**Concept:** Electric Potential Energy

Hey guys. In this video we're going to be talking about a stored energy given to charges by the electric force called the electric potential energy, let's get to it. If you have two charges, let's say they're both like charges and you hold them together they're not free to move but when you let them go and they are free to move they're going to do what, right? We know that like charges are going to repel. So, what they're doing is initially they're still and then they're moving apart. So, they gained this motion, this velocity that means that they've also gained kinetic energy, right? They weren't moving so they had no kinetic energy and I let them go and they were moving. So, they had kinetic energy, where did this kinetic energy come from? Well, clearly that energy wasn't invented out of nowhere so that energy had to be stored by the charges first. So, those two charges had some kind of stored energy between them and what do we call stored energy? We call it potential energy, the energy that has the potential to be converted into kinetic energy, okay? And just to remind you guys, we have this thing called energy conservation that says however much potential energy you lose, right? We have this negative sign to indicate this is potential energy lost has to equal the amount of kinetic energy gained. So, those charges each lost one Joule of potential energy, when I let them go and they started moving they each gained one Joule of kinetic energy once you will lost one Joule gained, okay? Well, what exactly is the potential energy stored by two charges, the electric potential energy? Well, if we look at this figure here, our kind of standard figure, when talking about electricity, we have two charges q1, q2 separated by distance r the potential energy is just going to be k q1, q2 over r, that's the electric potential energy. Now, really quickly I want to point out a couple of things about this equation that you guys need to bear in mind, one is that the equation is 1 over r you guys are going to be so used to writing down Coulomb's law k, q1, q2 over r squared that you guys are going to be tempted without even thinking to write one over r squared here, okay? Don't. Keep in mind for energy 1 over r, okay? The other thing is the sign of the charges, when we were talking about the electric force the sign really didn't matter, the sign for force just tells us what direction it goes and we know that like charges repel and opposite charges attract. So, we already took care of the direction of the force, for the potential energy the sign really matters so just make sure that you guys always include the sign, if you do these two things you guys will be fine, let's do a quick example, how far apart doing three, sorry, three micro Coulomb charge and a negative two micro Coulomb charge have to be so their potential energy is negative 100 milli joules, okay? Before, we even get started, this is a reminder, a micro Coulomb is 10 to the negative 6 coulombs and the millijoule is 10 to negative 3 joules, a millijoule like a millimeter. So, we know the equation for a potential energy is k, q1, q2 over r, if we want to find r, all we have to do is multiply r up to the other side and divide u over to the other side this means that r is k, q1, q2 over u which is 8.99 times 10 to the 9 times 3 microcoulombs times negative 2 microcoulombs, don't forget the negative sign, divided by negative 100 milli joules, okay? And this whole thing equals about 0.48 meters. Alright, really straightforward problem, really easy just an application of potential energy. Now, what if we have a group of charges, all of these charges are in the general vicinity of one another so they all have an electric force between one another and they all have a potential energy, okay? Well, let's consider a group of charges, what if we have two? We just have q1, q2? Well, that's easy, they each have a potential energy between each other and we've already talked about this, we'll call that u1 too, what if we add a third charge? we'll call it q3. Well, now there are two more potential energies here, there's the interaction between 1 & 3 and there's the interaction between 2 & 3, okay? And, if we were to add a fourth charge q4 there would be 3 more potential energies, right? There would be this potential energy to that interaction, this one and this one. Now, the total potential energy of the system, of this group of charges, is just going to be the sum of all the individual potential energies. So, we have u12 plus u23 plus u13, etc. This would also be right here, that be u34 u24, u14, etc. The only thing to watch out here is avoid what's called double counting, okay? Or over counting, right? We already have a potential energy right here at u23, we do not also want to have a potential energy u32, that's the same interaction as u23, we already considered that don't double count, let's do a quick example to illustrate this.

How much potential energy is carried by this system of charges, okay? So, we have 3 charges will call this charge1, charge2, charge3, we're going to have 3 potential energies, let's find them, u12 is where we'll start. Remember, equation k, q1, q2 over r, all we have to do is plug in our values, this is about 8, sorry, not about, this is 8.99 times 10 to the 9, this is 1 Coulomb, this is negative 2 coulombs, don't forget the negative sign here, negative 2 coulombs divided by 4 meters, right? And, this is about negative 4 times 10 to the negative 9 joules, sorry, 10 to the positive 9, okay? Don't forget about that negative sign, u23, going to be the same equation. So, we have 8.99 times 10 to the 9 negative two coulombs, 3 Coulombs over 3 meters and this is going to be about 16 times 10 to the 9 joules, and lastly, we have u13 which is going to be 8.99 times 10 to the 9, 1 coulomb, 3 Coulombs, and what about r? What's that distance between these two charges here? Well, notice for this triangle one of the legs is 3 meters one of the leg is 4 meters so this is just what we would call a 345 triangle the hypotenuse is 5 meters. Alright, so this is going to be over 5, multiplying all this together, this is going to be positive 4.8 times 10 to the 9 joules. So, now what's the potential energy of this system? It's just the sum of each of these individual potential energies, right? So, we got negative 4, we got negative 16 and we have positive 4.8, adding all these up, we're going to get about negative 15.2 times 10 to the 9 joules which is fine as written or we could call it negative 1.52 times 10 to the 10 joules, dealer's choice, either way is fine. Alright, guys that wraps up our talk about potential energy due to the electric force, thanks for watching.

If a negative charge (-Q) is placed at point P in the above diagram, which situation minimizes the electric potential energy?

A) Situation A

B) Situation B

C) Situation C

D) Situation D

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Three equal point charges, each with charge 1.35 μC, are placed at the vertices of an equilateral triangle whose sides are of length 0.250 m. What is the electric potential energy U of the system? (Take as zero the potential energy of the three charges when they are infinitely far apart.) Use ϵ_{0} = 8.85 x 10^{-12} C^{2}/Nm^{2} for the permittivity of free space.

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A system of charges are shown in the figure. What is the electric potential energy of the following system of charges?

1. *U* = *kQ*^{2 }/ ℓ

2. *U* = 0

3. *U* = *kQ*^{2 }/ 3ℓ

4. *U* = *kQ*^{2 }/ ℓ^{2}

5. *U* = - *kQ*^{2 }/ 2ℓ

6. *U* = - *kQ*^{2 }/ ℓ

7. *U* = - *kQ*^{2 }/ 3ℓ

8. *U* = *kQ*^{2}

9. None of these.

10. *U = **kQ*^{2 }/ 2ℓ

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Three point charges shown below are located at the vertices of an equilateral triangle. If the electric potential energy of two point charges is defined to be zero if the distance between them approaches infinity, the total electric potential energy of the system of those three charges is

A) Positive

B) Zero

C) Negative

D) All of the above

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A point charge q_{1} = 4.00 nC is placed at the origin. A second point charge q _{2} = -3.00 nC is placed on the x-axis at x = 20.0 cm. A third point charge q_{3} = 2.00 nC is placed on the x-axis between q_{1} and q_{2}. Assume that the potential energy of the three charges is zero when they are infinitely far apart. Determine the **electric potential energy of the system** of the three charges if q_{3} is placed at x = 10.0 cm.

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