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### Course: AP®︎/College Physics 2 > Unit 2

Lesson 3: Electric potential energy, electric potential, and voltage- Electric potential at a point in space
- Electric potential from multiple charges
- Electric potential: definition
- Electric potential: conceptual problems (part 1)
- Electric potential energy of charges
- Potential energy of a system of 3 charges
- Potential energy of a system of charges: Numerical
- Potential energy of a system of charges (intermediate)
- System of charges in an electric field

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# Electric potential at a point in space

Electric potential, a key concept in physics, is an abstract number associated with points in space. It's different from electric potential energy, but related. The units of electric potential are joules per coulomb, hinting at its connection to energy. The video explains how to calculate electric potential and its relationship with voltage. Created by David SantoPietro.

## Want to join the conversation?

- What is electric potential difference?(25 votes)
- Is it not true that whether a charged particle gains or loses potential energy between two points in an an electric field depends on what sign the charge has, and what the sign of the charged particle creating the field is? So say we have an electric field created by a positively charged particle, and point A is close to the charged particle creating the field, and point B is further away and the voltage difference is 10 between A and B. A negatively charged particle placed in the field at point A will gain 10 joules per coulumb by moving from A to B (moving it away from where it is 'inclined' to travel to) and it will lose 10 joules per coulomb of potential energy by travelling from B to A. Is this correct?(6 votes)

- how does this (voltage) work in actual electric devices?(17 votes)
- voltage is change in electric potential and when there is change in potential an electric field will be generated which causes the free electrons in the electric devices to move in a specific direction which is opposite to the direction of electric field i.e. from negative to positive direction and thus current is generated by the motion of electrons thus the electric devices work due to voltage.(30 votes)

- why dont we square the distance to find V like in electrostatic force or electric field?(16 votes)
- Also, when relating Voltage and work(energy), you find that by taking the equation Voltage = Electric Field * Distance and multiplying the entire equation by Charge to get Work = Electric Field * Distance * Charge and then taking the derivative of this equation (by distance), you get Force = Electric Field * Charge. In other words, this means that the equation you used to get Voltage(kQ/r) will similarly require differentiation and this will lead to the squaring of the distance(kQ1Q2/r^2) that we know as the electrostatic force equation.(2 votes)

- Does a 10V battery have more electric PE than a 1V battery because the difference in electric potential (voltage) b/w the positive and negative charges is greater in the 10V battery? Therefore, the 10V battery has more potential energy to do more work?(5 votes)
- An intuitive way to think about potential energy is to consider two water tanks. Tank one has twice as much water in it as tank two so that there exists more potential energy in it than tank two. If a pipe is now connected between the tanks there will exist unequal potential energies resulting in a flow of water from tank one to tank two until the water equalised. Another way to think of it is that tank one has a potential energy level that is 'looking' to find an equilibrium point that it can become equal to. This is what happens in electric circuits where electrons are moved from one point in a circuit that has a higher potential to some other point of a lower potential, thus enabling current flow. Of course, in the above example, if we then connected an outlet pipe from tank two to the ground there would be a further current flow due to the fact that there existed a potential difference between tank two and ground.(13 votes)

- My book (and wiki) says that electric potential is the work done to move a unit charge from
*infinity*to a point in space. But what does that actually mean? What has infinity got to do here? I like the definition which says that EP is a property of an electric field at a certain point in space. It makes sense to me.(5 votes)- Being infinitely far away from other charges just means the electric potential will be zero. If you move towards a positive charge, the potential will increase from 0 to some positive value getting more positive as you get closer to the positive charge. If you move towards a negative charge, the potential will decrease from 0 to some negative value getting more negative as you get closer to the negative charge. So the only place in space where the potential will be 0 is when you are infinitely far away from all charges or the net potential from multiple charges adds up to 0 at a particular location (ex. halfway between a positive and negative charge of the same magnitude).(8 votes)

- so when he says the charge has 200 Joules does he mean that it would take 200 Jouls of energy to move that charge from where it is to the Q charge?(5 votes)
- That dosent make sense ? So how we calculate the Electric potential at Q or very close point (just dr away) . Like lim goes to zero and infinite energy recuired ? Or for example in this video. We got the charge from very far with 200J to 9cm away another 100 required to get it 3cm away?(0 votes)

- does that mean that at the exact point where the charge is there is a infinite voltage(3 votes)
- Its a great question.

Yes, it would seem so huh?? But think about infinite in same way as you think about infinite distance....r. How far away is it? maybe only a few metres or even km at most. so maybe the voltage will also reach a realistic level. Also, remember issues such as

1) this equation is for a charge in isolation. and therefore unrelistic

2) other forces such as the strong force will come into play for some charged particles.

I like the way you are thinking about the situation....what is your response to my few suggestions here?(3 votes)

- If i have two similar metallic spheres having charges +q and -q seperated d cm apart frm each other in vaccum.what is the net electric potential at the midway of the line joining them?

Thanks (in advance)(2 votes)- Just calculate the potential from each one and then add them up.(4 votes)

- Where does the formula V=kQ/r come from??(4 votes)
- or multiplying coulomb's law by distance (same as r) and then dividing by q if you're algebra-based(1 vote)

- At9:10, it states that there would be a high potential energy assuming that q and Q are positive. Is the same true for when both charges are negative? What if q is positive, and Q is negative (vise versa as well)?(2 votes)
- The formula is the same and always works, as long as you put all the signs in correctly. So if Q is positive, it creates a positive electric potential V. If Q is neg, it creates a neg electric potential V. But then if you put a positive q in a positive V, it will have positive potential ENERGY (PE = qV). Same for a neg q near a neg Q (thus a neg V). But a positive q in a neg V or a neg q in a positive V will both have NEG potential energy.(4 votes)

## Video transcript

- Let's talk about Electric Potential V. This is confusing, this is one of, if not the most confusing
ideas in all of physics. For one it sounds just like
Electric Potential energy but it's not, this is different, it's related to Electric Potential energy but the Electric Potential V is different from Electric Potential energy. That was a poor choice of naming. And the other reason it's confusing is that Electric Potential V
is a number, just a number, that's it, associated
with points in space. So it's abstract, this
is an abstract idea, you can't go hold Electric
Potential in your hand, it's a number, an abstract
number at every point in space. Here is points in space, I just put circles around
empty spots on the screen, there's like nothing here, I just put circles here, these are circles around
empty points in space, just so we can talk about them explicitly. And, well, if there was no charge around, if you literally had an empty universe, the V value at every point in this universe would be zero. It'd be zero there,
it'd be zero over here, the number associated with every point in space would be zero. That'd be boring, and useless. How do we make it so that the V value, the Electric Potential value, is not zero? We just stick a charge in here, just stick a big ol' positive Q at some point in space over here, take a big ol' charge and
we'll stick it right there. Now, points in space around this charge will have a V value that's non-zero, and they'll be big if you're near this Q, so the V values around here
are gonna be really big, and then the V values way
out here will be smaller, the further way you go
the smaller it gets. And why do we care? Who cares? The reason we care is this. The units of Electric Potential
are Joules per Coulomb, so Electric Potential has
units of Joules per Coulomb, that gives you a hint
of why you should care, we care about Joules, Joules are energy so something about energy, that's useful, you can get work out of that or can turn it to Kinetic energy. And Joules per Coulomb,
that let's you know, alright, well, if this point over here happened to have, say
100 Joules per Coulomb, let's say the V value
at this point in space happened to be 100 Joules per Coulomb, what that means is, remember there's nothing there, but if there was something there, if we happened to take, say we had a positive two Coulomb charge and we took that charge
and we put it there at that empty point in space, before we put it there
the V value was 100, when we stick it here, who cares? Why do we care about this 100 value? 'Cause look at it, it's
100 Joules per Coulomb, that's what V, the Electric
Potential, is telling us. So if it's 100 Joules per Coulomb and I stick two Coulombs there, how many Joules of energy do
you think it's gonna have? It'll have 200. And that's the key, that's why we care
about Electric Potential 'cause it let's us find
Electric Potential energy, either PE, sometimes people write Electric Potential energy as U. So the formula is just Q, you take the Q that you
sticked at that point in space, in this case it was two Coulombs. Take whatever Q there is, multiply it by the value
of the Electric Potential and that tells you how
many Joules there would be for the charges in that region, so this Electric Potential
energy is between these two charges here, the charge that created the V and the charge that you
sticked at that point, and the V is a quick way to figure out how much Potential energy,
Electric Potential energy there will be. So in other words, in this case, since I have two Coulombs there, I take my two Coulombs and I multiply by 100 Joules per Coulomb 'cause that's the V value, and I get that there are 200 Joules of Potential energy now
stored between these charges. So that's why we care
about Electric Potential V, it's a way to figure out the
Electric Potential energy for a charge that's placed
at that point in space that has that V value. But, how do you get this V value? If I hadn't given you the
100 Joules per Coulomb we wouldn't have been
able to figure this out, we need a way to figure out the V value at points in space based on
the charges creating them, 'cause charges create the V value. There's a formula for it, and the formula says that
the V, Electric Potential, created by point charges equals K, K is the Electric constant
9 times 10 to the ninth, and it has units of Newton meter squared per Coulomb squared, that's always K. You take that K and you multiply by the charge that's creating the V value, so in this case is this Q, this positive Q here, whatever Q it is creating the V value that you wanna find, and that's key, if you plug in five Coulombs here you're finding the V created
by that five Coulombs, if you plug in negative three Coulombs you'll find the V created by
the negative three Coulombs, sometimes there's problems with multiple charges
in it, like this one, and this Q gotta be the
charge creating this V, not the charge you placed
at that point in space, and I'll put the two Coulombs up here, but the charge creating the
V value that I wanna find. And then you divide by the distance, so I divide by the distance between this charge and the point in space that I wanna figure out the V value at. Some people call this the radius, I don't like calling it radius, makes this sound like
there has to be a circle, it doesn't really have to be a circle, this r would be the distance from this point of charge
creating this V value to the point in space where I
wanna determine the V value, that's r. So this is r. So how do we determine this? Let me just give you some numbers, let's say the charge we stuck
here was one nanoCoulomb, nano is 10 to the negative ninth, so let's say that was one nanoCoulomb. And let's say the
distance from this charge to this point in space was, let's say it was nine centimeters. And I wanna know what's the V value, well I can solve for it now, we got our formula, the V would equal, alright my K is 9, always,
times 10 to the ninth, and it's Newtons meter
squared per Coulomb squared, and then I multiply by my charge, and I told you that the charge here was 10 to the negative ninth Coulombs, and my distance, I divide by the r value and the r value is nine centimeters, but be careful, everything's
gotta be in terms of meters, kilograms and seconds when you're doing physics with constants. Look at this is in terms of meters, so I've got to use meters here, so nine centimeters is .09 meters. And if I multiply all this out what you'll get is, 10
to the negative ninth cancels this 10 to the ninth, the powers are 10, these just go away, and then I have nine divided by .09, that gets equal 100. So I chose this so that we got the same answer down there. Okay, 100 Joules per Coulomb, you might be like, where
the Joules comes from? And how is this Joules per Coulomb? Well let's look at it, if we took, look at, one of these meters cancels
one of these meters, and one of these Coulombs
cancels one of those Coulombs, what are we left with? We're left with Newton
times meter over Coulomb, but Newton times meter,
that's force times distance, that's Joules, that's where
we get Joules per Coulomb. So this really does give
us the number of Joules there would be at a point in space per Coulomb of charge that you put there. And it works for any point, you pick any point, if I picked a point twice as close, it's half as far away, let's
say some point over here, let's say this r value here
was only 4.5 centimeters, well I'm dividing this by r, so if the r is half as
big this point over here will have a V value of
200 Joules per Coulomb and the closer I get,
if I went even closer, if I went to a point that
was three centimeters away, well this is a third as
much as this other distance, so if I'm only dividing by
a third as much distance as you get three times the result 'cause r is not squared, it's just r. So at this point, we'll have a V value of
300 Joules per Coulomb. This tells me, if I wanted to get a charge that have a whole
bunch of Potential energy, I should put that thing nearby, I should stick it over here, this will give me a lot
of Potential energy. Not quite as much, even less, the further I put my charge the less Potential energy it will have. There will be no Potential
energy until there is a charge, there'll just be Electric Potential. But once you place another
charge in that region to go with the first one, then you'll have Electric Potential energy and this will be a way to find it, Q times the V that you get
out of this calculation. You gotta be careful though, sometimes people get sloppy, and V looks, you know, we use V for Electric Potential and we use V for Voltage, what's the difference? Are they the same? Hmmm, not quite. Sometimes you can treat them as the same and you don't get into trouble, but sometimes you do and messes you up. Voltage is a, technically a change in Electric Potential between two points, this is the difference
in Electric Potential between two points in space, so it's got the same units 'cause the change in Electric Potential still gonna have units
of Joule per Coulomb, it's just, when it's a change in we give this a new title, we call the Joule per Coulomb unit a Volt. So Joules per Coulomb are Volts, but the word Voltage specifically refers to a difference in Electric Potential, what am I talking about? Well, look at, this point
is 300 Joules per Coulomb, this point over here
100 Joules per Coulomb, so the delta V, if I were to take delta V between these
two points right here and I ask, what's the difference in V? Well the difference in V is 200, 200 Joules per Coulomb, that means the Voltage between those two points
in space is 200 Volts, that's what it means. So, when you're talking about a difference in Electric Potential
between two points in space we call it a Voltage, when you're talking about just the Electric Potential
value at one point in space we call it the Electric Potential, and that's how they're related.