**Concept:** Electric Potential

Hey guys. In this video we're going to be talking about a new concept called the electric potential, while the electric potential is related to the electric potential energy those are two different things. Alright, let's get to it. The electric potential which we're often just going to call potential, okay? Some of these terms carry electric in front of them electric potential energy, electric potential, we know that we're talking about electricity in this context. So, just drop the word electricity, will often refer to them as just potential energy and potential, okay? So, be aware of that.

The potential is related to but different than the electric potential energy, okay? Which you notice we can also drop electric from that and just call it potential energy, this is the key theme that I want you to sort of take away from this video is that the potential is not the same as potential energy, their name very similarly but they are different quantities, okay? So, we want to develop sort of an intuitive understanding of what this potential is through an analogy to the electric field, okay? So, this is a refresher, the electric field is something that produces the electric force likewise the potential is something that produces a potential energy. Now, a single charge is responsible for producing the electric field, we have some single charge, which I'll say is q, sending out information in all directions, okay? What a field is, is it's just something that exists everywhere so the electric field exists everywhere, what is this information that is sending out? Well, the field tells a charge how much force to feel, okay? So, the information carried by the electric field is information about the force. So, we should think that the electric field is what we would call it a force field, is a field that carries information about the force, okay? Likewise the potential is produced by each single charge just like the electric field is produced by a single charge, a potential is also a field okay, if we have a single charge q it's also going to send the potential everywhere, okay? So, the potential is also a field, it exists everywhere, the information that the potential carries tell to charge how much energy to have, okay? So, we should think of this as an energy field, a field that tells us something about the energy, so this is the big distinction between the electric field and the potential is that the electric field that tells us information about the force, the potential tells us information about the energy, okay? If I were to drop a charge capital Q in the electric field, once there's a second charge there is a force, that second charge capital Q feels the electric field and feels a force due to it. So, let's say both of these were like charges it would feel a repulsive force, likewise for the potential, once there is a second charge in the potential, which I'll call capital Q, there is an energy, capital Q feels that potential as an energy. So, now it has some electric potential energy, okay? So, there are a lot of parallels to draw between these two fields, really quickly, just as a matter of being completely thorough, this is not an either-or situation, you don't either have an electric field or an electric potential, a single charge will also produce an electric potential, a single charge will produce both at the same time. So, when you drop a second charge it feels both an electric field and a potential due to the first charge, okay? So, the equation that tell us what the magnitude of the force is of a charge in an electric field is this equation, which we've seen before, when I drop a charge q into an electric field E it feels this force q, E, okay? Likewise, if I drop a charge q into a potential V, V is a symbol for electric potential, it will feel a potential energy of q times V, okay? So, these two equations are basically identical equations, some electric field E produces a force q, E some potential V produces a potential energy q, V, okay? And E is just the strength of the force field, right? The electric field and V is just the strength of the energy field, right? The potential. Now, in both of these equations the q that's being used is the charge that feels that the effect of that field, okay? It's the feeling charge in both cases, which I referred to by capital Q in these figures, this is the feeling charge in both cases, the charge is actually interacting with the fields, alright? So, the big thing to take away is this last bullet point.

The electric field E is a force field, a field that carries information about the force, the potential V is a field that carries information about the energy, alright? The unit of this field, of this potential fields is something called the volt and 1 volt given by the letter V is one Joule per Coulomb, okay? Now, something, that's important to. Notice here is that we need to be careful, V is both the symbol for electric potential and for the unit of potential, the volt, for example, this is a perfectly reasonable thing to say, that the potential at some point is 3 volts and if you notice, we have V on both sides of this equation, whoever decided that these should be the symbol for the volt and for the potential messed up, because now we have these things, these equations like this, that seem to make no sense, but this is the potential equal 3 volts. So, those two V's are technically different, alright? Let's do a quick question.

A 5 Coulomb and a 3 Coulomb charge are separated by some distance, if the 5 Coulomb charge feels 200 volts from the 3 Coulomb charge, what is the potential energy of the 5 Coulomb charge. Alright, the first question I want to ask is, which of these two charges, the 5 or the 3 Coulomb charge, is the feeling charge and which is the producing charge, okay? The 3 Coulomb charge is the producer and the 5 Coulomb charge is the feeler, okay? The 3 Coulomb charge is what's putting the 200 volts onto the 5 Coulomb charge, right? And the 5 Coulomb charge is feeling a potential energy, so the potential energy, which is just q times V, is going to be 5 coulombs, not 3 Coulombs, because it's the feeling charge times the 200 volts that it's in, which is 1000 joules, okay? That wraps up our discussion on electric potential, Thanks for watching.

**Example:** Movement of Charges in Potential Fields

Hey guys. In this video we're going to discuss how charges move in electric potentials. Alright, let's get to it. Now this is going to be super quick, super straightforward, alright? Positive charges always move towards low potentials, so the positive charge is an option to go from, let's say 5 volts or 10 volts, it's going to go to the 5 volts, it always moves to a low potential, negative charges do the opposite, they always move to a high potential. So, once again, if a negative charge has a 5 volts or a 10 volts option, it's going to go to the 10 volt option, alright? Let's do a couple examples.

An electron is at rest between two points, A at 10 volts and B at 0 volts which point will the electron move to. An electron is negatively charged. So, it's going to move to the high potential. It's going to move to point A, okay? Very easy, very straightforward.

Example two, a metal rod is placed in a uniform electric field as shown below which end of the rod is at a higher potential? Well, in this electric field where a positive charge is going to move? positive charges are going to move down the electric field lines, right? F equals q, E, if q is positive F points with E, negative charges are going to move up the electric field lines, if q is negative F points opposite to E, okay? Now, this is pretty straightforward, if the negatives are moving to the left, is at the high potential side of the low potential side? Remember, negative charges always moves to high potential. So, this has to be high potential, if the positive charges are moved to the right and positive charge is always going to low potential then the right side has to be low potential, okay? That wraps up our discussion on the movement of charges and electric potentials, thanks for watching.

**Example:** Potential due to Point Charges

Hey guys. In this video we're going to be discussing something called potential difference. Alright, let's get to it. Just as a reminder, electric potential, which we're often just going to call potential, is an energy field. Remember, that a field is something that exists everywhere and an energy field is a field that carries information about the energy, the particular information it carries is that if a charge q ito lands at a potential V the potential energy should be q times V, okay? Now, something, we can do is we can just divide the q over to the other side and represent the potential as the potential energy divided by q, what this means is mathematically the potential can be thought of as the electric potential energy per unit charge, right? The potential energy per unit charge, okay? Now, the next natural question is going to be, what is the potential due to a point charge? How can we calculate this? Well, imagine we have a point charge q and some point p that we want to know the potential at, and p is some distance r away from q. Well, then the potential due to this point charge will be k, q over r, this equation looks very similar to the equation for the electric field due to a point charge but there's a couple important differences.

First, this is over r, not r squared, your natural instinct is going to be to write k, q over r squared because you've written it a bunch of times for it, the electric field, just keep in mind when you're calculating potential is 1 over r, okay? The other thing is the sign of the charge, we typically ignore the sign of the charge when calculating the electric field, the sign of the charge is going to be very important for the potential, okay? Now, remember, because potential is a field it can exist everywhere. So, we can also measure the potential at some other point, if I called our original point 0.1, I also have another point 0.2, this will also be some other distance away, both of those points will potentials, I can see the potential 0.1 is sum V1 and the potential at point two is sum V2. Well, it turns out that it's important to know in physics, what the difference in potential between those two points are, which will say is Delta V which is V2 minus V1, the potential difference is just the difference in potential between two points, okay? Which, we call Delta V, what is the change in potential between two points? how does the potential change between two points? etcetera, that is the potential difference. Now, because there aren't enough names already this actually gives a unique name, okay? We typically call it the voltage, alright? Now, there's a problem with this name. Remember, that the units for potential are volts, okay? It's very, very important to remember. So, be careful that voltage is not the same thing as volts. Remember, that voltage is a measurement of potential difference, volts are the unit of potential, okay? So, we could for instance have a measurement V equals 3V or something like Delta V equals 4V and it's important to know all this means, this V is a potential, this V is the unit volts, this V is a potential difference or a voltage, this V is still the unit volt, okay? So, those two measurements are written V equals 3V or Delta V equals 4V, you have to understand what all of it means, okay? I know it's pretty confusing but sometimes physics is just confusing, alright? Now, let's do a quick example to wrap this up.

What is the potential half a meter away from a 2 Coulomb charge? What about 1 meter away? What is the potential difference between these two points? The voltage? Alright, so we have some point to 0.5 a meter away, there's some point to one meter away, if we say there the voltage of p1 is V1 then V1 is just K, q over r which is 8.99 times 10 to the 9 times q, which we said was 2 Coulombs, divided by the distance, which we said was half a meter, and that's going to be about 3.2 times 10 to the 10 volts. Now, the voltage at p2 which we call over V2, it's going to be the same equation. So, it's just 8.99 times 10 the 9, the charge is still 2 Coulombs, that didn't change, the distance changed, now it's one meter, and this is going to be about 1.6 times 10 to the 10 volts, okay? So, those are the potential at those two points. So, we answered the first question, we answered the second question. Now, the next question wants to know what the potential difference is. So, that's Delta V, which we'll just say is v1 minus v2 plus 3.2 times 10 to the 10, which is three, sorry, minus 1.6 times 10 to the 10, which is 1.6, times 10 to the 10 volts, okay? That's the answer 3.2 is just double of 1.6, alright? That answers the third question, the last question is what's the voltage? Well, remember, what is voltage? Voltage is just potential difference, okay? Remember, that voltage is potential difference. So, this just is the voltage because it is the potential difference. So, all four questions are answered, an important thing to keep in mind always, when you're working is that voltage is that change in potential, the potential difference, not the potential itself, okay? It is not this, right? That wraps up our discussion on potential difference, thanks for watching.

**Problem:** A -1uC and a 5uC charge lie on a line, separated by 5cm. What is the electric potential halfway between the two charges?

**Example:** Potential Difference Between Two Charges

Hey guys. Let's do an example using the electric potential. Two charges q and negative 3q lie on the line as shown below, what is the potential difference between point A and point B, okay? So, at point a each charge put some potential and the potential of point A is and with the sum of these two, at point B each chart puts a potential and the potential at point B is going to be the sum of those two, let's find those for individual potentials first. So, the potential due to q at A is just going to be k, q over x the distance between q and a, what about q or, sorry, the 3q charge at A? Well, this is going to be k times negative 3q over the distance between that negative 3q charge and A, if this total distance is s then that distance is s minus x where x is the distance between q and A, so this is just going to be over s minus x, okay? Now, what about for point B? Well, the potential for q at point B is just going to be k, q over the distance between q and point B like for the 3q charge in point A, this distance is s minus x, okay? So, this is over s minus x. Lastly, what about the potential for this 3q charge at point B? This is going to be k, negative 3 q divided by that distance which is just x, okay? These are our four potentials. Now, we need to add the two at A and add the two at B to find our potential at A and our potential at B. So, our potential at A is V, qA plus V of 3q, A, which is going to be k, q over x minus 3 kq over s minus x, in order to add these guys to have the same denominator, that means multiplying this denominator up into this numerator and this denominator up into this numerator. So, we have k, q, s minus x over x, s minus x minus 3 k, q, x over x, s minus x. Now, we can do the subtraction but first I want to distribute this term right here. So, this is going to be k, q, s minus k, q, x minus 3 k, q, x over x, s minus x and that leaves me, sorry, let me give myself a little bit of breathing room here to write the final answer, this leaves me with, we have negative K, q x plus negative 3k, q, x, this leaves me with k, q s minus 4k, q, x over x, s minus x, okay? So, that's 1. Now, we got to do the same thing for the other, point B, okay?

So V at B, Vq at B, which is going to be V3q at b plus V3, q with at B is going to be k, q over s minus x minus 3k, q over x, once again, we have to multiply the denominators across to get the least common denominator, this one to here, this one to here. So, this is going to be k, q, x over s minus x minus 3k, q, x minus x, sorry, this is x, s minus x, x, s minus x and we want to then combine the two fractions because they have the same denominator and we want to expand this term right here, okay? Let me hide myself. So, I have space. So, k, q, x minus 3k, q, s plus, okay? This negative and this negative cancel to be plus, 3k, q, x over x, s minus x, combining that we are going to get, this is a 3, this is a 1, 4k, q, x minus 3k, q s divided by x, s minus x, okay? So, these are two potentials at point A on the Left, point B on the right. So, what is the potential difference between point A and point B? Well, the potential difference which is VA minus VB is going to be k, q, s minus 4k, q, x over x, s minus x minus 4k, q, x minus 3k ,q, s over x, s minus x, right? They already have the same denominator. So, we can just combine them right away k, q, s minus 4k, q, x minus 4k, q, x plus 3k, q, s, okay? It's important, this negative and that negative become a positive, over x, s minus x and finally, we have a 3 here, we have a 1 here. So, that's 4k, q, s, we have a minus 4 and a minus 4. So, minus 8k, q, x over x s minus x and that is the final answer, you could simplify this a little bit more because those have a 4k, q, x, I'm sorry, a 4k q, right? This would become 2x this term will become 2x this term would just become s you could pull that 4k, q out but you don't have to simplify it any more than this, alright? Thanks for watching.

Two charged spherical conductors are connected by a long conducting wire. A total charge of q > 0 is placed on this combination of two spheres. Sphere 1 has a radius of r_{1} and sphere 2 has a radius of r_{2}, where r_{2} > r_{1}. If q_{1} represents the charge on sphere 1 and q _{2} the charge on sphere 2, what is the ratio q_{1} / q_{2} of the charges?

1. q_{1} / q_{2} = r_{2} / r_{1}

2. q_{1} / q_{2} = r_{1} / r_{1} + r_{2}

3. q_{1} / q_{2} = (r_{2} / r_{1})^{2}

4. q_{1} / q_{2} = r_{2} / r_{1} + r_{2}

5. None of these

6. q_{1} / q_{2} = 1

7. q_{1} / q_{2} = (r_{1} / r_{2})^{2}

8. q_{1} / q_{2} = r_{1} / r_{2}

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Three point charges are held on the circumference of a circle of radius 20 cm as shown in the figure. Assume that the electric potential is defined to be zero at infinity. Determine the electric potential at the center of the circle.

(a) +6.0 x 10^{4 }V

(b) +2.3 x 10^{5 }V

(c) -4.6 x 10^{5 }V

(d) –7.8 x 10^{5 }V

(e) zero volts

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Three charges of magnitude |q| = 3.0 x 10^{-9}C are placed along the circumference of a circle with radius R = 1.0m. A charge +q lies at x = -R, a charge -q at y = -R, and a charge -q at x = R.

a. Determine the electric potential at the origin of the circle.

b. If a negative charge with q = -1.5 x 10^{-9} C brought from r= ∞ and placed at the origin, what is the change in potential energy of the system of charges?

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Consider a square with sides *L =** *48*cm *and two negative charges *Q = - *2.5* μC *placed on the corners labeled with *Q *in the figure. What is the electric potential in the corner of the square labeled by B?

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Two particles, with charges of 20.0 nC and -20.0 nC, are located at the points with coordinates (0, 4.00 cm) and (0, -4.00 cm), as shown in the figure below. A third particle with charge 40.0 nC is located at the point (5.00 cm, 0).

Determine the **electrical potential **at the origin (0,0) due to the three fixed charges (40.0 nC, 20.0 nC, and -20.0 nC).

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Let: V = 0 at infinity. Three charges are arranged in the (x, y) plane (as shown in the figure below, where the scale is in meters). Find the electric potential Vo at the origin [coordinates (0 m, 0 m)].

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Which statement below, describing an electrostatic situation of a conductor, is **NOT correct**?

A) The electric field within a conductor is zero.

B) The electric potential within a conductor is equal everywhere.

C) The electric field outside of a conductor is always perpendicular to the surface.

D) Excess charges of a conductor only reside below the surface, not on the surface.

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Two particles, with charges of 20.0 nC and –20.0 nC, are located at the points with coordinates (0, 4.00 cm) and (0, – 4.00 cm), as shown in the figure below. A third particle with charge 10.0 nC is located at the origin (0,0). Determine the electric potential at the point (3.00 cm, 0) due to the three fixed charges (10.0 nC, 20.0 nC, and –20.0 nC).

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A sphere with radius 2.0 mm carries a 2.0 μC charge. What is the potential difference, *V _{B }- V_{A}*, between point B 4.0 m from the center of the sphere and point A 6.0 m from the center of the sphere? (The value of k is 9.0x10

A) 1500 V

B) -0.63 V

C) -1500 V

D) 170 V

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A 4.0-μC charge is situated at the origin of an xy-coordinate system. What is the potential difference between a point x = 4.0 m and y = -4.0 m because of this charge?

A) -18×10^{3} V

B) 18×10^{3} V

C) 0 V

D) 36×10^{3} V

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