PHY 102: GAUSS LAW

Added: 2026-05-21

[00:00:00]Good morning, everybody.

[00:00:01]My name is Elijah.

[00:00:03]I will be your physics tutor.

[00:00:05]But then before we proceed, I like to A lot of people actually find physics very difficult, especially university physics. But then I I realized that yes, indeed it's a little bit complicated. Why? Because the majority of the concepts discussed in physics are abstract. They are not really tangible. But then I tell you, I realized that there are actually two reasons why people find it a bit difficult. Number one is actually personal preference. There are a lot of people due to the fact that they might have fear about physics right from secondary school and ETC like that. So, a lot of people don't even have any interest. They don't find interest at all even in the simplest concepts of physics.

[00:00:51]They don't just have interest in it.

[00:00:53]That's the first reason. And number two is just that I discovered that a lot of people don't like physics because they don't have a very good tutor or a very good guide in physics. A lot of people they struggle with good teacher in physics. But then I tell you, you're welcome to Famoz Tutors where all tutors are actually experts in their field.

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[00:01:46]All right. So, let's go back to our topic for today.

[00:01:49]This is physics 102.

[00:01:53]We're going to be dealing with Gauss's law.

[00:01:56]Gauss's Gauss law.

[00:02:01]But like I said earlier, this is from your two toes.

[00:02:09]From your two toes.

[00:02:13]All right. Now let me talk about Gauss law.

[00:02:17]Gauss law is just an extension of electric field. Electric field, that's just what we talk about Gauss law. But the only issue is this. When we talk about electric field, remember that is meant to be e vector. And we are trying to say that electric field is going to decay of Q over what? R squared. That is what we call electric field. Where is your K?

[00:02:37]>> [clears throat] >> Where is your K?

[00:02:40]Where K is equal to 1 over 4 pi epsilon naught.

[00:02:44]That is the value of your K.

[00:02:46]Where is your K is also equal to 8.99 approximately, okay, times 10 raised to the power 9 or approximately be 9 times 10 raised to the power 9. That is the value of your K.

[00:02:59]But the only thing that happened that differentiate between this field uh that is the value of your K. But the only issue or why are we speaking this topic Gauss law is that your electric field, you are trying to talk about how charges relate to each other in free space.

[00:03:19]In an empty space. But when we get to this our Gauss law, we are just trying to say that, okay, if the charges react or relate with each other in free space, how would they react and how would they relate with each other in an enclosed surface or in a particular region? If you should count them together, if you should set a boundary, if you should put an imaginary boundary around them, how do your charges relate? That's actually what Gauss law is talking about. All right. So, let's go back what we call Charles Gauss's law. When I'm talking about your Gauss's law, Gauss's law states that the total Gauss's law law states the total electric flux is equal Okay? The total electric flux is equal to the total charge enclosed in a surface.

[00:04:42]And the name of your surface, because we're dealing with Gauss's law, is called your Gaussian surface.

[00:04:49]Gaussian surface.

[00:04:53]Enclosing a surface divided by by permittivity.

[00:05:05]Now, I'd like you to please pay attention. Gauss's law states that the total electric flux is equal to the total charge enclosed in a surface divided by permittivity. Now, let us try to take a look at the key words so that we can just take them one after the other. From this particular statement, I'll be getting my formula. Just calm down and just follow me. Is that okay? Now, Gauss's law states that the total electric flux And when we talk about electric flux, what is the sign? It is called phi.

[00:05:39]But because I said total, I can call it phi net.

[00:05:45]Is equal I the total charge.

[00:05:50]But what kind of charge is it? It is enclosed.

[00:05:53]So, for that reason I can call it Q E N C. May your E N C talk about enclosed charge. Remember that the letter Q is charge. Then I said divided by permittivity. So, all over permittivity, which is called your E naught. Allow me call this equation one.

[00:06:14]Now, I'd like you to really calm down now.

[00:06:18]You can do well to pause the video whenever you get if I am a bit too fast, so that you can pause it so that you can get it before you continue the video.

[00:06:28]But let us go on. How do we get If this is what we call Gauss's law. But then, take a look at it. I can get it from another means.

[00:06:37]In the previous class, I talked about what we call electric flux. I want to talk about electric flux. Your electric flux is a scalar product of electric field and your charge.

[00:06:51]But before that, remember we're talking about it must be in an enclosed surface. Let us assume Allow me split my board.

[00:07:00]Let us assume that the surface is a sphere.

[00:07:04]Why? Because we have different kind of Gaussian surface. It can be a planar surface. It can be a cylindrical surface. It can also be a spherical surface. But right now I want to make it a case of a spherical surface. Now, let us imagine that your charge is found at the center of your sphere.

[00:07:25]And because this is your charge Q, it extend an electric field outward.

[00:07:32]This is your electric field.

[00:07:34]But remember common mathematics. From your center to the circumference, we can call this radius R.

[00:07:43]But remember, this is just like a pizza.

[00:07:46]And what you have decided is that you decide to take a slice out of it. So, allow me to call that particular portion that you decide to consider. Instead of the total area to be A, allow me to call it a differential of A.

[00:08:00]To show it's just a small slice that you cut out of it. Is that making sense?

[00:08:05]If that is the case, you just follow.

[00:08:07]Like I said earlier, I taught in my previous just class what we call electric flux. And the electric flux by in the last class I discussed this. I said it is the scalar product, which is scalar product of your electric field and your what? Your area vector.

[00:08:27]Remember, this is still equation one.

[00:08:29]Then from there we can now continue by saying this.

[00:08:34]Because I'm dealing with a particular differential area. Therefore, remember we said that it is that enclosed.

[00:08:41]Therefore, I can now say that my flux is now going to be integral enclosed of what? E dot dA.

[00:08:52]There's nothing difficult in what I just said.

[00:08:55]Integration is for the entire. But because you are trying to talk about a closed surface, that is why I'm putting a coil around my integral sign. Just calm down. Nothing special.

[00:09:09]Oh, if that is the case, therefore. But there's something that we know. We know that my electric field E, yeah?

[00:09:18]This my electric field E is equal to what? K Q over what? R squared. If you remember, I discussed that earlier. So, if this is that, but I know that my K also is 1 over 4 pi E naught. Therefore, okay? Allow me use this place. I know that my K is equal to 1 over 4 pi E naught. Therefore, my electric field will become what now? I'll be having my Q over what? 4 pi E naught.

[00:09:56]That is just the value of my E.

[00:10:00]Therefore, do you know that this will miss my E dot integral of what? DA.

[00:10:10]What does that means now? It means that my flux is equal to what?

[00:10:15]What is my E? My E is now going to be what? Q over what? 4 pi E naught.

[00:10:23]Dot closed integral of DA. You remember, this is a sphere like I told you.

[00:10:31]What is the formula for total area of the sphere? Remember that the total area of the sphere, which is my DA, is equal to 4 pi r squared. Do you still remember all of that? So, if my DA integral DA is equal to 4 pi r squared, therefore, I can say that my flux is equal to Q over what? 4 pi E naught multiplied by Remember, the total integration of a sphere, you said it is 4 pi r squared.

[00:11:05]Oh, but wait. I'm so sorry. Remember, there's an r squared I forgot earlier.

[00:11:09]So, this becomes an r squared.

[00:11:12]This becomes an r squared.

[00:11:15]I'm sorry. Remember, there's an r squared here before, so I don't need to include it.

[00:11:19]So, now let's take it one after the other. So, automatically, what am I having? I'm having a 4 pi, so this can cancel out.

[00:11:28]An r squared, this can cancel out. Can you now see that my flux is still equal to what now? My Q over what? What am I left with?

[00:11:40]E naught.

[00:11:42]Is this not equal to what I have there?

[00:11:45]Sorry.

[00:11:49]You can see that it's the same thing with what I have up here.

[00:11:53]Which means that according to what we call Gauss's law, your Gauss's law which says that your electric flux, the total electric flux in an enclosed surface is just equal to the total charge enclosed in that surface divided by the permittivity.

[00:12:09]And from what we have so far, we have gotten our permittivity again directly from it. Now, allow me to wipe off my board. Gauss's law implies my flux is equal to my charge enclosed over what now? E naught. I have to really pay attention. Where this your E naught is called permittivity. And what is the value of your permittivity? Your E naught is always equals to 8.85 * 10 raised to power minus 12.

[00:12:45]That's the value of your E naught.

[00:12:47]Now, I'd like you to really pay attention.

[00:12:51]Most especially if you are dealing with um advanced level physics. We don't only major on calculations alone. You have to really pay attention to some statements questions. And how do we go about it?

[00:13:05]These are the two statements questions that you have to put in your mind.

[00:13:09]Because we are talking about an enclosed surface for Gauss's law, these are the two important things that you have to take a look at. The total charge inside this surface. Okay, let me write that.

[00:13:23]Note.

[00:13:25]You have to really pay attention to this. The total charge inside a closed surface is equal to zero.

[00:13:45]Number two, the electric field E also inside a closed surface is also equals to zero.

[00:14:04]This is just what you should just know in your heart. Due to the fact of equipotential in every conductor or in a surface, so for that reason, electric flux, electric field inside a closed surface is always equals to zero.

[00:14:21]Why? At the surface or outside the surface, they are never equals to zero.

[00:14:27]You have to You just know that. So, what you have been able to decipher so far is this formula that your flux is equal to Q enclosed over E naught. And then, we were able to also derive that your flux is also equal to what? E dot A.

[00:14:44]So, if I am mistaken, All right. After that, now, uh for the sake of convenience, what I will now be giving you is going to be formulas to remember.

[00:14:58]The ones we are done with the formulas to remember, then automatically we can now go with You just solve a very, very simple question, and then that will be all for today. Now, what are formulas to remember? Formula to remember.

[00:15:17]This is physics class.

[00:15:19]The first one says, remember according to Gauss's law, your flux must be equal to what? Q over what? E naught."

[00:15:28]I can call it Q enclosed.

[00:15:31]That's equation one. The second formula to remember, remember I told you that we have here that your flux is also equal to what? I said it is the scalar product between your electric field and your what? Your area vector.

[00:15:47]That is second formula.

[00:15:49]Therefore, if that is equal to that, number three, in most textbooks, this is the formula you're going to be seeing for your Gauss's law. And what is that?

[00:15:58]Your flux is equal to closed integral of your E dot d A.

[00:16:07]All right. If that is a third formula, then allow me say we can now link if your flux is equal to E dot A and the same flux is equal to Q naught over E naught, I can now allow me to just merge these two of them together by saying that your Q should be equal to Q. All right. If you agree, then automatically I would just say that, "Okay, let equation one, this equation one, equal to equation two."

[00:16:39]And what does your equation one say? It says, "Q enclosed over what now? E naught will now be equal to what now? What is equation two?

[00:16:49]E dot what? A."

[00:16:52]And that becomes your equation four.

[00:16:56]If you can actually remember these four formulas, I put it in your face, then there's no question on Gauss's law that you won't be able to solve.

[00:17:06]All right. Now that we have all of this, we have our formulas, we know the two statements that are actually very, very important in Gauss's law. We stated stated law, and then we have actually considered what Gauss's law is trying to talk about. Gauss's law is just trying to talk about how charges relate or react in a particular closed region. All right, having said all of that, now let us play with question so that we can apply all of these formulas. Are you ready? All right, I'll wipe off my board now so that we can start to work.

[00:17:38]Is that all I need to take my notes so that I can take your questions properly?

[00:17:42]Okay, question one says that a solid metal a solid metal surface metal surface of radius 0.1 m carries a total charge a total charge of 5.0 microcoulombs.

[00:18:18]Now, we're told to calculate the electric field at A, at the center of the sphere.

[00:18:36]B, a point 0.05 m inside the sphere.

[00:18:54]C, surface of the sphere.

[00:19:03]All right, so let us solve now.

[00:19:06]Now, let us take a look at the questions together.

[00:19:09]A solid metal surface of radius 0.1, which means that the first parameter we are given is radius, right? So I can say that the R Remember we're solving now. R is equals to 0.1 m. That's what they gave us. Now carries a total charge. Remember they said total charge. So what is the letter for charge? That is going to be Q. They said total. We can call it Q total. We can call it Q net. It is equal to they said 5.0 micro coulomb.

[00:19:41]Is that what they gave? But remember that so 5.0 micro coulomb and that's going to be what? 5 * 10 raised to the power minus 6. Remember that micro is 10 raised to the power minus 6 and I'm having my coulomb. All right. We are told to calculate the what now? Electric field.

[00:19:58]Electric field. So I'm looking for my electric field.

[00:20:02]It's question mark.

[00:20:04]Now, question A says center of the sphere, center.

[00:20:11]All right.

[00:20:13]Uh-huh.

[00:20:14]Remember I told you this is the scope.

[00:20:18]Flux inside, electric field inside the surface. Who can remind me? Can you still remember? Remember we agreed that in the notes I gave to you that electric field inside a surface and also the electric flux inside any surface is always equal to zero.

[00:20:35]So which means that for them to say at the center of the sphere center of the sphere, if this is my sphere it is at the center. Is it within this region? Yes, because it is within the region. Automatically, what is going to be my electric field inside? It is must be equal to zero.

[00:20:57]Why? Because it is within the surface.

[00:21:01]So let us go for question B.

[00:21:04]Yeah, as simple as that. Why? Because the [clears throat] electric field inside any scope is always equals to zero.

[00:21:11]As simple as that. All right. Now, let's go.

[00:21:14]Question B says at a point 0.05 m inside the sphere. So, which means that if this is my sphere, this is the center. Well, let's say we locate it now here. Which means that from here to here is 0.05 m.

[00:21:32]But, take a look at it.

[00:21:34]It is still within the sphere because the question said inside. So, a point at a point 0.05 m still inside.

[00:21:51]Because it is still inside, no matter the distance within the sphere, it is still equals to zero.

[00:21:59]Is that okay? All right. So, let's go question C now.

[00:22:03]Question C now said at the surface, which means that yeah.

[00:22:08]But, remember I told you within or inside zero. At the surface or outside, it is not equals to zero. Which means that because the question said at the surface, so I can solve this now. All right. So, let us go for this. So, they said at the surface of the something.

[00:22:25]Now, looking for remember electric field. And what do we agree about electric field? If you should recall, I told you that your electric field is always equals to what? Remind me. K, uh-huh, Q over what now? R squared.

[00:22:40]Isn't it? All right.

[00:22:42]But, remember I told you that your K, where your K is always equals to what? I told you 9 * 10 raised to the power of 9. Therefore, I can get my E by saying what is my K? 9 * 10 raised to the power of 9 multiplied by what is my Q? Remember they I you that the total charge is what? 5 * 10 raised to the power of what? -6. It was a 5 micro, but they have converted it into this. All over What is your radius? Your radius is what?

[00:23:14]0.1 raised to the power of 2 over 2.

[00:23:18]If that is taken, allow me wipe out this side.

[00:23:22]So, now let us go on. My E now is equal to that. But let's go on.

[00:23:29]Remember, I could have been I can actually solve this using indices. A 9 * a 5 gives a what? A 45.

[00:23:36]Times Remember, a 9 raised to the power of This is a 10. This is a 10. So, I can take one of them. The first 10 has the power of what? 9, based on indices. Now, according to indices, I can say that -6.

[00:23:47]Do you agree with me?

[00:23:48]All over. Now, remember this is 0.1. And this 0.1 raised to the power of 2 becomes what? What do you think the answer is? That becomes a what? A 0.01.

[00:24:01]That's going to be my answer. But remember that this 0.01 is also equal to what? 10 raised to the power of -2. Do you agree with me? Okay. You can actually use your calculator. Trust me.

[00:24:12]So, that becomes what now? A 45 * 10 raised to the power of 9 take away 6 becomes a what? A 3 over 10 raised to the power of -2.

[00:24:22]All right, if that is taken. And according to indices as well, my E becomes what? This is 45 times Remember, this is a 10 and a 10.

[00:24:31]So, I take one of them. The first one is saying what now? 3. But remember that according to indices, division is what?

[00:24:37]Minus. The second 10 is saying what? -2.

[00:24:40]And a minus times minus becomes what everybody? A plus. So, that my E now becomes what? A 45 * 10 raised to the power of 5. But remember, I'm talking about I'm talking about uh standard form. This is still not standard form. So, which means that I have to move it backward so that my electric [music] field now becomes what?

[00:25:02]It's 4.5 * 10 raised to power. Remember I moved it to once, so this becomes 2 + 1 to 8.

[00:25:08]So, raised to power 6 N/C. That is your final answer.

[00:25:14]So, we have come to the end of today's class. So, what I'm going to do now is going to be I'll wipe off my board now.

[00:25:19]You can actually pause the video so that you can just take what I've written. So, I'll wipe off my board now, then I'll give you a very good end of class assignment. So, do well Once you're done with the assignment, do well to drop your solution in the comment section. I will go through them. Then, that will be all. Pause it.

[00:25:39]All right. Now, this is going to be our assignment.

[00:25:43]A solid metal surface of diameter 10 mm.

[00:25:48]This is diameter, remember? Then, we used radius, but now we are making use of diameter. 10 mm. Carries a total charge of 2 nC.

[00:25:58]Remember what we used before was micro, a 5 micro, but now we are using nano.

[00:26:03]All right. Calculate the electric field.

[00:26:07]You have to really pay attention. I'm looking for electric field. Also, I'm looking for electric flux. Not just electric field, I'm also looking for what now? Electric flux at the what? At the surface of the sphere. Remember the [clears throat] surface of the sphere.

[00:26:22]Are they equal to zero or not equal to zero?

[00:26:25]Diameter equals to zero. So, once you're done with it, just drop the uh your solution in the comment section.

[00:26:32]So, we've come to the end of today's class. So, what we are going to be doing is just that in my next class, you can stay tuned to like, subscribe, and also on your notification of our YouTube tools. I'll be dropping a lot of materials and resources that will ease your academic journey. All right. So, in my next class, we're going to be talking about charge density, and then once I'm done with that, we'll talk about what we call the electric potential.