About this video
This video explores a charged conductor in electrostatic equilibrium. It shows that excess charge always sits on the surface, that the electric field inside the conductor must be zero (otherwise charges would still be moving), and that the field at the surface must be perpendicular to it. For a spherical conductor, the outside field looks exactly like a point charge at the center, while the inside field is zero; none of these results hold for insulators.
You'll be able to
- state that excess charge on a conductor resides on its surface.
- explain why the electric field inside a conductor in equilibrium is zero.
- state that the surface field of a conductor is perpendicular to the surface.
- describe the field of a charged spherical conductor inside and outside.
Key words
- Electrostatic equilibrium
- The state in which charges in a conductor have stopped moving.
- Conductor
- A material in which charges move freely, so excess charge rearranges until equilibrium is reached.
- Surface charge
- Excess charge on a conductor, which always settles on the outer surface.
- Perpendicular surface field
- At a conductor's surface in equilibrium the field points straight out or in, with no component along the surface.
Transcript
- [Instructor] Here's a neutral conductor. Remember, neutral means there are equal amounts of positive and negative charges inside of it, okay? Now, if I take such a neutral conductor and add some excess charge, for example, I add some extra electrons to it, so it gets some excess negative charge.
And let's say we add that excess electrons inside the material somewhere, okay? Then what's going to happen? Well, all these negative electrons are going to repel each other, so they would want to move away from each other, they'll start moving away from each other.
But what will happen eventually when all the charges have stopped moving? In other words, what happens when we reach electrostatic equilibrium? Electrostatic equilibrium just means that charges are not moving. So what does the distribution look like eventually when they have stopped moving?
And what does the electric field look like everywhere? Inside the conductor, outside the conductor? That's what we want to explore in this video. So let's begin. Okay, so these excess electrons are all mutually repelling each other. So they would tend to move as far away, move and settle as far away from each other as possible, right?
Well, that only happens when they have all reached the surface of the conductor, which means the excess charge on conductor always resides on the surface. Now, this is true even for non-spherical conductors, but for spherical conductors, the charge distribution would be uniform because of the symmetry, right?
Because of the spherical symmetry. Of course, if it's non-spherical, then the charge distribution will not be uniform. There will be more charges crowded whenever you have sharp corners. We don't have to worry about the details as to why and all of that happens.
But what's important is that for any conductor, the excess charge will always reside on the surface. Would the same thing happen for an insulator? Why don't you pause the video and think about this? Well, in insulator, remember charges cannot move easily. So if I add some excess charge in the bulk of an insulator, even though the charges are mutually repelling each other, since they cannot move that easily, they need not move, they will not reside on the surface.
So this is not true for an insulator. This is only true for a conductor. Okay, now comes the second question. What does the electric field look like everywhere? Let's start with electric field inside. What does the electric field inside look like? Let's look at one specific spot, let's say over here.
What is the electric field over here, the magnitude and the direction? How do we figure that out? Well, we know that each point charge produces a radial field. If it's a positive charge, it'll be radially outwards. If it's a negative charge, it would be radially inwards, right?
So the contribution from this one specific charge over here, point charge over here would be this way. The electric field would be in this direction. And we can now do the same thing due to all the other charges, we can calculate what the electric field looks like from all the other charges and then use the principle of superposition to add them up.
But that looks very tedious. I mean, the distances are all different and I don't even know how we could go about doing that, right? The good news is we don't have to do any of that. Why not? Think about it. If there was an electric field inside, then the charges, remember there are positive and negative charges, and inside any conductor, electrons can easily move.
Then the electrons would be affected by that field, that field would put a force on the electrons and the electrons would start moving. But we know that we are an electrostatic equilibrium and charges are not moving, which means there cannot be any electric field inside.
If there existed an electric field inside, then the charges inside, the electrons inside will move in such a way that eventually the electric field inside will become zero. And that should be the case everywhere inside. That's amazing, right? The fact that this is an electrostatic equilibrium demands that the electric field inside the conductor must always be zero as long as we're in electrostatic equilibrium.
And this is not just true for spherical conductors, this is true for conductors of any shape, conductors in general. As long as you're in electrostatic equilibrium, it doesn't matter what charge configuration is there, it doesn't matter what situation we are in, the fact that we're in electrostatic equilibrium means that the electric field inside the conductor must be zero.
The logic is very simple. If the electric field was not zero, then the charges inside would be moving and we wouldn't be in electronic equilibrium. That's amazing, right? I like that line of reasoning. Again, would this be the case for insulators? Why don't you pause and think about it?
The answer is no, because chargers cannot move easily inside insulators. So even if there was an electric field inside an insulator, it's possible for charges to be still at rest. So electronic equilibrium doesn't demand that electric field inside an insulator must be zero.
So this is again, a special case for conductors only. But now comes the next question, what does the electric field look like on the surface? Well, turns out that on the surface, the electric field should always be perpendicular to the surface. Remember, this is a negative charge and so the electric field would be inwards.
If this was a positive charge, it would be outwards. But what's important is that the field on the surface must be perpendicular to it. Why? Why can't it have any other angle like this or like this, for example? Why should it be perpendicular? Well, let's see.
If the electric field was at some angle, for example, then we could resolve that electric field into two components, the perpendicular and the tangential component. Now, the perpendicular component of the electric field will produce a push on this. It'll push the charge outwards.
Remember, this is a negative charge, so it experiences a force in the opposite direction of the electric field, right? Now, the charges cannot move outside the conductor, so this push is not going to make the charge move. But what about the tangential component of the electric field?
It's going to push the charge along the direction, you know, along the surface of the conductor, and the charge can move along the surface of the conductor, which means if the electric field existed at some angle, then the tangential electric field would push the charge and make it move, which means we wouldn't be in the electrostatic equilibrium.
So the fact that we are an electronic equilibrium means that this push shouldn't exist, right? So in electrostatic equilibrium, there should be no tangential force acting on the charges, which means there should be no tangential component of the electric field, which means the electric field should be perpendicular to the surface, and that should be the case everywhere.
Everywhere locally, the electric field on the surface must be perpendicular to it. And again, this is not specific to spherical conductors. Again, this is true for any conductor in general. Even if you had some weird-shaped conductor, locally the electric field should be perpendicular to it.
If it's a negative charge, then the electric field will be inwards. If it's a positive charge, it would be outwards. Okay, so again, this is a general result for conductors. Electric field on the surface is always perpendicular to the surface. But if the conductor is spherical, notice something interesting happens.
Any line that you draw perpendicular to the surface will pass through the center. In other words, these electric field lines end up becoming radial. So for a negatively charged spherical conductor, the electric field is radially inwards, and if it's a positively charged, it would be radially outwards.
And if there's no charge anywhere else in the surrounding, then it will just keep extending, right? The field lines would be radio everywhere. But hey, we know that radial electric fields are produced by point charges. So look at what this means. Suppose the total charge on this conductor was, say, -Q, then the electric field outside over here is identical to the field produced by a point charge at the center.
So this means the electric field due to a charged spherical conductor outside would be exactly the same as the electric field produced by a point charge of an equal magnitude kept at the center. And we know the expression for it. The electric field to a point charge at some distance r is given by, the magnitude of it is given by 1/4pi epsilon times the magnitude of the charge divided by r squared, the inverse square law, right?
And the direction is, well, if it's negative, it's radially inwards. If it's positive, it's radially outwards. And since the electric field to a spherical conductor is also radial, the exact same expression holds. But of course, it only holds as long as we are outside.
In other words, smaller is greater than the radius of the conductor. If you are inside, then the electric field is zero. And this is special condition for spherical conductors. If it's non-spherical, then the distribution will be different and then all bets are off.
Again, the final question is, would this result be true for insulators? The answer is no. For insulators, charges cannot move easily, so the electric field on the surface can be at any angle to the surface. So these are some criteria for conductors in electrostatic equilibrium.
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