Conservation of electric energy

0:00- [Instructor] Let's talk about electric potential energy and introduce a new unit of energy called electron volt. A quick recap, the electric potential difference between any two points is basically change in the electric potential energy per unit charge when you move it between those two points, right?

0:16Well, if we rearrange this, we can write the expression for the change in the electric potential energy. Now, you may be wondering what's the point? Well, the point is a lot of times we know the potential difference between the two points and we are more interested in computing how much the electric potential energy has changed.

0:38So for example, let's say we have a uniform electric field just to keep things simple. One question you could be having is where does uniform electric field even come from? Well, one way to generate that is by using a plate of charge. So you can have a positive plate of charge here, a negative plate.

0:52This is not a line of charge, it's a plate. So you can use that to generate the uniform electric field. But of course, it doesn't really matter how you generate the electric field, what matters is what the strength of the electric field is. So you have a uniform electric field, and let's consider two points, A and B, where the potential, electric potential over here is 100 volts and electric potential at this point is 40 volts.

1:16Remember that electric potential everywhere along this line is the same because when you travel perpendicular to the field, the electric potential does not change. Anyways, let's say we move this charge of plus 2 microcoulomb from point A to point B. Question is, what is the change in the electric potential energy of the system?

1:33How do we compute that? Well, we just need to calculate what the potential difference is multiplied by the charge and we get the change in electric potential energy. Great idea to pause the video and see if you can compute it yourself. All right. So what is the potential difference between these two points?

1:50Remember, potential difference is going to be final potential minus the initial potential. Since you're moving from A to B, it's gonna be VB minus VA, so it's going to be 40 minus 100, which is minus 60 volt. Okay? And we know the Q value is plus 2 microcoulombs, so delta U, the change in potential energy of the electric potential energy is just plus 2 microcoulombs times minus 60 volt, but it's gonna be minus 110, 20 microjoules.

2:21That is the change in the electric potential energy. And notice we didn't mention the path taken because the change in potential energy or the potential difference does not depend on the path. It only depends on the initial point and the final point. And notice we got a negative sign over here and signs can be very confusing.

2:38So let's focus on that. Does it make sense? Does it make intuitive sense that you get a negative sign? Well, let's see. What does a negative sign mean? It means that the electric potential energy of the system reduced. Does that make sense? Yeah. So my intuition is I just let that charge go.

2:56What'll happen? Well, there's an electric force to the right and it'll accelerate it towards the right. And so notice that it's gaining kinetic energy. Where does that energy come from? It must be coming from the potential energy of the system. And so the potential energy of the system must have dropped because some of it got connoted into kinetic energy and that's how this makes sense to me.

3:19And in fact, we can now see why this idea is so useful in the first place. What if we actually release this charge from rest right here? Then as we just said, it's going to accelerate. And when it comes to point B, it would have some speed. Let's say we're interested in figuring out what that speed is.

3:41How do we calculate that? Well, one way is to use Newton's laws. We can calculate the net force on the charge, and from there we can calculate the acceleration of that charge, and from there we can calculate the speed. But a much better way, a more elegant way is to use energy conservation.

3:58See, when the charger moves from point A to point B, if you assume that there are no external forces, so assume this is in vacuum and there are no resistive forces, then the only force which we're dealing with is the electrostatic force and that is a conservative force, which means the total mechanical energy is conserved, which means the change in the total mechanical energy is 0 as it goes from A to B.

4:19So the change in kinetic energy plus the change in electric potential energy, this is the total change in mechanical energy, that is 0. And so from here I can say the change in kinetic energy must be negative of the change in the electric potential energy and it should be negative q times delta V.

4:37And so in our case, the change in the kinetic energy is plus 120 microjoules. It's positive because negative times negative becomes positive. And of course intuitively, we just saw it makes sense, it gained kinetic energy of 120 microjoules. Now if I know the mass of the charge, then I can plug in the expression for kinetic energy.

4:58Changing kinetic energy is final minus initial. Initial kinetic energy was 0. So final kinetic energy is half MV squared. I can plug that in and I can figure out what the speed is. And by the way, if there was some resistive forces over here, then that 120 microjoules of electric potential energy could get transformed, some of it or all of it could get transformed into thermal energy.

5:18In that case, the gain in kinetic energy of course would be less than 120 microjoules. Okay. And just to make sure that we are comfortable with the signs, what if I had a negative charge and I let go of it? What's gonna happen? Well, a negative charge will accelerate towards the left naturally because there is an electric force in the opposite direction of the electric field, and therefore negative charge will gain kinetic energy as it moves towards the left.

5:44Again, the electric potential energy of our system will drop. And mathematically, again, you will see that makes sense because when the charge moves to the left, delta V is positive, it's going from lower potential to a higher potential, the change in potential is positive, but the charge will be negative and so we'll end up with a negative value.

6:05In contrast, if that negative charge was moving to the right, then notice there must have been an external force acting on the charge, otherwise it wouldn't move all by itself. Some external force must be doing work on it to move it to the right. And because of that, the potential energy of the system would increase.

6:24So change in electric potential energy of the system would increase. Okay, now consider two points in the electric field which are very close to each other and the potential difference is exactly 1 volt. And we take a proton, for example, which has a charge of plus E and we let it go.

6:45What's gonna happen? Well, again, just like before, there'll be a force acting on it and it'll accelerate the charge, it'll speed it up and it'll have some kinetic energy. So what is the kinetic energy gained by the charge? Or what is the electric potential energy lost now?

7:00We know the charge is E, which is 1.6 times 10 to the power minus 19 coulombs, and the change in the potential is negative 1, we are going towards lower potential. And if you multiply that, we just get negative 1.6 times 10 power minus 19 joules. Again, the negative sign is that the electric potential energy dropped and this is the amount of kinetic energy it gained.

7:24This energy is called an electron volt. I always got confused with the name, what exactly is an electron volt? It's a unit of energy. It's a very tiny unit of energy. And an electron volt, you can think of it as basically the energy gained or kinetic energy gained by an electron or a proton, and any charge which has a charge E when it's accelerated through an electric potential difference of 1 volt.

7:54Why do we introduce this new unit? Because whenever we're dealing with such small charges, it's usually not so convenient to continuously think in terms of 10 to the power minus 19 because it's such a small unit of energy, that's why we just call that as an electron volt.

8:09So in this particular case, we just say the potential energy, the electric potential energy system dropped by 1 electron volt and we just say the proton in this case gained a potential kinetic energy of 1 electron volt. Now, what if it was accelerated through a potential difference of 100 volts?

8:27Well, then it would have gained 100 electron volts. And these units get used all over physics, even when there are no potential differences involved, like in the case of radiation, so nuclear reactions, particle collisions. But this is where the unit comes from.

8:42And what does it mean? For example, we say nuclear reactions are of the order of mega electron volt. What does that mean? This means the energy released in the nuclear reactions is equivalent to the energy that an electron or proton would gain if it were accelerated through a potential difference of a million volt.

9:00Similarly, in particle accelerators, we have energies of the order of giga electron volt. It means that it's of the same order of energy released when an electron or proton is accelerated through a potential difference of a billion volt and so on. All right, let's take some example problems.

9:17We have a point charge with mass capital M and charge capital Q. And let's say it's at a point right now where the electric potential is V1, then it has some speed u1. Now there's some complicated electric field you can imagine because of which it accelerates and it ends up somewhere over here some distance R.

9:36And at this point the electric potential is V2. And so now the question is what is the speed u2 at this point? We don't care about the direction. We only care about the speed. And in this scenario, there are no external forces. So we are only dealing with electrostatic forces.

9:54Okay. How do we find this speed over here? Well, at first it seems like a very complicated problem, right? Like I don't even know how the electric field is. The velocity directions are all over the place. I don't know. How do we figure this out? How do we find an expression for u2?

10:10Well, we can use energy conservation. We know as the charge moves from here to here, the change in kinetic energy plus the change in potential energy, electric potential energy must be 0. So can you pause the video and see if you can plug in the expressions for the kinetic energies and the electric potential energies and see if you can figure out an expression for u2?

10:28Great idea to pause and try. All right, what is the change in kinetic energy? It's the final kinetic energy minus the initial kinetic energy, which is half Mu2 squared minus half Mu1 squared. Plus what is a change in potential energy? Well, that is basically Q times the change in potential, potential difference, which is V2 minus V1.

10:49So that should equal 0. So now we can rearrange. We can pull out the half M and then since we are interested in u2 squared, we just have to rearrange this. So u2 squared minus u1 squared gives us this. So u2 squared equals this much, and if you take squared on both sides, we get the expression for u2.

11:10Look at this. It looks so complicated, the final expression, but we computed it just by using energy conservation. That is how powerful this idea is. All right, another question. We have a ball of charge with total charge plus Q spread uniformly throughout its volume.

11:28And since conductors cannot have any excess charge inside, this must be an insulator. And we have a tiny charge plus small q outside it. And we move that charge into the center, right? The charge moves into the center. The question is, what happens to the potential, electric potential energy of the system?

11:50Does it go up, does it go down, or does it stay the same? It'll be a great idea to pause the video and see if you can answer this first intuitively and then confirm that mathematically as well. All right. So we know that the electric field everywhere outside and even inside for that matter is radially outwards.

12:09So the electric force on this is also outwards, which means I have to put an external force inwards to push it and move it inwards. And when doing so, the external force will be doing a positive work and it'll be transferring energy into the system and that will increase the electric potential energy of the system.

12:36So we would expect the electric potential energy of the system to go up. But let's also confirm this mathematically. We should get change in electric potential energy to be positive number. Let's see if we get that. So we know that the potential difference between these two points is the final potential, which is the potential of the center, minus the initial potential, which is the potential outside.

12:56Now, where is the potential higher? Here's a simple way to think about it. The electric field points radially outwards, right? And we know the field always points towards lower electric potential, which means as we move outwards, the electric potential should reduce and therefore the electric potential on the outside should be lower than the electric potential at the center.

13:21So delta V is positive. And we know that the change in potential energy is q times delta V and q is also a positive number. So we're getting positive times positive, which is positive, which is more than 0, which means the electric potential energy of the system increases exactly as our intuition told us.