4.6 Insulated Conductor
Let’s consider an insulated conductor. Earlier we have studied that whenever we place an excess amount of charge inside of a conducting medium, it immediately moves to the surface due to the Coulomb repulsive force and we end up with no net charge enclosed inside of the conducting medium. So all the excess charge that we place inside of a conductor immediately moves, under the influence of this repulsive Coulomb force, to the surface of the conducting medium and it redistributes itself along the surface. Therefore, as we remember or recall, q enclosed was always equal to 0.
And from Gauss’s Law, which is e dot da integrated across a closed surface is equal to q enclosed over Epsilon 0. Since in the interior region any Gaussian surface that we choose will not enclose any net charge, then the electric field inside of a conducting medium we conclude that that will always be 0. Since q enclosed is 0 inside of a conducting medium, then as an important property we concluded that electric field inside of a conducting medium, or inside of a conductor, is equal to 0.
Well if we recall the potential difference expression v sub f minus v sub i, which is equal to minus integral of E dot dl integrated along a path l from some initial point to final point. And if we’re inside of a conducting medium, inside of a conductor, now we know that because of the lack of q enclosed electric field is 0, so this integral is going to be equal to 0 for a conductor or let’s say a region inside of a conductor is 0. Then v sub f minus v sub i is going to be equal to 0 and from here we can conclude that v sub f will be equal to v sub i.
So this tells us then, inside of the conducting medium the potential at every point will be the same. They are going to be equal to one another, and the potential difference between any two points inside of a conducting medium is going to be equal to 0. So as a second property then we can say potential of all points in or on a conductor are at the same potential. In other words, the potential difference in any two points will always be equal to 0 for the inside or on the surface of a conductor.
And this also leads us to the third property. We can say that surfaces of charged conductors represent equal potential surface. Therefore when we charge a conductor, all the charge is going to be uniformly distributed throughout the surface of the conductor, and furthermore the potential at the surface will be equal to the potential inside of the conductor. They will all have the same value. And the potential difference between any two points in or on the surface of the conductor will be equal to 0.
And also as you recall, earlier we stated that the electric field at the surface of a conductor is always perpendicular or normal to the surface. So in this sense if we have an electric field in the medium, for example, that it is pointing from left to right something like this, an external electric field and if we place a conducting object inside of this electric field, the electric field lines will bend and take positions such that they will be perpendicular to the surface, something like this, at every point.
And as we know, a conducting medium, we assume that this is a piece of metal, has abundance of free electrons and these free electrons rearrange themselves along the surface of this conducting medium such that the net electric field inside of the conducting medium will become 0.
Now if we consider this spherical charge example, let’s say spherical solid conducting charge. Assume that we have a solid copper ball with radius r and so we charge this ball positively and all the charge will be collected at the surface and for such a ball the inside will be equal to 0 since q enclosed will be 0. And as you recall, spherical charge distribution behaves like a point charge for all the exterior regions and e outside will be equal to, if the net charge is q on this ball, will be equal to q over 4 pi Epsilon r squared, and it’s going to be in a radially outward direction.
So in vector notation, we multiply this by the end vector r, and the electric field is going to be pointing radially out, originating from the source, going to the infinity.
On the other hand, if we look at the potential of this insulated charge conducting sphere, v is going to be equal to, the net charge is q, and we know that from the electric field that it is behaving like a point charge for all exterior regions, therefore v out at a specific point is going to be equal to q over 4 pi Epsilon 0 r. At the surface it’s going to be equal to q over 4 pi Epsilon 0 times big R.
Now inside of this charged conducting sphere, every point has the same potential and the value of that potential is equal to the value at the surface, and that will be q over 4 pi Epsilon 0 r. And as you see this of course constant and inside also potential will be constant. Electric field inside of this charged metallic sphere is 0, but the potential is equal to a constant quantity and its value is equal to at the surface.
If we plot these quantities first for the electric field and also for the potential electric field as a function of radial distance r, inside, this is the radius distance, is 0 because q enclosed is 0 and then at the surface it will have the value of q over 4 pi Epsilon 0 big R square and then it will decrease and go to 0 as proportional to 1 over r square.
In a similar way have a look at the behavior of potential of this charged sphere, potential is a function of radial distance, inside of the sphere potential is constant and it is equal to the value at the sphere at the surface and that is equal to q over 4 Pi Epsilon 0 big R, and once we go out of the sphere then the potential will decrease and it will go to 0 as proportional to 1 over r.