- Example 1: Potential of a point charge
- Example 2: Potential of an electric dipole
- Example 3: Potential of a ring charge distribution
- Example 4: Potential of a disc charge distribution
4.2 Equipotential Surfaces
Now we will introduce an important concept associated with the electric potential which is called equal potential surfaces. These are the surfaces which are represented by the points that they have the same potential. The surfaces represented by the points which have the same potential. Now as you know from geometry, three points represent a surface. Assuming that we just measure the potential in space… By the way, the instrument, basically, which measures the potential difference between two points is called voltmeter.
Let’s assume we know the potential of every point in the region that we are interested with, and if we look at the surfaces represented by the points that they have the same potential, then we know all the equal potential surfaces in that region. Let’s assume that we have several equal potential surfaces. Something like this… That all the points on surfaces is at volt v1. All the points on the second surface is at volt v2. And then the other one as at volt v3 and so on and so forth. Knowing the potential in a region means that we know all the equal potential surfaces in that region.
When we look at some important properties of these surfaces, the first one is stated as no net work is done as the charge moves between two points on the same equal potential surface by the electric field. It means that you pick just… Take any charge, move it along a path over the surface of the same equal potential surface from some initial point to some final point. No work will be done by the electric field on that charge.
We can mke the proof of this property, this feature, easily. Since potential difference is equal to negative of the work done in moving the charge from some initial point to a final point divided by the charge, and since for the same equal potential surface, every point at the same potential v sub f will be equal to v sub i. In this case it is equal to v1, if we are talking about that first equal potential surface over here.
Therefore we will have v1 minus v1 is equal to minus work done per unit charge. Since the left-hand side of this equation is going to now add up to 0, then the work done in moving this charge along the same equal potential surface will be equal to 0.
So if you have rq, which is moved along this path on the same equal potential surface then the work done by the electric field becomes equal to 0. This is in a way an equivalent case. If we move an object with a mass m along a surface that is parallel to the ground, for example over the table surface, the work done by the gravitational force for that case is going to be equal to 0. Because if you visualize the situation…
If this is the table surface for example, and if you move any object of mass m, let’s say from this initial point to this final point, the displacement vector dl over here pointing along this path, and the gravitational force is going to be always pointing in this case in downward direction, mg. And the work done, or the incremental work is going to be equal to f dot dl. The total work done will be the integral or sum of the incremental works along this incremental path, which eventually makes the whole path from the initial point to final point.
That will be equal to integral of mg, magnitude of f times magnitude of dl times cosine of the angle between these two vectors. That angle in this case is 90 degrees. Cosine of 90… Since cosine of 90 is 0, then the total work done moving the particle along this tabletop by the gravitational field will add up to 0.
In the case of equal potential surfaces, of course they don’t have to be flat like the tabletop surface or flat to the ground. They just simply form from the points that every point on the surface has the same potential. For such a surface, when we move a charge along the surface, the electric field will always be equal to 0.
The second property is also in a way linked to this property. It simply states that the electric field is always perpendicular to the equal potential surfaces. We can prove this case also by looking or by considering the situation that this statement is not true. Assume e is not perpendicular to the equal potential surface. If this is the equal potential surface, and if the electric field is not perpendicular, therefore it is going to be making an angle with the surface theta, which is different than 90 degrees.
Then under this condition, we can always resolve the electric field vector into two components by taking its projection along the surface, which we will call that component as e parallel. The other one which is perpendicular to the surface, it is going to be represented by this symbol.
Now, if this is the case then if we place a charge on this equal potential surface, q0, a test charge, let’s say that’s positive, then it will be immediately be under the influence of a Coulomb force generated by this parallel component, which is going to be equal to q0 times e parallel. It will be in the same direction with the electric field.
This force, f, which is q times e parallel will accelerate the charge. In other words, it is going to do a work on this charge. A work will be done. But from property one, when a charge moves on the same equal potential surface work done by the electric field should be 0. Of course, this is possible if e parallel is 0. It implies, therefore, electric field should always be perpendicular to the equal potential surface. V is perpendicular to the surface. Equal potential surfaces are very important, and we should always remember these two conditions associated with the properties of equal potential surfaces.