Make sure the energy can be transferred from the source to the load.

确保能量可以从源传输到负载。

And in our second subsection,

在我们的第二小节中，

we have known what is an electrical charge. and usually we have two types. one is positive, and the other is negative charge.

我们已经了解了什么是电荷。通常有两种类型。一种是正电，另一种是负电。

And if the charges have different polarity, they will attract. but if they have the same polarity, they will repel each other.

如果电荷的极性不同，它们会相互吸引。但如果它们的极性相同，它们会相互排斥。

And electric charge is actually the reason that we have the current. So if the electric charge moves, it will produce current.

而电荷实际上就是我们产生电流的原因。所以，如果电荷发生移动，就会产生电流。

And the direction of the flow is from negative to positive.

电流的方向是从负极流向正极。

And we know the current is existing because because of the charge. So the current can be expressed by the charge divided by the time.

我们知道电流的存在，是因为有电荷在移动。所以，电流可以用电荷除以时间来表示。

And we also introduce the equation to compute the power. based on the voltage and the current. If you know the current and voltage, you can compute how much power. how much power is consumed. by the circuit.

我们还引入了一个计算功率的公式。它是基于电压和电流的。如果你知道电流和电压，就可以计算出功率是多少。也就是消耗了多少功率。由电路消耗的。

And yeah, so here is the famous Ohm's law. the voltage is equivalent to the current times the resistance.

是的，这里就是著名的欧姆定律。电压等于电流乘以电阻。

And sometimes the power is not a constant value, it's varying. so in order to compute the energy, you have to do the integration.

有时候，功率并不是一个恒定值，而是会变化的。所以，为了计算能量，你必须进行积分运算。

Yeah, we'll start the subsection if you have introduced what a resistor is. And here you can see different symbols or different types of resistors. Some resistors have fixed value. and some resistors can be changed. And they are variable resistors. Some resistors are dependent on external conditions, such as heat. like focus, etc.

好的，在开始这一小节之前，我们先来介绍一下什么是电阻。在这里，你可以看到不同的符号，或者不同类型的电阻。有些电阻的阻值是固定的。而有些电阻的阻值是可以改变的。它们就是可变电阻。还有一些电阻的阻值会受外部条件的影响，比如温度。还有光线等等。

And conductance, conductance is the inverse. one divided by the resistance is the conductance.

还有电导，电导就是电阻的倒数。也就是1除以电阻，就是电导。

and we know the conductor exactly is actually has a length, a certain length, right? and also has a cross section. Based on the length and the cross section area, how can we compute the resistance? and here is an equation. Here is the equation. So rho here is the resistivity. It's the only given.

我们知道，导体实际上是有长度的，有一定的长度，对吧？而且还有一个横截面积。根据长度和横截面积，我们该如何计算电阻呢？这里有一个公式。就是这个公式。这里的ρ就是电阻率。这是唯一给定的参数。

So here is a table for the resistance.

所以，这里有一个电阻的表格。

And also we know the resistance can be changed because of the temperature. If the temperature increased, the resistance will also be changed. But how can we compute?

我们还知道，电阻会随温度的变化而改变。如果温度升高，电阻也会发生变化。但是，我们该如何计算呢？

the resistance at a certain temperature. and here you can use this equation. to compute the resistance at a certain temperature. Here is alpha zero and r zero. Here we use zero degree as the reference. But sometimes we also use 20 degree. If you use 20 degree as the reference, the equation will be changed. it will change correspondingly. So here we use alpha 20 and r 20. instead of alpha zero and r zero.

在特定温度下的电阻？这里我们可以使用这个公式。来计算特定温度下的电阻。这里有一个α0和R0。这里我们以0度作为参考点。但有时我们也会使用20度。如果用20度作为参考，公式就会发生变化。它会相应地改变。所以这里我们使用α20和R20。而不是α0和R0。

And we know some different coordinate system for the resistance. as long as the color band is given. so you can understand how much resistance actually used for the resistor.

我们知道，对于电阻有不同的坐标系。只要给出了色环。这样你就能明白实际的电阻值是多少。这个电阻的阻值。

And finally, we have to know the resistor can be connected in series and also can be connected in parallel. and how can we compute the corresponding voltage and current? for example, once they are connected in parallel, they have the same voltage. And the current, you can add the current together. to be equivalent to the total current.

最后，我们需要知道，电阻可以串联连接，也可以并联连接。那么，我们如何计算相应的电压和电流呢？举个例子，如果它们是并联的，那么它们的电压是相同的。而电流的话，你可以把电流加起来。这样就等于总电流。

And here you can see if the resistors are connected in series. they have the same current. And if we add the voltage, you will get the total voltage.

这里我们可以看到，如果电阻是串联的。那么它们的电流是相同的。如果我们把电压加起来，就能得到总电压。

Great. That's all what we have learned in our last

很好。这就是我们在上一节课中学到的全部内容。

And today we want to introduce a very famous law.

今天，我们想介绍一个非常著名的定律。

it's called Kirchhoff.

它叫做基尔霍夫定律。

Kirchhoff is actually a German

基尔霍夫实际上是一位德国

physicist. and he introduced this law.

物理学家。就是他提出了这个定律。

So the law is also defined by his name.

所以，这个定律也以他的名字命名。

And he observed that there are two general conditions. to form the network analysis of any electrical network.

他观察到，存在两个普遍的条件。用于分析任何电气网络。

And these conditions are stated as follows. The first is Kirchhoff's current law.

这些条件如下所述。首先是基尔霍夫电流定律。

The definition is that at any instant, the algebraic sum of the current at a junction

其定义是，在任意时刻，电流的代数和在节点处等于零。

but the current have to have different direction. so the current have different signs. so different signs are allocated to currents. either to flow towards the junction or to flow away from it. so if you define the current flow towards a junction as positive, then

但是，电流必须有不同的方向。所以电流有不同的符号。因此，不同的符号被分配给不同的电流。它们要么流向节点，要么从节点流出。所以，如果你定义了电流，并将流向节点的电流定义为正，那么

the current flow away the junction will be negative.

从节点流出的电流就是负的。

but if you define the current flow away the junction as positive, then the current flow towards the junction will be so it depends on how you define the direction.

但如果你定义从节点流出的电流为正，那么流向节点的电流就是负的。所以，这取决于你如何定义方向。

But the key sentence here is the algebraic sum of the current at a junction is zero. So this is the key sentence here. This is also the first law. And second law is the voltage law. its definition is that at any instant, in a closed loop, the algebraic sum of the emf, you know emf, right? electromotive force. it's a source. It can be either voltage source or a current source. the algebraic sum of the emf, electromotive force, acting around the trip is equal to the algebraic sum of the vds. vds is potential difference.

但这里的关键句子是：电流的代数和在节点处等于零。所以，这就是这里的关键句子。这也是基尔霍夫第一定律。第二定律是电压定律。它的定义是，在任意时刻，在一个闭合回路中，电动势的代数和等于零。你知道什么是电动势吧？就是electromotive force。它是一个电源。可以是电压源，也可以是电流源。沿回路一周，所有电动势的代数和等于电压降的代数和。Vds就是电位差。

if equal to the algebraic sum of the potential difference round the loop. it might be a little bit difficult to understand the definition. might be a little bit difficult to understand what it's actually mean. but no worries, we will use some examples to explain how can we use these two definitions. how can we use these two laws.

也就是说，它等于沿回路一周所有电位差的代数和。这可能有点难理解，就是这个定义。可能有点难理解它到底是什么意思。不过别担心，我们会用一些例子来解释一下如何使用这两个定义。也就是这两个定律。

okay, here is an example. to illustrate the first law, which is called a current law.

好的，这里有一个例子。为了说明第一定律，也就是电流定律。

here the definition again. the sum of currents leaving any node equals the sum of the currents entering the node. so if you define, for example, i1 is entering. right? i2 is leaving. i3 is leaving. I4 is entering. and i5 is entering.

这里再重复一下定义。流出任意节点的电流之和，等于流入该节点的电流之和。所以，如果你定义，比如，I1是流入的。对吧？I2是流出的。I3是流出的，I4是流入的。I5也是流入的。

so if you, based on the definition, the current entering the node is equivalent to the current leaving the node. in this case, the current i2 and i3 are leaving. i1, i4 and i5 are entering. so the algebraic sum of the current entering the node is equivalent to the current leaving the node.

那么根据定义，流入节点的电流就等于流出节点的电流。在这种情况下，电流I2和I3都是流出的。而I1、I4和I5是流入的。所以，流入节点的电流的代数和等于流出节点的电流。

that's the reason we have the final equation at the bottom.

这就是为什么我们得到了底部的最终方程。

also, for the second law, here's an example to explain. explain what the second law mean. and if you have a look at the definition again, the algebraic sum of all voltage around a closed loop

另外，对于第二定律，这里有一个例子来解释。说明第二定律的含义。如果你再看一下定义，所有电压的代数和在闭合回路中等于零。

and here we have in this circuit we have two loops. the first one is loop a. the second is loop b. so actually we can use the voltage law twice. because we have two loops. so we can use the voltage law twice. so for loop a, if we follow the definition, the algebraic sum of is around a closed loop is zero. and if we take conventionally, in normal cases, we use the electric motive force. electric motive force is the source. or a current source. we take as positive. the resistive voltage drops.

在这个电路中，我们有两个回路。第一个是回路A。第二个是回路B。所以实际上，我们可以两次应用电压定律。因为有两个回路。因此，我们可以使用两次电压定律。那么，对于回路A，如果我们按照定义来，所有电压的代数和等于零。沿着一个闭合回路。通常情况下，如果我们按照惯例来考虑，我们会使用电动势。电动势就是电源。或者是电流源。我们将其视为正值。而电阻上的电压降则是负值。

so for the first loop a, you have a voltage Vs, right? and you have v1. v1 is the voltage drop. and they are negative. and you add them together. because based on definition, you have add them together, right? this is the algebraic sum. algebraic sum of all voltages. in a closed loop is zero.

所以，对于第一个回路a，我们有一个电压源Vs，对吧？还有一个V1。V1是电压降。而且它们是负值。然后把它们加在一起。因为根据定义，你需要把它们相加，对吧？这就是代数和。所有电压的代数和。在一个闭合回路中，结果等于零。

and the electric motive force is positive, Es. v1 is negative, v2 is negative, v4 is negative.

而电动势是正的，即Es。V1是负的，V2是负的，V4也是负的。

because the voltage drops. and you add them together. and the final voltage or the total voltage will be zero. this is the exact definition. definition of the Kirchhoff law. and this equation is derived from loop a. and similarly, you can derive a second equation. from loop b.

因为电压是下降的。然后把它们相加。最终的电压，也就是总电压，将等于零。这就是确切的定义。基尔霍夫定律的定义。这个方程是从回路a推导出来的。同样地，你也可以推导出第二个方程。是从回路b推导出来的。

i will introduce another way. another way to define the sign. of the voltage. so here you can see there are two loops, right? the loop also have direction. they are clockwise direction. like a clock.

我来介绍另一种方法。就是定义符号的方法。即电压的符号。大家可以看到，这里有两个回路，对吧？每个回路也有方向。它们是顺时针方向。就像时钟一样。

so if the current is, for example, for loop a, if the current has the same direction as the loop direction,

那么，如果电流——比如对于回路a来说，如果电流的方向与回路的方向一致，

you will get a positive.

那么结果就是正的。

And that's the reason we do it in loop b. so you see the direction of b2 is the same as the direction of loop b. so in loop b, we have positive b2. and minus v3. v3 is negative because the direction is different.

这就是原因，我们在回路b中也是这样做的。大家可以看到，I2的方向与回路b的方向是一样的。所以在回路b中，我们有正的I2。还有负的V3。V3是负的，因为它的方向不同。

from the loop direction.

与回路的方向相反。

any questions?

有什么问题吗？

and here are some very traditional equations. to compute the voltage and current. in a circuit. so for example, in this case, the resistors, the two resistors are connected in series. and how can we compute the current? if you know the voltage and you know the resistance, for each resistor. so you can use the ohm's law. to get the circuit current.

这里有一些非常传统的公式。用来计算电压和电流。在电路中。比如，在这个例子中，这两个电阻是串联连接的。那么我们该如何计算电流呢？如果你知道电压和电阻，对于每个电阻来说。那么你就可以使用欧姆定律。来计算电路中的电流。

and also you can get the voltage for each resistor.

此外，你还可以计算出每个电阻两端的电压。

because you first compute the current, right? once you know the current and you know the resistance, you can get the voltage.

因为你首先需要计算出电流，对吧？一旦知道了电流和电阻，你就可以算出电压了。

okay, i think it's quite simple. so i will not speak so slowly.

好的，我觉得这很简单。所以我不会说得那么慢。

and similarly, so if we draw the conductance, we know conductance is actually the inverse. right? so here we have two conductors. conductance is actually also kind of resistor, but it's a different form. to express resistance.

同样地，如果我们画出电导，就会知道，电导实际上是电阻的倒数。对吧？所以这里有两个电导。电导其实也可以看作是一种电阻，但形式不同。它是用来表示电阻的。

and we know the total conductance of the network is g1, plus g2.

我们知道，该网络的总电导是G1，加上G2。

and g1 plus g2 is this is the total conductance.

G1加G2就是总电导。

and based on the ohm's law, right? the total conductance is actually the inverse. the inverse resistance.

根据欧姆定律，对吧？总电导实际上就是倒数。也就是电阻的倒数。

so the current divided by the total conductance is the voltage. because this equation is actually equivalent to voltage equals to current times the resistance. because the resistance is the inverse.

所以，电流除以总电导就等于电压。因为这个公式实际上等同于电压等于电流乘以电阻。因为电导是电阻的倒数。

and once you know the total current is i, you can compute the current. you can compute the current for each conductor.

一旦你知道了总电流是I，你就可以计算出电流了。你可以计算出每个导体的电流。

any questions? i think this will be the last slide. i think if i will break the record.

有什么问题吗？我想这应该是最后一张幻灯片了。如果我没记错的话。

if there's no questions, that's all for lecture four.

如果没有其他问题的话，第四讲就到这里。

so basically in lecture four, i'll focus on basic definition. or concept.

基本上，在第四讲中，我会重点介绍一些基本的定义。或者说是概念。

very basic definition or concept for analog circuit.

这些是关于模拟电路的最基本定义或概念。

now let's do the exercise for

现在，我们来做一道练习题，来

lecture four.

巩固一下第四讲的内容。

let's first solve the first seven questions. one to seven. i will give you maybe five minutes.

我们先来解答前七个问题。从一到七。我给大家大概五分钟的时间。

是哪一个来着

是哪个来着？

是哪一个来着

嗯？是哪一个来着？第四章吧。

怎么跳到跳到四了 我退一下

怎么直接跳到第四章了？我退一下。

那个电流定律 电流的那个定律是只有在导线上 还是在哪儿都是这样

那个电流定律，是不是只适用于导线上？还是说在任何地方都适用？

电路里面都成立的

在电路里都是成立的。

在电源也成立吗

那在电源上也成立吗？

电源

电源？

它必须是必须是有那种 直流

它必须是那种直流电。

它必须是有那种死路可回路的地方

它必须有一个闭合的回路。

直流就成立

所以直流电是成立的。

不是，它不是在那个，你说那个第一定律是吧？

不是，你说的不是那个第一定律吗？

电流定律必须是在一个节点 节点是有直流 有直流汇入有直流汇出的 基本上成立 如果没有直流，它只有一条线的话，它没有直流。

电流定律必须是在一个节点上。节点才有直流。有电流流入，也有电流流出。基本上是这样。如果没有闭合回路，只有一条线的话，就没有电流。

如果它没有分支的话 你看我们平常使用的 在高中学的也是，如果两个电阻并联的话，是不是

如果它没有分支的话，你看我们平时用的高中也学过，如果两个电阻并联的话，是不是

一半的电流是吧？ 其实就相当于用这种方法来一个比较简单 如果电阻特别多的话，就是分得特别多块。

电流会分一半？其实就相当于用这种方法，比较简单。如果电阻特别多的话，就会分成很多份。

电荷量都是一样的 但是我就会想就是如果是一个电

电荷量都是一样的。但是我就在想，如果是一个电流

the first one is you have the potential difference.

首先，是有电势差的。

do you know the ohm's law? the question one will be kind of simple. let's just have a look at question two.

你知道欧姆定律吗？第一题会比较简单。那我们来看一下第二题。

Make sure you have an understanding of what electrostatics is about. In our second subsection, we will look at what electric charge is. Usually, we have two types of electric charge: one is positive charge, and the other is negative charge.

确保你们理解静电学的内容。在第二个小节中，我们将学习什么是电荷。通常，电荷有两种类型：一种是正电荷，另一种是负电荷。

If charges have different polarities, they will attract each other; if they have the same polarity, they will repel each other. Electric charge is actually the reason we have current. If an electric charge moves, it produces current.

如果电荷具有不同的极性，它们会相互吸引；如果具有相同的极性，则会相互排斥。电荷实际上是我们拥有电流的原因——当电荷移动时，就会产生电流。

The direction of the flow will be from negative to positive. We know that the current exists because of the charge. So the current can be expressed by the charge divided by the time.

流动的方向是从负极到正极。我们知道电流的存在是因为电荷。因此，电流可以表示为电荷除以时间。

We also introduced the equation to compute the power based on voltage and current. If we know the current and voltage, we can compute how much power is consumed by the circuit. Here is the famous Ohm's law: the voltage is equivalent to the current times the resistance.

我们还介绍了基于电压和电流计算功率的公式。如果我们知道电流和电压，就可以计算电路消耗了多少功率。这就是著名的欧姆定律：电压等于电流乘以电阻。

Sometimes the power in a resistor is not a constant value; it is varying. In order to compute the energy, you have to do the integration. Now we start the subsection on resistors. Here you can see different symbols for different types of resistors. Some resistors have fixed values. Some resistors can be changed; they are variable resistors. Some resistors are dependent on external conditions such as heat, light, focus, etc., and conductance.

有时电阻中的功率并非恒定值，而是变化的。为了计算能量，必须进行积分运算。现在我们开始电阻这一小节。这里可以看到不同类型电阻的不同符号。有些电阻具有固定阻值。有些电阻可以改变，称为可变电阻。有些电阻依赖于外部条件，例如热、光、聚焦等，以及电导。

Conductance is the inverse of resistance; one divided by the resistance gives the conductance. We know that a conductor has a certain length and a cross-section area. Based on the length and the cross-section area, how can we compute the resistance? Here is the equation. Rho here is the resistivity, which is unit-dependent. So here is a table for the resistors. And also we know the resistance.

电导是电阻的倒数；用一除以电阻即得电导。我们知道导体具有一定的长度和横截面积。根据长度和横截面积，如何计算电阻？以下是该方程。这里的ρ是电阻率，其数值取决于单位。这里有一张电阻器的表格。此外我们还知道电阻……

Resistance can be changed by temperature. If the temperature increases, the resistance increases; if the temperature decreases, the resistance also changes. But how can we compute the resistance at a certain temperature? Here you can use this equation to compute the resistance at a certain temperature. Here is alpha-zero and R-zero. We use zero degrees as the reference, but sometimes we also use twenty degrees as the reference. If you use twenty degrees as the reference, the equation will change correspondingly. So here we use alpha-twenty and R-twenty instead of alpha-zero and R-zero. And we know different coefficients of resistance, as long as the constant…

电阻会随温度变化。如果温度升高，电阻增大；如果温度降低，电阻也会改变。那么如何计算某一温度下的电阻呢？这里可以使用这个方程来计算某一温度下的电阻。这里有α零和R零。我们用零度作为参考，但有时也用二十度作为参考。如果用二十度作为参考，方程会相应改变。所以这里用α二十和R二十代替α零和R零。我们知道不同的电阻系数，只要常数……

The power band is given, so you can understand how much resistance is actually being used for the resistor. And finally, we have the resistors can be connected in series and also can be connected in parallel. How can we compute the corresponding voltage and current? For example, once they are connected in parallel, they have the same voltage, and you add the currents together to be equivalent to the total current. And here you can see if the resistors are connected in series, they will have the same current.

功率带已经给出，因此你可以了解电阻器实际使用了多大的电阻。最后，电阻器可以串联连接，也可以并联连接。我们如何计算对应的电压和电流？例如，当它们并联连接时，它们具有相同的电压，将各电流相加即等于总电流。这里你可以看到，如果电阻器串联连接，它们将具有相同的电流。

In our last lecture, we discussed voltage and current. Today, I want to introduce a very famous law. It is called Kirchhoff's law.

在上一节课中，我们讨论了电压和电流。今天，我想介绍一条非常著名的定律，它叫做基尔霍夫定律。

Kirchhoff is actually a German physicist. He introduced this law, so the law is also identified by his name. He observed that there are two general conditions fundamental to the analysis of any electrical network, and these conditions

基尔霍夫实际上是一位德国物理学家。他提出了这一定律，因此该定律也以他的名字命名。他观察到，对于任何电路网络的分析，存在两个基本的普遍条件，而这些条件……

The first law is Kirchhoff's Current Law. The definition is that at any instant, the algebraic sum of currents at a junction in a network is zero.

第一条定律是基尔霍夫电流定律。其定义为：在任意时刻，网络中任一节点的电流代数和为零。

The sum of the currents at a junction in a network is zero. The currents have different directions, so the currents have different signs. Different signs are allocated to currents flowing towards the junction and those flowing away from it. If you define the current flowing towards a junction as positive, then the current

网络中某一节点处各电流之和为零。由于电流方向不同，它们具有不同的符号。流向节点的电流与流出节点的电流被分配了不同的符号。如果将流向节点的电流定义为正，那么该电流

If the current flows away from the junction, it will be negative. But if you define the current flow away from the junction as positive, then the current flow towards the junction will be negative. So it depends on how you define the direction. The key sentence here is: the algebraic sum of the current at a junction is zero. This is the first law. The second law is Kirchhoff's voltage law. Its definition is that at any instant in a closed loop, the algebraic sum of the EMF—electromotive force—is equal to...

如果电流从节点流出，则为负值。但如果你将从节点流出的电流定义为正，那么流向节点的电流就为负。所以这取决于你如何定义方向。这里的关键一句话是：在任一节点处，电流的代数和为零。这就是第一定律。第二定律是基尔霍夫电压定律。其定义是：在任一时刻，沿闭合回路，电动势（EMF）的代数和等于……

A source can be either a voltage source or a current source. The algebraic sum of the EMF, electromotive force, acting around the loop is equal to the algebraic sum of the PDs, potential differences, around the loop. These two definitions might be a little difficult to understand what they actually mean. No worries, there will be some examples to explain how we can use these two laws.

电源可以是电压源，也可以是电流源。沿回路作用的电动势（EMF）的代数和等于沿回路的电位差（PD）的代数和。这两个定义可能有点难以理解其实际含义。不用担心，后面会有一些例子来说明如何使用这两条定律。

Here is an example to illustrate the first law, Kirchhoff's Current Law. The definition: the sum of the currents leaving any node equals the sum of the currents entering the node. For example, I₁ is entering the node, I₂ is leaving, I₃ is leaving, I₄ is entering, and I₅ is entering.

下面用一个例子来说明第一条定律——基尔霍夫电流定律。其定义为：离开任一节点的电流之和等于进入该节点的电流之和。例如，I₁进入节点，I₂离开，I₃离开，I₄进入，I₅进入。

The current entering the node is equivalent to the current leaving the node. In this case, the currents I2 and I3 are leaving; I1, I4, and I5 are entering. So the algebraic sum of the currents entering the node is equivalent to the current leaving the node.

流入节点的电流等于流出节点的电流。在本例中，电流I2和I3是流出的；I1、I4和I5是流入的。因此，流入节点的电流的代数和等于流出节点的电流。

That's the reason we have the final equation at the bottom of the slide. Also, for the second law, here is an example to explain what the second law means. If you look at the definition again, the algebraic sum of all voltages around a closed loop is zero.

这就是为什么幻灯片底部有最终的方程。另外，关于第二定律，这里有一个例子来解释第二定律的含义。如果我们再看一遍定义，沿闭合回路所有电压的代数和为零。

In this circuit, we have two loops. The first one is Loop A; the second is Loop B. Since we have two loops, we can apply Kirchhoff's Voltage Law twice. For Loop A, if we follow the definition, the algebraic sum of all voltages around a closed loop is zero. By convention, in normal cases, we take the electromotive force—whether it is a voltage source or a current source—as positive, and the resistive voltage drops

在这个电路中，我们有两个回路。第一个是回路A，第二个是回路B。由于有两个回路，我们可以对基尔霍夫电压定律应用两次。对于回路A，按照定义，沿闭合回路所有电压的代数和为零。按照惯例，在通常情况下，我们将电动势——无论是电压源还是电流源——取为正值，而电阻上的电压降

In the first loop A, you have voltage source V_s and voltage drop V_1. The voltage drops are taken as negative. Based on the definition, you add them together—this is the algebraic sum of all voltages in a closed loop, which equals zero.

在第一个回路A中，有电压源V_s和电压降V_1。电压降取负值。根据定义，将它们相加——这就是闭合回路中所有电压的代数和，其结果等于零。

V₀ is positive; V₁ is negative, V₂ is negative, V₄ is negative, because the voltage drops. When you add them together, the total voltage will be zero. This is exactly the definition of Kirchhoff's Voltage Law. And this equation is derived from Loop A. Similarly, you can derive a second equation from Loop B.

V₀为正，V₁为负，V₂为负，V₄为负，因为电压是下降的。将它们相加，总电压为零。这正是基尔霍夫电压定律的定义。该方程由回路A推导得出。类似地，你可以从回路B推导出第二个方程。

I will introduce another way to define the sign of a voltage. Here you can see there are two loops. The loops also have direction. They are clockwise, like a clock.

我将介绍另一种定义电压正负号的方法。这里可以看到有两个回路。回路也有方向，它们是顺时针方向，就像时钟一样。

If the current has the same direction as the loop direction, you will get a positive sign. That is the reason V2 in loop B is positive. The direction of V2 is the same as the direction of loop B.

如果电流方向与回路方向相同，则取正号。这就是回路B中V2为正的原因。V2的方向与回路B的方向相同。

In Loop B, we have positive V2 and minus V3. V3 is negative because its direction is different from the loop direction. Any questions? Here are some traditional equations that compute the voltage and current in a circuit. For example, in this case, two resistors are connected in series. How can you compute the current if you know the voltage and the resistance for each resistor?

在回路B中，V2为正，V3为负。V3为负是因为它的方向与回路方向不同。有问题吗？下面是一些用于计算电路中电压和电流的传统方程。例如，在这种情况下，两个电阻串联连接。如果你知道每个电阻的电压和电阻值，如何计算电流？

You can use Ohm's law to get the circuit current. And also, you can get the voltage for each resistor. Because you first compute the current; once you know the current and you know the resistance, you can get the voltage for each resistor.

你可以使用欧姆定律来求电路电流。同时，你也可以求出每个电阻上的电压。因为你先计算出电流，一旦知道了电流和电阻值，就能求出每个电阻上的电压。

I will speak slowly since this is quite simple. Similarly, if we draw the conductance, we know that conductance is actually the inverse of resistance. We have two conductors here. A conductor is actually also a kind of resistor, but we use different forms to express resistance.

这个内容比较简单，我会讲得慢一些。类似地，如果我们画出电导，我们知道电导实际上是电阻的倒数。这里有两个导体。导体实际上也是一种电阻器，但我们用不同的形式来表达电阻。

We know the total conductance of the network is G1 plus G2. Based on Ohm's law, the total conductance is actually the inverse of resistance. So the current divided by…

我们知道该网络的总电导为 G1 加 G2。根据欧姆定律，总电导实际上是电阻的倒数。因此，电流除以……

If you add up the total conductance, this equation is actually equivalent to V equals current times resistance, because the resistance is the inverse of conductance. And once you know the total current is I, you can compute the current for each conductor.

如果你把总电导加起来，这个方程实际上等价于 V 等于电流乘以电阻，因为电阻是电导的倒数。一旦你知道总电流为 I，你就可以计算每个导体的电流。

Any questions? I think this might be the last slide. If there are no questions, let's move on to Lecture 4.

有没有问题？我想这可能是最后一张幻灯片了。如果没有问题，我们就进入第四讲。

Basically, a vector field is based on definition or concept, or analog circuit. Now, let's do the exercise for vector field.

基本上，向量场是基于定义或概念，或者模拟电路。现在，我们来做向量场的练习。

Let's first solve questions one through seven. I will give you thirty minutes.

我们先来做第一到第七题。我给大家三十分钟时间。

[Mostly ambient noise: shuffling, coughing, indistinct background chatter. No coherent lecture content is present in this segment. A student asks:] Teacher, which one was it?

【本段主要为环境噪音：翻动声、咳嗽声、模糊的背景交谈声，无连贯的讲课内容。一名学生提问：】老师，是哪一个来着？

Student: Does Ohm's law apply only to conductors, or does it hold everywhere in a circuit? Instructor: It holds throughout the entire circuit. Student: Does it also apply to the power source? Instructor: A power source has internal resistance, which produces a voltage drop across it.

学生：欧姆定律是只适用于导线，还是在电路中都成立？ 教师：在整个电路中都成立。 学生：在电源上也成立吗？ 教师：电源有内阻，会在其两端产生电压降。

Kirchhoff's Current Law must be applied at a node. A node is a point where branches meet — currents flow in and currents flow out. The law holds at such a junction. If there is no branching, meaning only a single wire with no split, then the law does not apply. Consider a familiar example from high school: when two resistors are connected in parallel and have equal resistance, the current divides equally between them.

基尔霍夫电流定律必须在节点处应用。节点是支路交汇的点——有电流汇入，也有电流流出。该定律在这样的交汇处成立。如果没有分支，即只有一条导线而没有分叉，则该定律不适用。回想一个高中熟悉的例子：当两个电阻并联且阻值相等时，电流在两者之间平均分配。

The instructor explains that this method serves as a relatively simple example for dealing with current. If there are many resistors, one divides the circuit into multiple sections and sums them up. A student raises a question regarding Kirchhoff's Current Law: since the law states that the amount of charge flowing in equals the amount flowing out, does it also apply to a capacitor or a power source? The instructor clarifies that it does not. Kirchhoff's Current Law applies specifically at a junction—an intersection or a point where current flows through. The student confirms understanding, noting that in such a case, it is simply a conductor.

教师说明，这种方法可以作为处理电流的一个较为简单的例子。如果有很多电阻，就将电路分成多个部分再求和。 一位学生就基尔霍夫电流定律提出疑问：既然该定律指出流入的电荷量等于流出的电荷量，那么它是否也适用于电容或电源？ 教师澄清：不适用。基尔霍夫电流定律专门适用于节点——即电流经过的交汇处或通路点。学生确认理解，指出在这种情况下它就是导线（导体）。

No discernible instructional content detected in this segment. The audio consists of ambient classroom noise including paper shuffling, chair movement, and indistinct background murmuring with no identifiable lecture speech.

该时间段内未检测到可辨识的教学内容。音频仅包含教室环境噪音，如翻动纸张、移动椅子以及模糊的背景低语，无可识别的讲课语音。

[No intelligible lecture content detected in this segment. The audio consists of ambient classroom noise, movement sounds, and indistinct background murmuring without discernible speech.]

[该片段未检测到可辨识的讲课内容。音频仅包含教室环境噪音、移动声及不可辨别的背景低语，无可转录的教学信息。]

No discernible lecture content is present in this segment. The audio consists of ambient classroom noise, coughing, and movement sounds without intelligible speech.

该片段中无可辨识的课堂讲授内容。音频仅包含教室环境噪音、咳嗽声和移动声，无清晰可辨的语音。

The questions are quite simple, so let's do them quickly. The first one: you have three resistors. The first is 2 Ω, the second is 3 Ω, and the third is 5 Ω. They are connected in…

这些题目相当简单，我们快速过一遍。第一题：你有三个电阻，第一个是 2 Ω，第二个是 3 Ω，第三个是 5 Ω。它们以……方式连接。

In series, the current is 2 amperes. We have to compute the potential difference. P dot D dot is potential difference; potential difference is the voltage. We want to compute the potential difference across each resistor and the total supply voltage. So we have computed V1, V2, V3, and also the total V. As I said, if you know the Ohm's law, question one will be quite simple. Let's just look at question two. Question two: the lamps.

在串联电路中，电流为2安培。我们需要计算电势差。P点D点的电势差就是电压。我们要计算每个电阻两端的电势差以及总电源电压。因此我们已经求出了V1、V2、V3以及总电压V。正如我所说，如果你了解欧姆定律，第一题就非常简单了。我们来看第二题。第二题：灯泡。

Lamps in a set of Christmas tree lights are connected in series. If there are 20 lamps, each with a resistance of 25 ohms, we can compute the total resistance. Since they are connected in series, you simply add them together to get the total resistance. The second question is to calculate the current. The voltage is 230 volts.

一串圣诞树彩灯中的灯泡是串联连接的。如果有20个灯泡，每个灯泡的电阻为25欧姆，我们可以计算总电阻。由于它们是串联连接，只需将它们相加即可得到总电阻。第二个问题是计算电流。电压为230伏特。

The total resistance, you use Ohm's law to get the current. Question 3: three lamps are connected in series across a 120-volt voltage supply and take a current of 1.5 amperes. The voltage is given, the current is given, so you can get the total resistance based on Ohm's law. And once you know the total resistance, you just minus the resistance of the two lamps, then you get the resistance of the third lamp. Question 4: the fixed coil.

总电阻，你可以用欧姆定律来求电流。第三题：三盏灯串联接在120伏电源上，通过电流为1.5安培。电压已知，电流已知，所以你可以基于欧姆定律求出总电阻。一旦知道总电阻，只需减去两盏灯的电阻，就能得到第三盏灯的电阻。第四题：固定线圈。

A DC generator has a field coil with a resistance of 60 ohms, supplied from a 240-volt source. The current in the coil is to be limited to 2 amperes. Find the resistance of a resistor to be connected in series with the coil.

一台直流发电机有一个电阻为60欧姆的励磁线圈，由240伏电源供电。要求将线圈中的电流限制在2安培以内。求需要与线圈串联的电阻的阻值。

But the maximum current is limited to two ampere, and the voltage is two hundred and forty. The voltage is two hundred and forty, and you can get the total resistance is one hundred and twenty ohms. One hundred and twenty ohms. And for the coil, it has sixty ohms, so the resistance should be at least sixty. Okay, question five is connected in parallel. You have three resistors; they are all connected in parallel. In parallel, and you have to come...

但最大电流限制为两安培，电压为二百四十伏。电压为二百四十伏，由此可得总电阻为一百二十欧姆。一百二十欧姆。对于线圈，其电阻为六十欧姆，因此电阻至少应为六十。好，第五题是并联连接。你有三个电阻，它们全部并联连接。并联，然后你需要……

The first question is the current in each branch. The resistors are 6 ohm, 9 ohm, and 13 ohm. Just use Ohm's law because they are connected in parallel, so each resistor has the same voltage—the line voltage. Divide the line voltage by 6, by 9, and by 13 to get the current for each resistor. Once you know the current for each resistor, add them together to get the total current.

第一个问题是求各支路电流。电阻分别为6欧姆、9欧姆和13欧姆。由于它们并联连接，每个电阻两端电压相同，即线电压。直接用欧姆定律，将线电压分别除以6、9和13，即可得到各电阻的电流。求出各电阻电流后，将它们相加即可得到总电流。

Total current is here. And C is the total resistance. There are many different ways to compute the total resistance. Once you know the total voltage and the total current, use Ohm's law; you can get the total resistance. Or use the equation. The total resistance can be obtained by this equation. So you can get the total resistance as well using this formula. The question says: the effective resistance of two resistors connected in parallel.

总电流在这里。C 是总电阻。计算总电阻有很多不同的方法。一旦知道总电压和总电流，利用欧姆定律即可求得总电阻。或者使用这个方程，总电阻可以由该方程得出。因此，利用这个公式同样可以得到总电阻。题目说的是：两个电阻并联时的等效电阻。

And here we have two resistors connected in parallel. In this case, the total resistance is 8 ohms, right? The total is 8, and one of them is 12 ohms. How can we compute the second one, the resistance of the second one? Similarly, use this equation, and you can get the second resistance. And once you know the second resistance, when they are connected in series—question B is connected in series, right—so you can compute the total resistance.

这里有两个电阻并联连接。在这种情况下，总电阻是8欧姆，对吧？总电阻是8，其中一个电阻是12欧姆。我们如何计算第二个电阻的阻值呢？同样地，使用这个公式，就可以得到第二个电阻的阻值。一旦知道了第二个电阻的阻值，当它们串联连接时——问题B是串联连接，对吧——就可以计算总电阻了。

After Question 7, we can have a break. Question 7 is a parallel circuit with three resistors: 4 Ω, 8 Ω, and 16 Ω. The current in the 8 Ω resistor is 3 A. We want to find the currents in the other resistors. Here you know the resistance and the current, so you can get the voltage. The voltage is the same for each resistor in parallel. So here it should also be 24 volts. Once you know the voltage,

第七题之后我们可以休息一下。第七题是一个并联电路，包含三个电阻：4 Ω、8 Ω 和 16 Ω。8 Ω 电阻中的电流为 3 A。我们要求其他电阻中的电流。这里已知电阻和电流，因此可以求出电压。并联电路中每个电阻两端的电压相同，所以这里也应该是 24 伏。一旦知道了电压，

If you know the voltage and the resistance, you can get the current.

如果你知道电压和电阻，就可以求出电流。