we also times i dt. so we get i square here. and here we get i, here i. dt can be removed. so we get vi.

对于这个公式，如果你把i dt乘以等式的左右两边，然后进行积分，对于第一项，在等式左边，就是v乘以i dt。对于右边的每一项，我们也乘以i dt。这样我们在这里就得到了i的平方。这里我们得到i，这里也是i。dt可以约掉。所以我们得到vi。

and we do the integration for both left and right side together.

然后我们对等式的左右两边同时进行积分。

And for each term, there's a physical meaning.

对于每一项，都有物理意义。

This is the energy supplied. Because E is the voltage, right? So voltage times the current is the power.

这就是提供的能量。因为E是电压，对吧？所以电压乘以电流，就是功率。

This is the energy supplied by the battery. to the circuit. And this is the energy dissipated.

这就是电池提供的能量。输送给电路的。而这一项是耗散的能量。

because of the resistance. The current square times the resistance is the power. This is the energy dissipated in the resistance. And the final term is the energy stored.

这是因为电阻的存在。电流的平方乘以电阻，就是功率。这就是在电阻上耗散的能量。最后一项是储存的能量。

stored in the magnetic field. magnetic field of the coil.

储存在磁场中。也就是线圈的磁场。

And this part represents the energy stored in the. So here we do the integration. And finally, we get the energy can be expressed by half of the inductance times current square.

这一部分代表了储存的能量。所以，我们在这里进行积分运算。最后，我们得到能量可以表示为电感的一半乘以电流的平方。

Okay, it's also very similar. to the capacitor, right? The energy stored in the capacitor is half times capacitance times voltage square. And here for the inductor is half of the inductance times current square.

好的，这也非常相似。就像电容器一样，对吧？电容器中储存的能量是电容的一半乘以电压的平方。而对于电感来说，就是电感的一半乘以电流的平方。

And also we can compute the instantaneous power. to the inductor. If you use the u times the current, what you find out is the power.

我们还可以计算瞬时功率。对于电感来说。如果你用电压乘以电流，得到的就是功率。

And the voltage can be expressed by the inductance times the rate change of the current.

而电压可以用电感乘以电流的变化率来表示。

So you can also use this equation to get the instantaneous power.

所以，你也可以用这个公式来计算瞬时功率。

at each time frame.

在每个时间点。

last part, time constant of RC, RL circuit, sorry, RL circuit. we have introduced the RC circuit has time constant, right? and for the RL circuit, it also has a time constant. the conclusion is that the time constant is L divided by R. it's not RC, R times C. It's the time constant in RC circuit. But in the RL circuit, time constant is R. it's L divided by R.

最后一部分是RC和RL电路的时间常数，抱歉，是RL电路。我们之前已经介绍过，RC电路有时间常数，对吧？而对于RL电路，它也有时间常数。结论是，时间常数等于L除以R。不是RC，也不是R乘以C。那是RC电路中的时间常数。但在RL电路中，时间常数就是R。是L除以R。

And we have introduced this curve for many, for multiple times. So let me close the speech. and watch it when it gets sunny. and current. This is the charging process, right? current will increase slowly from zero. to steady state.

我们已经多次介绍过这个曲线了。那么，我就结束今天的演讲了。等天气晴朗的时候再来看吧。还有电流。这就是充电过程，对吧？电流会从零开始缓慢增加。直到达到稳态。

So at time t1, the speech closed. and the current must be zero. since the inductance opposes any change. The inductance opposes any change in the current.

所以在t1时刻，演讲结束了。此时电流必须为零。因为电感会阻碍电流的变化。电感会抵抗电流的任何变化。

And we know we can use this to express the voltage, right? the inductance times the rate of change of current is the voltage.

我们知道，可以用这个公式来表示电压，对吧？电感乘以电流的变化率，就是电压。

And the initial rate is not twice. of current in this circuit is

而初始的变化率，是两倍。在这个电路中，电流的变化率是

and it can be calculated to change from one to the base. So the rate change of the current is equals to the voltage, right? and the inductance.

可以从1变化到e。所以，电流的变化率等于电压，对吗？再除以电感。

But at time t2, t2 is some point. and in the middle before the steady state. so at any time frame t2, how can you get the current? the current through the inductor has increased to i2. because the final steady state is is the voltage divided by the resistance.

但是在t2时刻，t2是一个特定的时间点。它处于稳定状态之前的中间阶段。所以在t2这个时间范围内，你怎么得到电流呢？流过电感的电流，已经增加到i2了。因为最终的稳定状态是电压除以电阻。

and the voltage across the inductor, can now be expressed by

而电感两端的电压，现在可以表示为

and this is the voltage across the inductor, right? and this is the voltage drop because of the resistance.

这就是电感两端的电压，对吧？而这是电阻造成的电压降。

But if we assume we know the final current, it's I final. and time t2, the current is i2.

但是，如果我们假设知道最终的电流，也就是I_final。而在时间t2时，电流是I2。

so the delta i, or di, can be expressed by the i final minus i2. it's the change of current, right? and dt is also the change of the time. so we can further expand. the equation. di is replaced by i final minus i2. it's the change of the current. and dt is replaced by the bigger t. big t is the change of the time. from t2 to the final steady time.

所以，电流的变化量Δi，或者di，可以表示为I_final减去I2。这就是电流的变化量，对吧？而dt也是时间的变化量。所以，我们可以进一步展开。这个方程。di被替换为I_final减去I2。这就是电流的变化量。而dt被替换为大的T。大的T就是时间的变化量。从t2到最终的稳定时间。

And based on this equation,

基于这个方程，

we know the final current. the final current is the e divided by the resistance. so we change the formula to this. we can get this equation. i final times the resistance is the final is the voltage source.

我们就知道了最终的电流。最终的电流等于E除以电阻。所以我们把公式改成这样。我们可以得到这个方程。I_final乘以电阻，就是最终的电源电压。

at this point in the equation, you can remove the same term from both sides. if we just remove it, we are going to get the resistance is the

在这个等式中，你可以把相同的项从两边都去掉。如果我们把它去掉，就能得到电阻等于

v final divided by the i. and we change a little bit. so time constant is the inductance divided by the resistance.

最终电压除以电流。我们稍微调整一下。所以，时间常数等于电感除以电阻。

any questions?

大家有什么问题吗？

the final process, any question? the time constant t or here we use tau. it indicates the speed at which the changes can take place.

最后的步骤，还有问题吗？时间常数用t表示，或者这里用τ。它表示变化发生的速度。

the speed at which the changes can take place. for example, if we have a larger L or smaller R, we will get larger tau. which means it will take longer

也就是电路中变化发生的快慢。例如，如果电感L更大，或者电阻R更小，那么时间常数τ就会更大。这意味着，变化需要更长的时间。

to reach the final state.

才能达到最终状态。

which means the current increase very slowly. to finally reach the final value.

也就是说，电流增加得非常缓慢。最终才能达到最终值。

And if we know the constant value t or tau, we can draw the transient current as it grows.

如果我们知道常数t或tau的值，就可以画出电流随时间增长的瞬态曲线。

because the transient curve is always can be derived using the series of the straight line.

因为瞬态曲线总是可以通过级数推导出来的。也就是直线。

Because the straight line is the tangent. It's the tangent of the curve, right? If we have many straight lines, we can draw the curve.

因为这条直线就是切线。它是曲线的切线，对吧？如果我们有很多条直线，就可以画出这条曲线。

And we know the current increase slowly. and finally reach the steady right? But how can we use the equation to express this curve? This is exactly the solution. It also use the exponential function. And how here is the time constant? It's the inductance divided by R. So this is the equation to express the curve. When the current increase from zero to steady value.

我们知道，电流是缓慢增加的。最终达到稳定状态。对吧？所以这是一条曲线。但是，我们如何用方程来表示这条曲线呢？这正是解。它也使用了指数函数。那么，这里的時間常数是多少呢？它是电感除以电阻。所以，这就是表示这条曲线的方程。当电流从零增加到稳定值时。

And also we can use this exponential function to express the voltage. across the inductor. The voltage is decaying slowly. until zero, like a checker. because finally the inductor is in short circuit. so the value finally is zero.

我们也可以用这个指数函数来描述电压。在电感两端的电压。电压正在缓慢衰减。直到变为零，就像这样。因为最终电感处于短路状态。所以最终的值是零。

It is from here zero. And after one times time constant value, the voltage has already dropped 36%.

它是从这里开始为零的。经过一个时间常数后，电压已经下降了36%。

Okay, series and parallel inductance.

好的，串联和并联电感。

For the inductors, it's similar to what we have introduced for resistor. If they are connected in series, we just add them together. And if they are connected in parallel,

对于电感来说，这与我们之前介绍的电阻的情况类似。如果它们是串联的，我们只需将它们相加。如果是并联的，

we use this equation to compute the equivalent.

我们就用这个公式来计算等效值。

Okay, here's a table for comparison. We have a resistor.

好的，这里有一个对比表格。我们有电阻。

We have introduced resistor is just introduced at the very beginning. and inductance here. The unit is Henry. We use H to represent. And we have a capacitance. right? The unit is farad. And voltage for the resistor, we use Ohm's law.

电阻是在最开始就介绍过的。而这里是电感。它的单位是亨利。我们用H来表示。还有电容。对吧？它的单位是法拉。电压对于电阻，我们使用欧姆定律。

to receive or to get the voltage. It's the current times the resistance. And for the inductance, the voltage is the inductance time the rate change and for the capacitance, voltage is the integration of the current.

来计算或得到电压。就是电流乘以电阻。而对于电感，电压等于电感乘以电流的变化率。对于电容，电压等于电流的积分。

divided by the capacitance.

再除以电容。

And we also introduce, we know how to compute the power for the system, right? how to compute the current for the resistance. and for the inductance, for the inductor,

我们还介绍了如何计算系统的功率，对吧？以及如何计算电阻上的电流。而对于电感，也就是电感器，

the current

电流

because the voltage is the derivation of the current. so the current is the integration of the voltage.

是因为电压是电流的导数。所以，电流就是电压的积分。

and also we introduced on how to compute the power. stored in the inductor.

此外，我们还介绍了如何计算功率。存储在电感器中的功率。

is the instantaneous power.

这是瞬时功率。

and similarly, here voltage is the integration of the current. so the current is the derivation of the voltage. and also you can get instantaneous power.

同样地，这里的电压是电流的积分。所以，电流是电压的导数。而且，你还可以得到瞬时功率。

for the capacitor.

对于电容器来说。

okay, here's another table for comparison. here we also given the energies stored in the capacitor.

好的，这里还有一个对比表格。这里我们也给出了能量存储在电容器中的情况。

if the energy stored in a capacitor is half times of the capacitance voltage square. and the energy stored in an inductor is half of the inductance times current square. And we know how to compute the equivalent resistance. or equivalent capacitance. or equivalent inductance. If they are connected in series,

如果电容器中存储的能量等于电容的一半，乘以电压的平方。而电感器中存储的能量等于电感的一半乘以电流的平方。而且我们知道如何计算等效电阻。或者等效电容。或者等效电感。如果它们是串联连接的，

and if they are connected in parallel. In parallel, how to compute the equivalent capacitance and the equivalent inductance?

如果是并联连接的。那么，在并联情况下，如何计算等效电容和等效电感呢？

and for the DC direct current circuit, the capacitor, when it reach the steady state, it can be considered as open circuit. And for the inductor,

对于直流电路，当电容器达到稳态时，可以将其视为开路。而对于电感器，

when it reach the steady state,

当它达到稳态时，

it can be considered as short circuit.

可以将其视为短路。

And here is an R-L circuit. that side is R-L source free circuit, present is RC.

这里是一个RL电路。那边是RL无源电路，这边是RC电路。

and we know for the R-L circuit, the current will be get slowly to reach finally zero, right? because the R-L, the inductor will be finally

我们知道，对于RL电路，电流会慢慢衰减，最终变为零，对吧？因为对于RL电路，电感最终会

be finally considered as a short circuit. So it's a wire, it has zero current. And this is the equation to express this part.

被视为短路。所以它相当于一根导线，电流为零。这就是表达这一部分的公式。

And tau is the time constant by itself.

这里的tau就是时间常数。

the inductance divided by the resistance. And for the RC circuit, we get the expression for the voltage.

它是电感除以电阻。而对于RC电路，我们得到了电压的表达式。

These two equations looks very similar. and similar to each other. Just replace the current is voltage. And here the tau, time constant, is the resistance times the capacitance.

这两个公式看起来非常相似。彼此非常相似。只需要把电流换成电压就可以了。这里的tau，也就是时间常数，等于电阻乘以电容。

That's all for today.

今天就讲到这里。

If you have any questions,

如果大家有任何问题，

just

尽管提出来。

Suspend a magnet permanently in free space on Earth. It will align itself in such a way that one end points towards the north and the other towards the south. This is the principle of our compass. If you use the compass for navigation, the magnetic bar inside the compass has one end pointing towards the north and the other end pointing towards the south, and you use this direction to navigate.

将一块磁铁永久地悬挂在地球上的自由空间中。它会自行排列，使一端指向北方，另一端指向南方。这就是我们指南针的原理。如果你使用指南针进行导航，指南针内部的磁棒一端指向北方，另一端指向南方，你利用这个方向来导航。

So why does this happen? It happens because the magnet has a magnetic field, represented by magnetic field lines. As this figure shows, the dashed lines are magnetic field lines. A bar magnet produces magnetic field lines. We use H to represent the magnetic field, or B for magnetic flux density.

那么这是为什么呢？这是因为磁体具有磁场，用磁力线来表示。如图所示，这些虚线就是磁力线。条形磁铁产生磁力线。我们用 H 表示磁场强度，用 B 表示磁通密度。

A bar magnet always tries to align one end towards the other because the magnetic field lines try to align their axis with the Earth. So the Earth is actually a very giant magnet. We have north, we have south.

条形磁铁总是试图使其一端朝向另一端，因为磁感线会试图与地球的轴对齐。所以地球实际上是一块巨大的磁铁。我们有北极，有南极。

The Earth has a south pole and a north pole. So the Earth is also a magnet. If you have a small magnet, this small magnet will try to align its direction with the Earth. We have introduced magnetic field lines; they are shown as dashed lines.

地球有南极和北极，因此地球本身也是一块磁体。如果你有一块小磁体，这块小磁体会试图使其方向与地球对齐。我们已经引入了磁感线的概念，它们以虚线表示。

Outside the magnet, the magnetic field lines travel from North to South. This can be traced using a small compass—we use a compass for navigation. But inside the magnet, the magnetic field lines travel from South to North. So finally, you get a closed loop.

在磁体外部，磁力线从北极指向南极。这可以用小指南针来追踪——我们用指南针进行导航。但在磁体内部，磁力线从南极指向北极。因此，最终你会得到一个闭合回路。

Magnetic field lines travel from north to south on the outside. Inside, they travel from south to north. Magnetic field lines always exist in loops—closed loops. Outside, they travel from north to south; inside, they travel from south to north. So it is a loop, a closed loop. There is no beginning and no end. You do not know where it starts and where it ends. And they never intersect each other. There are many magnetic field lines, but they never intersect.

磁感线在外部从北极指向南极，在内部从南极指向北极。磁感线总是以闭合回路的形式存在。外部从北极到南极，内部从南极到北极，因此它是一个闭合回路。它没有起点也没有终点，你无法判断它从哪里开始、到哪里结束。而且它们永远不会相互交叉。虽然有很多条磁感线，但它们从不相交。

They never intersect themselves. There is one magnetic field line that propagates along an upper line, but they never intersect themselves. And the geodesic triangle shortens geodesics. So how can you find the shortest way from north to south? From north to south has to be the shortest way. And the geodesic triangle shortens geodesics to find the shortest path from north to south outside.

它们永远不会自相交。有一条磁力线沿上方传播，但它们永远不会自相交。测地线三角形会缩短测地线。那么，如何找到从北极到南极的最短路径？从北极到南极必然是最短路径。测地线三角形通过缩短测地线来找到从北极到南极外部的最短路径。

Magnetic field is generally represented in vector notation. You have already been introduced to the symbol B, which is used to represent the magnetic field.

磁场通常用矢量符号表示。大家已经接触过符号 B，它用来表示磁场。

The magnetic field has both magnitude and direction, so it is a vector. You know the equation: B times current times the length of the wire gives you the force. If you use BIL, you will get the force, because B is the magnetic field. The magnetic field has direction and also has magnitude, so it is a vector. Sometimes the magnetic field is also defined as magnetic flux density.

磁场既有大小又有方向，因此它是一个矢量。你们知道这个公式：B 乘以电流再乘以导线长度就等于力。如果你使用 BIL，就能得到力，因为 B 就是磁场。磁场有方向，也有大小，所以它是矢量。有时磁场也被定义为磁通密度。

Also, we define magnetic flux density. Magnetic flux density is a physical quantity to measure the magnetic field at a given point—how strong the magnetic field is. It can be used to quantify or measure the magnetic field. And the unit of magnetic field or flux density...

另外，我们定义磁通量密度。磁通量密度是一个物理量，用于测量给定点处的磁场——即磁场的强弱。它可以用来量化或测量磁场。而磁场或磁通量密度的单位……

The unit of magnetic field H is Tesla. You can also use Newton per Ampere-meter. Why? Because you know force: F = BIL. I is the current, measured in Amperes; L is the length. Force has the unit Newton. So Newton divided by I and L gives you the unit of Tesla.

磁场强度H的单位是特斯拉。你也可以用牛顿每安培米来表示。为什么？因为你知道力的公式：F = BIL。I是电流，单位是安培；L是长度。力的单位是牛顿。所以牛顿除以I和L就得到特斯拉这个单位。

From F equals I L B, you get B equals F divided by I times L. Force has units of Newtons, I is Ampere, L is length in meters. So force divided by current and length gives you the magnetic field B. It also has units: Newton per Ampere meter. The Earth is also a very big magnet, and its magnetic field is—

由 F 等于 I L B，可得 B 等于 F 除以 I 乘以 L。力的单位是牛顿，I 是安培，L 是长度（米）。因此，力除以电流再除以长度就得到磁场 B。它也有单位：牛顿每安培米。地球也是一个非常大的磁体，它的磁场是——

The Earth is also a magnet, but its magnetic field is very small. Here is a typical conclusion for magnets: the North pole attracts the South pole. This is known as unlike poles attract, like poles repel.

地球也是一块磁体，但其磁场非常微弱。关于磁体的一个典型结论是：北极吸引南极。这被称为异名磁极相互吸引，同名磁极相互排斥。

Here you can see that when we put the loss bar closer...

大家可以看到，当我们把损失条放得更近时……

Opposite poles attract — a north pole and a south pole try to move toward each other. The magnetic field lines also travel from north to south between the two magnets. Like poles repel each other: if both north poles are placed close together, they will repel.

异名磁极相互吸引——北极和南极会试图靠近彼此。磁感线也在两个磁体之间从北极指向南极。同名磁极相互排斥：如果两个北极靠在一起，它们会互相排斥。

Here, the magnetic field lines travel from north to south outside the magnet. Inside, they travel from south to north. When two magnets attract each other, the north pole of one magnet also has magnetic field lines traveling to the south pole of another magnet. But at the same point, there are no magnetic field lines traveling from one magnet to another.

这里，磁感线在磁体外部从北极指向南极，在磁体内部则从南极指向北极。当两个磁体相互吸引时，一个磁体北极的磁感线会延伸到另一个磁体的南极。但在同一点上，并没有磁感线从一个磁体直接连到另一个磁体。

Magnetic induction is a process by which a material is magnetized by an external magnetic field. Here is an example: we have a U-shaped magnet. If you put a material between the U-shape, the material will be

磁感应是指材料在外磁场作用下被磁化的过程。这里举一个例子：我们有一个U形磁铁。如果在U形之间放入一种材料，该材料将会被

Here is an example using magnetite in place of soft iron. Recall that H is the symbol for the magnetic field that we introduced at the beginning.

这里举一个用磁铁矿代替软铁的例子。回顾一下，H 是我们一开始引入的磁场符号。

This is a magnetic field. When people put a soft iron block in a magnetic field, the flux lines pass through the iron and temporarily magnetize it due to magnetic induction. And there is a very important law: there is no line through the core.

这是一个磁场。当人们把一块软铁放入磁场中时，磁力线穿过铁块并由于磁感应使其暂时磁化。还有一条非常重要的定律：没有磁力线穿过铁芯。

The field lines travel around the hole; they never go through the hole directly. That means if there is an object inside this hole, the object will not be magnetized. Here we introduce the direction of the magnetic field. The magnetic flux Φ is a measure of the total magnetic field.

磁力线绕过孔洞传播，从不直接穿过孔洞。这意味着如果孔洞内部存在一个物体，该物体不会被磁化。这里我们引入磁场的方向。磁通量 Φ 是对总磁场的度量。

The total magnetic field which passes through a given area. If you define the area, you can use the magnetic flux to measure how many magnetic field lines go through this area. Then you get the simple equation to compute the magnetic flux Φ. Φ is the magnitude of the strength of the magnetic field.

穿过某一给定面积的总磁场。如果你定义了该面积，就可以用磁通量来衡量有多少磁感线穿过这个面积。然后你得到一个简单的方程来计算磁通量 Φ。Φ 是磁场强度的大小。

Magnetic field B, or flux density, times the area. Theta is the angle between the direction of the magnetic field and the area. The area sometimes has an angle with the magnetic field. The unit of magnetic flux is weber. Based on this equation, we know the unit of magnetic field is tesla, T. The area is meter squared.

磁感应强度B，即磁通密度，乘以面积。θ是磁场方向与面积之间的夹角。面积有时与磁场存在一个夹角。磁通量的单位是韦伯。根据这个公式，我们知道磁场的单位是特斯拉，即T。面积的单位是平方米。

Weber can be expressed as Tesla times meter squared. Alternatively, one Tesla equals one Weber divided by meter squared. Now let us discuss the relation between magnetic fields and electric fields. A magnetic field can produce current, and current also...

韦伯可以表示为特斯拉乘以平方米。或者换一种说法，一特斯拉等于一韦伯除以平方米。现在我们来讨论磁场与电场之间的关系。磁场可以产生电流，而电流也……

A current can produce a magnetic field. This is the famous Ampère's law, which we will introduce in detail later. If a wire carries an electrical current I, it will produce a magnetic field H around the wire. Its direction can be given by the right-hand screw rule.

电流可以产生磁场。这就是著名的安培定律，我们稍后会详细介绍。如果一根导线通有电流 I，它会在导线周围产生磁场 H，其方向可由右手螺旋定则确定。

Here is the right-hand thumb rule. Because the current flows away from the reviewer, if you use the right hand, your thumb is going inside, so the direction of the four fingers is the direction of the magnetic field lines. An electrical current can produce a magnetic field, and we can use the right-hand thumb rule to determine the direction of the H-field or magnetic field.

这是右手定则（右手拇指定则）。由于电流方向是远离观察者的，如果你使用右手，拇指指向纸面内部，那么四指的方向就是磁力线的方向。电流可以产生磁场，我们可以利用右手拇指定则来判断H场或磁场的方向。

In another example, if you have a current going through this wire, it will produce a magnetic field around this wire, in circles around the wire. If you have a coil, just like this example shows, the relationship between the current and the H field is found using the right-hand rule. We have introduced this in the previous slide. You can use the right-hand thumb rule to determine the direction of the magnetic field.

在另一个例子中，如果电流通过这根导线，它会在导线周围产生磁场，磁场呈环形分布。如果你有一个线圈，正如这个例子所示，电流与H场之间的关系可以用右手定则来确定。我们在上一张幻灯片中已经介绍过这一点。你可以使用右手拇指定则来判断磁场的方向。

Here is an example. If you have a coil like this and you use the right-hand rule, your four fingers point in the direction of the current, and your thumb indicates the direction of the magnetic field lines inside the magnet.

这里举一个例子。如果你有一个这样的线圈，使用右手定则，四指指向电流方向，拇指则指示磁体内部磁力线的方向。

Magnetic field lines travel inside a magnet from south to north. This means you can use this method to determine which end is the north pole and which end is the south pole. Conversely, if the north and south poles are given, you can also use this method to determine the direction of the current.

磁感线在磁体内部从南极指向北极。这意味着你可以利用这一方法判断哪一端是北极、哪一端是南极。反过来，如果已知南北极的位置，你也可以用这一方法确定电流的方向。

If the current direction is given, you can use this method to determine where is the north pole and where is the south pole. Now we have this equation again. If you put a wire in a magnetic field, it is current I. Because of the magnetic field, it will produce force on the wire. The magnitude of the force is the magnetic field times…

如果电流方向已知，你可以用这个方法判断哪端是北极、哪端是南极。现在我们再次看到这个方程。如果把一根导线放入磁场中，其中通有电流 I。由于磁场的存在，导线会受到力的作用。这个力的大小等于磁场乘以……

The force on a current-carrying wire involves the current going through the wire times the length of the wire. If you have a single moving charge, you can use qvB to compute the force applied to the charge.

载流导线所受的力涉及通过导线的电流乘以导线的长度。如果只有一个运动电荷，你可以用 qvB 来计算作用在该电荷上的力。

Q is the symbol for electric charge. V is the velocity of the charge, that is, how fast the charge moves. B is the magnetic field. From Q, V, and B, you can get the force applied to the charge.

Q 是电荷的符号。V 是电荷的速度，即电荷运动的快慢。B 是磁场。由 Q、V 和 B，你可以得到作用在电荷上的力。

When you apply a current I through a wire, it produces a force on the wire. This force makes the wire move.

当你在导线中通入电流 I 时，会产生一个作用在导线上的力。这个力使导线运动。

Here is an example. You have two magnets. If you put a material or a foil between the two...

这里有一个例子。你有两块磁铁。如果你在两者之间放入一种材料或一片箔……

If there is no current, the material or the wire will not move. But if you close the switch, there will be current flowing through the coil, and the coil will move. This is the principle of an electric motor. Many countries now have many electric cars on public roads. They use the electric motor to drive the car. In an electric motor, if you give it current, because of the magnet, the material will be rotated.

如果没有电流，材料或导线就不会运动。但如果你闭合开关，就会有电流流过线圈，线圈就会运动。这就是电动机的原理。许多国家现在公共道路上有很多电动汽车，它们使用电动机来驱动汽车。在电动机中，如果给它通入电流，由于磁体的作用，材料就会旋转。

Here we have two parallel wires. For each wire, if we put them in a magnetic field, there will be a force on each wire. And if the current has the same direction, the wires will attract. If the current has a different or opposite direction, the wires will repel each other, because for each wire there will be a force.

这里有两根平行导线。将每根导线置于磁场中时，每根导线上都会受到一个力的作用。如果电流方向相同，两根导线相互吸引；如果电流方向不同或相反，两根导线相互排斥，因为每根导线上都会受到一个力。

You can determine the direction of the force based on the right-hand thumb rule. If you know the direction of the magnetic field and the current direction, you can use the right-hand thumb rule to determine the force for each wire. Once you determine the force for each wire, you can then find whether they attract each other or repel each other. The conclusion is that if the currents have the same direction, they will attract each other; if the currents have different directions, they will repel each other. Now, Ampere's Law — the famous scientist who coined the new unit.

你可以根据右手定则判断力的方向。如果你知道磁场方向和电流方向，就可以用右手定则确定每根导线所受的力。一旦确定了每根导线的受力方向，你就能判断它们之间是相互吸引还是相互排斥。结论是：如果两根导线中的电流方向相同，它们会相互吸引；如果电流方向不同，它们会相互排斥。接下来是安培定律——那位以他命名新单位的著名科学家。

He was inspired towards science through reading. He read a lot of books. Among those books, he found a certain one that had produced magnetic fields. During the time of Faraday, the source of electricity was chemical cell batteries. At that time, there is also...

他通过阅读而受到科学的启发。他读了很多书。在这些书中，他发现有一本涉及磁场的内容。在法拉第的时代，电的来源是化学电池。在那个时期，还有……

There are lots of salts—washing salts or final salts—and they are typically chemical boundaries. Faraday's main contribution is that he found the relation between magnetic fields and electrical currents.

存在大量盐类——洗涤盐或终盐——它们通常是化学边界。法拉第的主要贡献在于他发现了磁场与电流之间的关系。

The left figure shows that a current can produce a magnetic field. The right figure shows that a magnetic field can produce a current.

左图表明电流可以产生磁场，右图表明磁场可以产生电流。

De Broglie achieved his results. For sure, he had done a lot of experiments and some papers.

德布罗意取得了他的成果。可以肯定的是，他做了大量实验并发表了一些论文。

Failure is common, and only one succeeds. Success is the final result behind thousands of failures.

失败是常态，只有一个人能成功。成功是数千次失败背后的最终结果。

The first observation is that if we change the current, it can produce a magnetic field. And also the other way around: if we change the magnetic field, it can produce a current. This is the final conclusion — a changing current can produce a magnetic field, and a changing magnetic field can produce a current. Now, if the magnet is static, you will have zero voltage output. But if we move the magnet into the coil...

第一个观察结果是：如果我们改变电流，它可以产生磁场。反过来也一样：如果我们改变磁场，它可以产生电流。这是最终结论——变化的电流可以产生磁场，变化的磁场可以产生电流。现在，如果磁铁是静止的，你将得到零电压输出。但如果我们把磁铁移入线圈……

When the voltmeter is not zero, it is because you moved the magnet into the coil, and it appears as a voltmeter reading. This observation is that if you change the magnetic field, you can produce current. A magnet moving into the coil changes the magnetic field; that is why you have current and the voltmeter is not zero. And if you move the magnet outside or out of the coil, you can also—

当电压表读数不为零时，是因为你将磁铁移入了线圈，这表现为电压表的读数。这一观察表明，如果你改变磁场，就能产生电流。磁铁移入线圈时磁场发生变化，因此产生了电流，电压表读数不为零。而如果你将磁铁移出线圈外部，你同样可以——

Received voltage in the reverse direction. Here is a brief summary: a change in magnetic flux linking a circuit induces an electromotive force in that circuit. So if you change the magnetic field, it can produce a current. The magnitude of the induced electromotive force is proportional to the rate of change of the flux. Here is the equation to express this description.

接收到的电压方向相反。以下是一个简要总结：穿过电路的磁通量发生变化时，会在该电路中感应出电动势。因此，如果你改变磁场，就可以产生电流。感应电动势的大小与磁通量的变化率成正比。以下是用来表达这一描述的方程。

The rate of change of the flux, dΦ/dt, is the rate of change of the magnetic flux. If the magnetic field changes, then you can get current. Here is the voltage. When they combine these two statements together, they can find a change in magnetic field.

磁通量的变化率，即 dΦ/dt，就是磁通量的变化率。如果磁场发生变化，就会产生电流。这里就是电压。当把这两个表述结合起来，就能求出磁场的变化。

A changing magnetic flux in one coil also produces a voltage across that coil. Here is an equation to express this concept. Phi is the magnetic flux. d phi by dt is the rate of change of flux.

一个线圈中变化的磁通量也会在该线圈两端产生电压。这里给出一个表达这一概念的方程。Phi 是磁通量，d phi 比 dt 是磁通量的变化率。

Magnetic flux Φ can also be produced by current. Since Φ represents magnetic flux, the magnetic field can also be produced by current. And if you change the current, the magnetic field can also be changed.

磁通量Φ也可以由电流产生。由于Φ代表磁通量，磁场也可以由电流产生。如果你改变电流，磁场也会随之改变。

The direction of the induced electromotive force will always tend to set up a current that opposes the motion.

感应电动势的方向总是倾向于产生一个阻碍该运动的电流。

We can use Lenz's description to determine the direction of the induced current, because the induced current will always try to prevent the motion. Let's take a short break.

我们可以利用楞次定律的描述来判断感应电流的方向，因为感应电流总是试图阻碍运动。我们休息一下。

The application of Lorentz force in electromagnetic induction.

洛伦兹力在电磁感应中的应用。

Here you have a voltmeter. If you change the magnetic field or the flux, you can see the pointer rotate. That means it produces voltage. And you can use a voltmeter to measure voltage. And there are many other use cases. An electrical generator produces electricity. If there is no electricity, an electrical generator can produce electricity. How can it produce? It changes the flux here. Changing the flux here will produce electricity.

这里有一个电压表。如果你改变磁场或磁通量，你可以看到指针发生偏转。这意味着它产生了电压。你可以使用电压表来测量电压。还有许多其他的应用场景。发电机可以产生电能。如果没有电，发电机可以产生电。它是如何产生的？它在这里改变了磁通量。在这里改变磁通量就会产生电。

How can we change the flux? If this coil rotates, the flux will always be changed. Because we know that the flux is the magnetic field times the area times cosine theta, right? Cosine theta — theta is the angle between the magnetic field and the coil. So if the coil rotates, different angles will produce different magnetic fields.

我们如何改变磁通量？如果这个线圈旋转，磁通量将始终发生变化。因为我们知道磁通量等于磁场强度乘以面积再乘以余弦θ，对吧？余弦θ——θ是磁场与线圈之间的夹角。因此，如果线圈旋转，不同的角度将产生不同的磁场。

Finally, you might get a current from the generator. We introduced the RC circuit in our previous lecture. Here is the RL circuit. Here is the inductor. The RL circuit also has a typical time constant.

最后，你可能会从发电机获得电流。我们在上一节课中介绍了RC电路。这里是RL电路。这里是电感器。RL电路也具有典型的时间常数。

The time constant is L divided by R. For a capacitor, the time constant is R times capacitance. For an RC circuit, the time constant is R times capacitance. But now the time constant is L divided by R, where L is the inductance. Next, let us consider what happens when we close the switch. If we close the switch, what about V1? If the switch is open, the circuit is open, so there is no current at that voltage.

时间常数为L除以R。对于电容，时间常数为R乘以电容值。对于RC电路，时间常数为R乘以电容值。但现在的时间常数是L除以R，其中L是电感。接下来，我们考虑闭合开关时会发生什么。如果闭合开关，V1会怎样？如果开关断开，电路处于开路状态，因此该电压下没有电流。

The x-axis is the time. When you close the switch, the current rises slowly from zero to attain a stable value.

横轴为时间。闭合开关后，电流从零缓慢上升，直至达到稳定值。

This is the final current, because the inductor can be finally regarded as a wire. So it is a wire; there is no resistance here. The final stable current is just the voltage divided by the resistance. The current increases slowly until it reaches a stable state. As for the voltage, when the switch is closed—

这是最终电流，因为电感最终可以被视为一根导线。所以它是一根导线，这里没有电阻。最终稳定电流就是电压除以电阻。电流缓慢增加，直到达到稳定状态。至于电压，当开关闭合时——

The inductance wants to understand the voltage of the source, and it decays slowly and tends to zero. Because finally it's a wire, right? A wire behaves like a short circuit, looks like a short circuit, so there's no voltage here finally. And the voltage can be expressed as the inductance times the rate of change of the current.

电感想要理解电源的电压，它缓慢衰减并趋于零。因为最终它就是一根导线，对吧？导线表现得像短路，看起来像短路，所以这里最终没有电压。而电压可以表示为电感乘以电流的变化率。

If you have…

如果你有……

If we just consider one turn of an N-turn coil, the contribution of this one turn to the magnetic flux is Φ₁. And Φ is proportional to the current flowing through the coil. So if the current is larger,

如果我们只考虑N匝线圈中的一匝，这一匝对磁通量的贡献为Φ₁。而Φ与流过线圈的电流成正比。所以如果电流更大，

They are produced by an operator and they are eigenfunctions. k is our coefficient; it is a constant value. But if you have n terms, the total flux phi will be n times k, n times i, where i is the current.

它们由算符产生，是算符的本征函数。k 是我们的系数，是一个常数值。但如果你有 n 项，总磁通量 φ 将等于 n 乘以 k，即 n 乘以 i，其中 i 是电流。

Based on Faraday's law, the voltage is minus N times the rate of change of flux. The flux can also be expressed by N times I. Because N is a constant, we just put it outside the derivation. So finally we get minus N squared times the rate of change of current.

根据法拉第定律，电压等于负的N乘以磁通的变化率。磁通也可以用N乘以I来表示。由于N是常数，我们直接将其提到微分运算之外。因此最终得到负的N的平方乘以电流的变化率。

L squared is a capital value, defined as the inductance of the coil. We use L here. The voltage is the inductance times the rate of change of the current. And don't forget the sign is minus, because at the differential level, the produced voltage always tries to prevent the change.

L 的平方是一个大写字母值，定义为线圈的电感。这里我们用 L 表示。电压等于电感乘以电流的变化率。别忘了符号是负的，因为在微分层面上，产生的电压总是试图阻止变化。

The induced voltage prevents the change of the flux, and that is represented by the minus sign. The unit of inductance is the Henry, and we use the symbol H to represent it. The Henry can also be expressed as volt times second divided by ampere.

感应电压阻止磁通量的变化，这由负号表示。电感的单位是亨利，用符号H表示。亨利也可以表示为伏特乘以秒除以安培。

Based on this equation, we can find that the voltage across an inductor is proportional to the time rate of change of current. It is proportional because L is a constant value, and di/dt is the time rate of the current. So the voltage is proportional to the time rate of change of the current. And if we draw the equation using this figure, the x-axis is the time rate of change of the current, while the y-axis is the voltage. So we can get a linear line. The slope is the inductance L.

根据这个方程，我们可以得出电感两端的电压与电流的时间变化率成正比。之所以是正比关系，是因为L是一个常数值，而di/dt是电流的时间变化率。因此，电压与电流的时间变化率成正比。如果我们用这张图来绘制该方程，横轴是电流的时间变化率，纵轴是电压。这样我们就能得到一条直线，其斜率即为电感L。

Inductance is a property of the inductor. It always tries to prevent the change of current. It also determines how much electromotive force is generated per unit change of current.

电感是电感器的一个属性。它总是试图阻止电流的变化。它还决定了每单位电流变化所产生的电动势的大小。

Here is an example. If a coil has an inductance of one Henry and its current changes at one ampere per second, then based on the equation, the induced electrical voltage across it is also one volt. That is opposite the change in current—opposite this change in current.

这里举一个例子。如果一个线圈的电感为一亨利，其电流变化率为一安培每秒，那么根据公式，线圈两端产生的感应电压也为一伏特。这个电压的方向与电流的变化相反——与电流变化的方向相反。

Producing an electric monopole current can prevent discharge. As you know, voltage is the integration of current. If you perform the integration, you can obtain the current using the voltage.

产生电单极电流可以防止放电。如大家所知，电压是电流的积分。如果对电压进行积分运算，就可以得到电流。

And the impedance is k s squared. We change the equation a little bit, and we can finally get phi times n divided by i.

阻抗为 k s 的平方。我们对方程稍作变换，最终可以得到 phi 乘以 n 除以 i。

Also defined as flux linkage. So this is the total flux linking the coil. Total flux linking the coil equals the flux linking a turn times the number of turns. The number of turns is N. The total flux linking the coil is Φ. So N times Φ is the flux linkage.

也定义为磁链。这是穿过线圈的总磁通量。穿过线圈的总磁通量等于每匝的磁通量乘以匝数。匝数为N。穿过线圈的总磁通量为Φ。因此N乘以Φ即为磁链。

The inductance can also be described as the flux linkage per unit current. This is very similar to capacitance. In capacitance, we know that capacitance is the charge Q divided by voltage, that is, charge Q per unit voltage. Similarly, here we have inductance as the flux linkage per unit current.

电感也可以描述为单位电流的磁链。这与电容非常相似。在电容中，我们知道电容是电荷Q除以电压，即单位电压的电荷Q。类似地，这里电感就是单位电流的磁链。

C is Q divided by V. And here we have phi times delta I by inductance. And here again, the equation—how can we calculate the inductance? The inductance is the flux linkage, phi times N divided by I, and it changed from linkage.

C等于Q除以V。这里我们有磁通量φ乘以电流变化量ΔI除以电感。再看这个方程——我们如何计算电感？电感就是磁链，即φ乘以N除以I，它由磁链定义而来。

You can get n squared times pi divided by n times i. Here we introduce another new variable defined as reluctance. Reluctance represents the opposition to magnetic flux. It depends on the geometry and material of the coil. So reluctance is defined as N squared A divided by the

可以得到 n 的平方乘以 π 除以 n 乘以 i。这里我们引入另一个新变量，定义为磁阻。磁阻表示对磁通的阻碍作用，它取决于线圈的几何形状和材料。因此，磁阻定义为 N 的平方乘以 A 除以

Flux phi. The unit is ampere-turns per weber. Because here MMF is NI, right? MMF is equal to NI. N is turns, I has amperes, so the unit is ampere-turns. So the reluctance has unit ampere-turns per weber. Flux has unit weber.

磁通量Φ。其单位是安匝每韦伯。因为这里磁动势MMF等于NI，对吧？MMF等于NI。N是匝数，I的单位是安培，所以MMF的单位是安匝。因此磁阻的单位是安匝每韦伯。磁通的单位是韦伯。

We just replace n_i here. And here we change the form a little bit. We know the impedance is n squared times pi, divided by pi times...

我们在这里替换 n_i。这里我们对形式做一点改变。我们知道阻抗是 n 的平方乘以 π，再除以 π 乘以……

Reluctance R. And the flux can be written as l divided by reluctance. And reluctance can also be expressed by μ₀, μᵣ, A, and l. I finally get a form to compute the inductance, which is very similar to capacitance. Instead of using ε₀ and εᵣ, here we use μ₀ and μᵣ. A is the same.

磁阻R。磁通可以写成l除以磁阻。磁阻也可以用μ₀、μᵣ、A和l来表示。我最终得到了一个计算电感的公式，它与电容的公式非常相似。这里不使用ε₀和εᵣ，而是使用μ₀和μᵣ。A是相同的。

A is the same, and A_i is also the same. For rank-2 tensors we have n squared, because here we have n terms—many terms. For a scalar there are no terms. For rank-2 tensors we have n terms, so here we have n squared. Now, μ₀—what is μ₀? μ₀ is also a constant value, a constant quantity. It is the permeability of free space. If you remember, ε₀...

A 是相同的，A_i 也是相同的。对于二阶张量，我们有 n 的平方，因为这里有 n 个分量——很多分量。对于标量则没有分量。对于二阶张量有 n 个分量，所以这里是 n 的平方。那么 μ₀ 是什么？μ₀ 也是一个常数值，一个常量。它是真空磁导率。如果你还记得，ε₀……

Epsilon zero is the permittivity of free space, and mu zero here is the permeability of free space. Mu r is relative permeability. If you have this equation at the top on the left side, if you have a coil, and on the right side it is a circle, then for this case the equation is applicable at the top. But here we have to pay attention to the area. The area is the circular cross-sectional area.

ε₀ 是自由空间的介电常数，μ₀ 是自由空间的磁导率。μᵣ 是相对磁导率。如果等式左边上方是一个线圈，右边是一个圆，那么对于这种情况，该等式在上方是适用的。但这里我们需要注意面积，面积是圆形截面的横截面积。

The cross-sectional formula. Here, A is the cross-section of the circle, and AI is the mean or average circumference.

截面公式。这里，A 是圆的截面积，AI 是平均周长。

If you consider it as a circle, it is not the cross-sectional area of the wire; it should be the cross-sectional area of the circle. This equation is very similar to what we introduced in the capacitor lecture. The only difference is that we replace ε₀ and εᵣ with μ₀ and μᵣ.

如果将其视为一个圆，那么它不是导线的横截面积，而应该是圆的横截面积。这个方程与我们在电容器讲座中介绍的非常相似，唯一的区别是我们用 μ₀ 和 μᵣ 替换了 ε₀ 和 εᵣ。

μ is the permeability of free space, μ₀. It is also a constant value, similar to ε₀. And μᵣ is the relative permeability. Here we analyze inductors in circuits. We know that inductors try to prevent change. They oppose any change. A circuit with an inductor will try to oppose and delay any changes in its voltage or current.

μ是自由空间的磁导率，即μ₀。它也是一个常数值，类似于ε₀。μᵣ是相对磁导率。这里我们分析电路中的电感器。我们知道电感器试图阻止变化，它们会抵抗任何变化。含有电感器的电路会试图抵抗并延迟其电压或电流的任何变化。

Here is an example. Consider a circuit with resistor R in parallel with a coil of inductance L and resistance RL. RL is very small—smaller than R—so the branch is nearly a short circuit. Current I1 jumps almost instantaneously.

这里举一个例子。考虑一个电路：电阻R与一个电感为L、内阻为RL的线圈并联。RL非常小，比R小得多，因此该支路近似短路。电流I1几乎瞬间跳变。

The steady-state value is voltage divided by R. We know the voltage and we know the resistance R. So the current I₁ jumps instantaneously to its steady state. As for I₂, it takes much longer to reach its steady state. According to the negative figure, it increases slowly and then finally reaches its steady state. It takes much longer because the coil has back EMF. Back EMF opposes the change.

稳态值等于电压除以R。我们已知电压，也已知电阻R。因此电流I₁瞬间跳变到其稳态值。而I₂则需要更长时间才能达到稳态。根据负向图形，它缓慢增加，最终达到稳态。之所以需要更长时间，是因为线圈存在反电动势，反电动势会阻碍变化。

And that's related to the change of the current. So the L means if you want to change the current, the coil wants the change to be slow, so it reaches steady state much more slowly. And finally, it also reaches a steady state, which is the voltage divided by the resistance R. In this case, it also stores energy. When you reach the steady state, it has a steady current. But when we open the switch, you can see the current decays slowly. And in this case, the inductor is considered as a source.

这与电流的变化有关。L 意味着如果你想改变电流，线圈会使变化变慢，因此它达到稳态的速度要慢得多。最终，它也会达到一个稳态，即电压除以电阻 R。在这种情况下，它还储存能量。当达到稳态时，它有稳定的电流。但当我们断开开关时，可以看到电流缓慢衰减。在这种情况下，电感被视为一个电源。

The inductor provides energy to the circuit. When the switch is open, the inductor acts as the source and provides current to the circuit. The current decays slowly and finally reaches zero when all the energy is released. Here you can see a similar figure. The current increases slowly; this is a storage process, also defined as the charging process. The energy is stored in the inductor.

电感器向电路提供能量。当开关断开时，电感器充当电源，向电路提供电流。电流缓慢衰减，当所有能量释放完毕后最终降为零。这里可以看到一个类似的图示。电流缓慢增大，这是一个储能过程，也称为充电过程。能量存储在电感器中。

In the open circuit, the inductor releases its energy and the current decays from steady state to zero. At steady state, for direct current, the inductor can be considered as a wire, a short circuit.

在开路中，电感释放其储存的能量，电流从稳态衰减至零。在稳态下，对于直流电，电感可视为一根导线，即短路。

At steady state, an inductor can be considered a short circuit. As I mentioned before, a capacitor at steady state is an open circuit. And as we have explained before, the current through an inductor cannot be changed suddenly, because the current always tries to prevent change and must delay the change. So the current in the inductor cannot be changed suddenly. The coils are used to provide high—

在稳态下，电感可被视为短路。如我之前所提到的，电容在稳态下是开路。正如我们之前解释过的，通过电感的电流不能突变，因为电流总是试图阻止变化，必须延迟变化。因此，电感中的电流不能突变。线圈用于提供高——

Inductors provide strong protection to semiconductor devices because the current always tries to prevent change and delays the changing. So if in your circuit there is a spark, you can use an inductor to prevent the spark and do the protection.

电感器能够为半导体器件提供强有力的保护，因为电流总是试图阻止变化并延迟变化的发生。因此，如果你的电路中出现火花，你可以使用电感器来防止火花并实现保护。

We don't want sparks. If you are using this inductor, it won't spark. The inductor can also be used to store energy. But how much energy is actually stored in the inductor? Here is a typical RLC circuit connected with a voltage source. If the switch is closed, it means the inductor is being charged, so energy is stored in the inductor. But how much energy? Based on...

我们不希望产生火花。如果你使用这个电感器，它不会产生火花。电感器也可以用来储存能量。但电感器中实际储存了多少能量？这里是一个典型的RLC电路，连接了一个电压源。如果开关闭合，意味着电感器正在被充电，因此能量储存在电感器中。但储存了多少能量？基于……

Based on Ohm's law, we have the resistor voltage and the inductor voltage. If we add them together, we get the voltage source. The voltage of the resistor can be expressed by Ohm's law: it is the resistance times the current. The voltage of the inductor is the inductance times the rate of change of the current.

根据欧姆定律，我们有电阻电压和电感电压。将它们相加，即可得到电压源。电阻上的电压可以用欧姆定律表示：即电阻乘以电流。电感上的电压等于电感乘以电流的变化率。

If we multiply both sides by i dt and do the integration, for the first term on the left side, it is e times i dt, and for each term on the right side, we also multiply by i dt. So we get i squared here, and here we get di, since dt can be removed. We do the integration for both the left and right sides together. For each term, there is a physical meaning. This is the energy supplied, because e is the voltage, so voltage times current—

如果我们在等式两边同时乘以 i dt 并进行积分，左边第一项为 e 乘以 i dt，右边每一项也乘以 i dt。因此这里得到 i 的平方，这里得到 di，因为 dt 可以被消去。我们对左右两边同时进行积分。每一项都有物理意义。这是供给的能量，因为 e 是电压，所以电压乘以电流——

This is the power. This is the energy supplied by the battery to the circuit. This is the energy dissipated because of the resistance. Current squared times resistance is the power. This is the energy dissipated in the circuit.

这是功率。这是电池供给电路的能量。这是因电阻而耗散的能量。电流的平方乘以电阻就是功率。这是在电路中耗散的能量。

And the final part is the energy stored in the magnetic field of the coil. And this part represents the energy stored in the inductor. So here we do the integration, and finally we get the energy can be expressed by half of the inductance times current squared.

最后一部分是储存在线圈磁场中的能量。这一部分代表储存在电感器中的能量。在这里我们进行积分运算，最终得到能量可以表示为电感的一半乘以电流的平方。

Times current squared. It's also very similar to the capacitor, right? For the capacitor, the energy stored in the capacitor is half times capacitance times voltage squared. Here for the inductor, it's half of the inductance times current squared. And also we can compute the instantaneous power delivered to the inductor. It is V times the current. Voltage times current is the power. And the voltage can be expressed by the inductance times the rate of change of the current.

乘以电流的平方。这与电容非常相似，对吧？对于电容，存储在电容中的能量等于二分之一乘以电容值再乘以电压的平方。这里对于电感，能量等于二分之一乘以电感值再乘以电流的平方。我们还可以计算瞬时传递给电感的功率，即电压乘以电流。电压乘以电流就是功率。而电压可以表示为电感值乘以电流的变化率。

You can also use this equation to get the instantaneous power at each time frame. Now, what is the time constant of an RL circuit? We have introduced that the RC circuit has a time constant. The RL circuit also has a time constant. The conclusion is that the time constant is L divided by R. It is not RC.

你也可以使用这个方程来求得每个时间帧的瞬时功率。那么，RL电路的时间常数是什么？我们已经介绍过RC电路具有时间常数。RL电路同样也有时间常数。结论是，时间常数等于L除以R，而不是RC。

RC is the time constant in the RC circuit. But in the RL circuit, the time constant is L divided by R. We have introduced this current from previous times. So when we close the switch, the voltage will decay slowly, and the current will increase slowly from zero. This is the charging process, until it reaches steady state.

RC是RC电路中的时间常数。但在RL电路中，时间常数是L除以R。我们之前已经介绍过这个电流。所以当我们闭合开关时，电压会缓慢衰减，电流会从零缓慢增大。这就是充电过程，直到达到稳态。

At time t₁, the switch is closed. When we close the switch, the current passes through zero. Current passes through zero since inductance opposes any change in current. So inductance opposes any change in current. And we know we can use this to—

在时刻 t₁，开关闭合。当开关闭合时，电流经过零值。电流经过零值是因为电感会阻碍电流的任何变化。因此，电感阻碍电流的任何变化。我们知道可以利用这一特性来——

Express the voltage. The inductance times the rate of change of current equals the voltage. The initial rate of rise of current in the circuit can be given. So the rate of change of the current equals the voltage divided by the inductance. At time t₂, which is some point before the steady state, how can you get the current?

表示电压。电感乘以电流的变化率等于电压。电路中电流的初始上升率可以给出。因此，电流的变化率等于电压除以电感。在时间 t₂，即稳态之前的某个时刻，如何求得电流？

The current through the inductor has increased to I₀. The final steady state is driven by the resistance. The voltage across the inductor can now be expressed by L·di/dt. This is the voltage across the inductor, and this is the voltage drop because of the resistor.

通过电感的电流已增大至 I₀。最终稳态由电阻驱动。电感两端的电压现在可以用 L·di/dt 来表示。这是电感两端的电压，而这是由于电阻产生的电压降。

If we assume we know the final current is I_final, and at time t₂ the current is I₂, then the change in current, ΔI or dI, can be expressed as I_final minus I₂.

如果我们已知最终电流为 I_final，且在时刻 t₂ 时电流为 I₂，那么电流的变化量 ΔI 或 dI 可以表示为 I_final 减去 I₂。

dt is some sort of change over time. So we can further expand the equation. di is replaced by i final minus i initial; this is the change of the current. And dt is replaced by big T, where big T is the change of time from t zero to the final steady time. Based on this equation, we know the final current is E divided by the resistance.

dt 是某种时间上的变化量。因此我们可以进一步展开该方程。di 被替换为 i_final 减去 i_initial，即电流的变化量。dt 被替换为大 T，其中大 T 是从 t 零到最终稳态时间的变化量。基于这个方程，我们知道最终电流等于 E 除以电阻。

By changing the voltage a little bit, you can get this equation: I final times the resistance is the voltage source. At this point in the equation, you can remove the same term from both the left and right sides. If we just remove it, we find that the resistance is the inductance divided by T. So the time constant is the inductance divided by resistance.

通过稍微改变电压，可以得到这个方程：最终电流乘以电阻等于电压源。在方程中，可以从左右两边同时消去相同的项。消去之后，我们发现电阻等于电感除以T。因此，时间常数等于电感除以电阻。

Any questions about the process? Any question? No time.

关于这个过程，有什么问题吗？有问题吗？没时间了。

The time constant, denoted as T or τ (tau), indicates the speed at which changes can take place within the circuit. For example, if we have a larger area, we get a larger τ. A larger τ means it will take longer to reach the final state — that is, the current increases very slowly before finally reaching the final value.

时间常数，记作 T 或 τ（tau），表示电路中变化发生的快慢程度。例如，如果面积较大，则 τ 较大。较大的 τ 意味着达到终态所需的时间更长——也就是说，电流增加得非常缓慢，最终才达到终值。

A constant value T or tau can join the transient parameters as it grows in a circuit, because the transient curve can always be derived using a series of straight-line approximations. Because a straight line is—

一个常数值T或tau可以随其在电路中的增长而连接瞬态参数，因为瞬态曲线始终可以用一系列直线近似来推导。因为直线是——

It is the tangent of the curve. If you have many straight lines, they can enclose the curve horizontally. We know the current increases slowly and finally reaches the steady state. So it is a curve. But how can we use an equation to express this curve?

它是曲线的切线。如果你有很多直线，它们可以水平地包围这条曲线。我们知道电流缓慢增加，最终达到稳态。所以它是一条曲线。但我们如何用方程来表达这条曲线呢？

This is the exact solution. It also uses an exponential function. And tau here is the time constant. Tau is the inductance divided by R. So this equation expresses the current when the current increases from zero to its steady value. And also you can use this exponential function to express the voltage across the inductor. The voltage decays slowly from t equals zero, because finally the inductor...

这是精确解，它同样使用了指数函数。这里的 tau 是时间常数，tau 等于电感除以 R。因此这个方程表达的是电流从零增加到稳态值的过程。你也可以用这个指数函数来表达电感器两端的电压。电压从 t 等于零开始缓慢衰减，因为最终电感器……

In a short circuit, the final value is zero. It decays from E-zero to zero. After one time constant, the voltage has already dropped thirty-six percent. Now, series and parallel inductors—for inductors, it is similar to what we have.

在短路情况下，终值为零，从E零衰减到零。经过一个时间常数后，电压已经下降了百分之三十六。接下来是电感的串联与并联——对于电感，情况与我们之前所学类似。

If we introduce a resistor, when connected in series, we just add them together. When connected in parallel, we use this equation to compute the equivalent inductance. Here is a table for comparison. We have a resistor, introduced as resistance. We have inductance; the unit is Henry, and we use H to represent inductance. Then we have capacitance; the unit is farads.

如果引入一个电阻，当它们串联时，直接将它们相加即可。当它们并联时，我们使用这个方程来计算等效电感。这里有一张对比表。我们有电阻，即电阻值。我们有电感，其单位是亨利，用H表示电感。然后我们有电容，其单位是法拉。

For the resistor, we use Ohm's law: the voltage equals the current times the resistance. For the inductance, the voltage equals the inductance times the rate of change of the current. For the capacitance, the voltage equals the integral of the current divided by the capacitance. We also introduce another power—

对于电阻，我们使用欧姆定律：电压等于电流乘以电阻。对于电感，电压等于电感乘以电流的变化率。对于电容，电压等于电流的积分除以电容值。我们还引入了另一个功率——

How do we compute power in polar systems? For the inductor, the voltage is the differentiation of the current, and the current is the integration of the voltage. We also introduced how power is stored in the inductor.

如何在极坐标系统中计算功率？对于电感，电压是电流的微分，电流是电压的积分。我们还介绍了功率在电感中如何储存。

This is the instantaneous power in the inductor. Similarly, here voltage is the integration of current, and current is the differentiation of voltage. And also you can get the instantaneous power for the capacitor.

这是电感中的瞬时功率。类似地，这里电压是电流的积分，而电流是电压的微分。同样，你也可以得到电容的瞬时功率。

Here is another type of comparison. We are also given the energies stored in the capacitor and the inductor. The energy stored in a capacitor is one-half times the capacitance times voltage squared, and the energy stored in an inductor is one-half of the inductance times current squared. And we know how to compute the equivalent resistance, or equivalent capacitance, or equivalent inductance.

这是另一种比较方式。我们还给出了电容器和电感器中存储的能量。电容器中存储的能量等于二分之一乘以电容再乘以电压的平方，电感器中存储的能量等于二分之一乘以电感再乘以电流的平方。我们也知道如何计算等效电阻、等效电容或等效电感。

If they are connected in series and if they are connected in parallel, how to compute the equivalent capacitance and the equivalent inductance. And for the DC, direct current circuit, the capacitor, when it reaches the steady state, it can be considered as an open circuit. And for the inductor, when it reaches the steady state...

如果它们串联连接，如果它们并联连接，如何计算等效电容和等效电感。对于直流电路，电容器在达到稳态时，可以视为开路。而对于电感器，当它达到稳态时……

Here is an RLC circuit. The left side is an RLC series circuit; the right side is an RC series circuit. For the RLC circuit, the current, when it gets loaded, will eventually reach zero because the inductor will be...

这里是一个RLC电路。左侧是RLC串联电路，右侧是RC串联电路。对于RLC电路，当它被加载时，电流最终会降为零，因为电感将会……

At time t equals zero, the capacitor acts as a short circuit, so in the wire it has zero current. This is the equation to express this curve. How is the time constant? The time constant is the capacitance divided by the resistance. For the RC circuit, we get the expression for the voltage. These two equations look quite similar to each other—just replace current with voltage.

在 t 等于零时，电容器相当于短路，因此导线中电流为零。这是描述该曲线的方程。时间常数如何定义？时间常数等于电容除以电阻。对于 RC 电路，我们得到电压的表达式。这两个方程看起来非常相似——只需将电流替换为电压即可。

And here the tau time constant is the resistance times the capacitance. That's the last one for today. If you have any questions or you don't understand, you can just come to me.

这里，时间常数 tau 等于电阻乘以电容。这是今天的最后一个内容。如果你们有任何问题或者没有理解，可以直接来找我。