Electric Potential - Online Tutor, Practice Problems & Exam Prep

Electric Potential

Alright, guys. So in this video, we're going to be talking about the electric potential. We talked about the electric potential energy between two charges. And even though those two things sound similar, we're going to see how in this video they are different. Let's check it out. Electric potential is sometimes just simply called potential, and electric potential energy is sometimes just simply called potential energy. It's kind of assumed that we're talking about electricity. So a lot of people just don't even say electric potential energy. We just say potential and potential energy. But one important distinction to make is that even though these two things are related and they sound the same, they actually represent different things. So we have to be very careful about our word choices and how we use these terms. Now, the best way to understand the difference between potential and potential energy is to go back and talk about electric fields. So, we said that basically a single charge, whether it was a positive or negative charge, let's assume it's a positive charge right here, emitted these field lines, these electric field lines, and basically, these field lines were just information, and this information or field told charges that were in the vicinity how much force to feel.

What happens is that a single charge alongside producing an electric field also produces something called an electric potential, and it's very similar to how it works. So, if you have a positive charge at the same time it's producing this electric field outwards, telling other charges how much force to feel, a positive charge is also emitting some information, a field called a potential, which, by the way, has the symbol V. And this symbol basically tells charges how much energy to have or how much energy to feel. And the thing is, we were talking about single charges in which this was like a big Q. We said that that was the producing charge and then it emitted these field lines and basically nothing happened unless you actually dropped a second charge inside of here, a small q. And once you had a second charge, all of a sudden, there is a force on this thing because if you have some electric field lines that point in this direction, and if you drop a charge here, then, for instance, if it's a positive charge, then this is going to feel a repulsive force like this. And we have the relationship between the force and the electric field that was just given as F=qE.

Similarly, for potential, this single charge sets up a field called an electric potential, and once you drop a second charge inside of this potential, now there is energy. These two things are different, but the mechanisms in which they sort of set up these force fields and energy fields are very similar. We said that the force that we could calculate on a charge from an electric field is just given by this equation: F=qE. Well, it's very similar for energy. What happens is once there's a second charge that you put in between these two things. Now it basically creates some electric potential energy, which we know the equation for, and this electric potential energy is given as U=qV in which, in the case of Coulomb's law in the case of electric fields, this E represented the strength of the electric field that this feeling charge was put inside of. Well, this V is the strength of the energy field that is put inside of.

Another way you might see that is actually the potential, sometimes the potential field. So basically, we know that the q that this little q here always corresponded to the q that was feeling the field that it was put inside of. So, in other words, this q is always the feeling charge. Well, it's the same way with the potential in this formula right here: U=qV. This q always represents the thing that is feeling the potential at that specific spot.

Alright, so I just want to go ahead and wrap up everything really quickly, once more. So you have a single charge. It produces something called an electric field, and that field tells charges that are inside of it, how much force to experience. And once that second charge is put there, there's force called Coulomb's Law, and it's given by this equation or kqq2/r2 where r is the strength of the electric field? And that q is the feeling charge. Well with the potential, it basically does the same exact thing because it's for energy and the equation is slightly different.

A single charge produces an electric potential out here. And once you are in that potential it tells charges inside of it how much energy to have. And once there is a second charge, all of a sudden now there's energy. There's potential energy between these two charges that potential energy is given as U=qV or U=kQq/r where that V is the strength of the energy field or the potential field. And this little q is also the feeling charge. Okay, so the unit of this electric potential is called the Volt, and it's given by the letter V. And this V is actually defined as one Joule per one Coulomb.

Now we have to be very careful here because this V is the symbol for both the electric potential and the unit. So, for example, it would be perfectly sensible to have an equation like this: V = 3 volts. This would be perfectly sensible. It was just some guy who decided hundreds of years ago that the symbol for the letter and the unit was going to be the same. So this right here is the symbol for electric potential. Whereas this right here is the unit. So just so you know, don't get confused between those two. And that's basically all we need to know about the electric potential. Let's go ahead and check out an example. We have a five and a three Coulomb charge that are separated by some distance right here.

So if the five Coulomb charge feels 200 volts from the three Coulomb charge, what's the potential? Great. So we have these two charges right here. This is going to be a five. Coulomb charge, and now we're supposed to figure out what is the potential energy on the five Coulomb charge. So in other words, we're trying to figure out what U is. We know that U is just going to be K Q1 Q2 / r. Here's the problem though: we could use this, we could try to use this potential energy right here to figure out what the potential energy of this five Coulomb charge is. The problem is, we actually don't know what this r distance is, so we can't use this potential energy formula.

Instead, we're going to have to use a different potential energy formula, which is that U is equal to qV. So we have with the charges. This charge corresponds to the feeling charge, and we know that the five Coulomb charge is feeling 200 volts from this charge over here. So, in other words, this is actually the producing charge Q, and this is actually the feeling charge little q. So this is actually going to be the charge that we use in this formula. And what we're doing is we're basically saying this producing charge here is producing some potential field, some potential out in this field here. And this little q is feeling it.

Okay, So that means that this potential energy is just going to be the five Coulombs times the potential, which is 200 volts at the specific point. So at right here, the potential is 200 volts. So that means that the potential energy is going to be five times 200 which is equal to 1000, and that's it. So basically, just let me know if you guys have any questions, let's go ahead and do some examples.

Movement of Charges in Potential Fields

Hey, guys. So now that we've talked about electric potentials, it's time to quickly discuss how charges move inside of potential fields. There's not a whole lot of calculations that you'll need, but there are definitely some conceptual points you need to know. So let's check it out. So basically, the whole deal is that positive charges will always move towards lower potentials and negative charges, accordingly being the opposite of those, will always move towards higher potentials. The way that I like to think about it is to think back to when we were talking about energy. If you have a roller coaster like this, then mass will always try to move towards the lower potential. It's kind of like that, but that's basically the whole entire game, and it's because potential, when we're talking about electric charges, is a field that sort of provides the motivation for charges to move, and it gives them potential energy, so positive charges always want to reduce that potential and negative charges always want to do the opposite of that. All right, so before we talk about how electric fields are going to be sort of incorporated into potentials, let's go ahead and just do this quick example. We have an electron that's at rest, and we have two points. We've got this point right here, A, and we're told that it is at 10 volts, and then we have another point B, that is at zero volts. So somehow we know what the potentials are at both of these points. And then we're going to have to place an electron inside of here and figure out which direction that electron is going to move to. Well, that electron is a negative charge, and negative charges always move towards higher potentials. So that means that this electron is always just going to go to the left, towards A, and that's the answer.

Alright, so let's move on to this next question here because now we're actually going to be relating the way that charges move and potentials with electric fields. So you have a metal rod, and that's actually important because that metal rod means that it is a conductor. It's placed in a uniform electric field and I'm just going to say that this electric field has a strength of E. And we're supposed to figure out which end of this rod is at a higher potential. Okay, so basically, we know how charges in electric fields move. So if we have a positive charge inside of this, we know that the force on those charges always wants to go with the flow, right, because of F = QE. So we know that the force, if we have a charge in an electric field, if that charge is positive, always wants to move in the same exact direction, or it's the opposite for negative charges. Any negative charge basically wants to go against the flow. So we can say if this charge right here, which is negative, wants to go against the flow and we know that negative charges always move towards higher potentials, then that means that the left side of this electric field has to be the one with a higher potential, whereas the right side is going to be the one with lower potential. So basically, it kind of follows our discussion of charges inside of electric fields, and those were basically the answers. This is high potential, and this is low potential. Alright, guys, let me know if you have any questions.

Potential Due To Point Charges

Hey, guys. So in the last video, we saw how charges that are put inside of potentials experience electric potential energy. But we also said that charges produce their own potentials. So we're going to check out what the potential of a point charge is in this video. We're also going to see a very important quantity called potential difference. Let's check it out. So we've got this electric potential, which is mostly just called potential, and basically, what it is. It's like an energy field. So if a charge produces some field that tells other charges around it how much energy to experience we saw with the equation? For that it was U = Q V. And one of the things we can do is we can rearrange to actually solve for this potential right here. We just move the Q over, and we know that the potential is just U ÷ Q. So, in other words, one way to think about this potential is it's basically the amount of electric potential energy per unit charge. It's basically how much potential energy something is going to give another charge divided by how many coulombs it is. That's kind of one way to think about this.

So we saw that if you have a charge like this, so, for instance, you have a producing charge which we'll call Q. And you wanted to figure out what the potential is at a specific point. Let's say right over here, all you need to know is Q and you need to go the distance. And basically, what you'll be figuring out here is that at point A, for instance, you'd be figuring out what the potential is at the specific location. But remember that this potential basically exists everywhere, which means I could drop another point like this. And basically, if I know that distance, I was gonna call this R₁ and R₂ Then if I wanted to figure out what the potential is at Point B, then basically, all we just need to know is these two quantities right here, and I'll be figuring out the potential at this location. Now this potential has an equation, and it's basically K × Q ÷ r. One of the ways we can see where that equation comes from is directly from this V = U ÷ Q. We know that U = K × Q × q ÷ r. So if what happens is if you divide this formula by the feeling charge, basically, you get back to this formula. So it's like the amount of energy that something is going to produce per unit charge of that producing charge, if that makes sense.

All right, now this potential is often referred to or is often associated with energy rights. Basically, how much energy something's going to give another charge. And when we're talking about energy in physics, usually we're interested in the difference in energy because that tells us something about its motion. So that means that there is an actual quantity here, which is basically the difference in the potentials at this location. And this potential difference, which is just the difference in the potential of two points, has a symbol, ΔV. And if things weren't already complicated enough, this symbol has a special name called Voltage, and that name just sucks because it's already sort of similar to something that we've seen before, which is volts. But remember that volts is basically just the unit for potential. And what I want to point out is that these two things are not equal to each other. So you can't just say that the voltage is equal to volts. So I just want to reiterate that again, voltage is not the same thing as volts. Voltage is the difference in potential between two points and volts is just the potential at one location. Okay, so there's actually a difference. There literally is a difference. And basically, this, uh, this potential difference is, if you want to measure it from point A to point B, then this would just be V_B - V_A. So it's in other words, it's basically final minus initial. So this potential difference right here is always going to be the final minus initial. You'll normally see it like a charge is moving from here to here. That will be your point A to point B. But if you're not given that, if you're just asked what the difference in the potential between two points is, it's just going to be the magnitude.

All right, so one thing that all these equations mean is that a charge, so, for instance, if I put a little Q over here at this charge at this location point A and it moves to point B, it's moved through this potential difference. In other words, it's actually gained or lost some energy. And one way we can see that is through this formula right here. If U = Q V, that means that ΔU The change in energy is equal to Q × ΔV, which is the potential difference. So if a charge moves from one potential to another, it experiences a potential difference, and now it has gained or lost some energy. And really, all that depends on is the magnitude of the charge. If it's positive and the potential difference is positive, then it's gained, and vice versa. Right? All right, that's basically it. So let's go ahead and take a look at some examples and see how this stuff works out.

So we've got this point charge right here, and it could produce a potential at 0.5 m away, so this is P₁ and this is equal to 0.5 m. So we're supposed to figure out what the potential is at this specific location, and we know what this Q is. So in other words, we've got for part A. We're just gonna figure out what V₁ is equal to. Now we know V₁ is just going to be K Q ÷ r_₁, which is just the initial distance, which is this guy right here. So all we have to do is just plug that in. We've got 8.99 times 10 to the ninth, and now we've got the two, and this is micro coulombs, so this is going to be times 10 to the minus six and then divided by 0.5. If you go and work this out, you're going to get 3.6 times 10 to the fourth. Now, what I also want to point out is that the signs here are actually very important because we are talking about potential differences here, the signs are actually very, very important. So go ahead and plug in. So if this was a negative microcoulomb charge we would have plugged in a negative sign. Okay, So part B is saying what is the potential 1 m away? So in other words, if this other distance here is just 1 m and by the way, it doesn't have to be necessarily in the same direction. It could be in any other direction because it's not a vector. Then what is the potential at this location? So it was V₂, and this is going to be V₁. So now the two are just going to be K × Q ÷ r_₂. In other words, we've got 8.99 times 10 to the ninth, now you've got two micro coulombs divided by 1 m away. And so one way to think about this is this was double the distance, which means that this should be half the number. So in other words, 1.8 times 10 to the fourth. And that's volts. All right, so that is volts. So I want to point out these units right here, are volts and, uh, this part C right here. What is the potential difference from 0.1 to 0.2? So, in other words, what I like to do is just draw a little arrow right here. So I know which one is the final and which one is the initial. So for part C, the potential difference ΔV, that's the voltage is going to be V_₂ - V_₁. And that's just going to be equal to 1.8 times 10 to the fourth, minus 3.6 times 10 to the fourth. So that means that the potential difference from 0.1 to 0.2 is just equal to negative 1.8 times 10 to the fourth. The negative sign is very important, and this is the voltage. So I want to write out that as well. So that is the voltage. So remember, voltage is ΔV, not V. That's very important. And the other thing is that the final minus initial is also important. If I were asked what the potential differences from point 2 to 1, then I would get the same number, but the sign would just be opposite. And that's important. We are going to see why that is later.

Alright, guys, let me know if you have any questions, and I'll see you guys in the next one.

How far from a 5μC charge will the potential be 100 V?

• A. 4.5×10⁻³ m
• B. 450 m
• C. 45,000 m
• D. 450,000 m

A −1μC and a 5μC charge lie on a line, separated by 5cm. What is the electric potential halfway between the two charges?

• A. 1.44×10⁴ V
• B. 7.19×10⁵ V
• C. 1.44×10⁶ V
• D. 2.16×10⁶ V

Potential Difference Between Two Charges

Alright, guys. So we've got these 2 charges. They're lying on this line, and we're supposed to figure out what the potential difference is between point a and point b. So in other words, what is the potential difference between these two points? Another thing they could have asked us is, instead of this, they could have asked us for the voltage, because those things mean the same exact thing. In any case, we know that the potential difference from point a to point b is just going to be V_b - V_a. So basically, we know that at these points right here, we're going to have 2 contributions to the potential. We're going to have the negative 3Q charge and the Q charge producing potentials at both of these locations.

Basically, what we just need to do is we need to figure out what these things are. Now I just want to warn you, the math here and the algebra is going to get really messy, but the procedure is fairly straightforward. We know that at point b, we're going to have 2 contributions: the potential from Q at this point, plus the potential from the negative 3Q at this point.

We know our equation. We know that V is just equal to kQ/r. So that means the potential from Q at point b is just going to be: kQs-x.

From the negative 3Q charge, we just have this distance x over here. So this is going to be: k(-3Q)x.

One of the things that we can do in this case is cross multiply so that they have the same denominator. So what this ends up being, and this is where the algebra gets a little messy, is this is going to be: kQxsx(s-x) and the next term,-3kQ(s-x)xs(s-x).

This is what the potential looks like at point b. Now all we have to do is basically the same exact thing for the potential at point a. At a, this is going to be: kQx and -3kQs-x.

The final result is delta V = 8kQxxs(s-x) minus 4kQsxs(s-x). That is the potential.

And then all you would need to do to actually solve for numbers is use these potential differences between point charges. Let me know in the comments if any one of these steps sort of tripped you up or got you confused. I'd be happy to explain it. Just let me know, guys. Alright.