The Potential of a Sphere

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  • 0:01 What Is Potential?
  • 0:58 Gauss' Law
  • 2:03 Potential as a Graph
  • 3:04 Lesson Summary
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Lesson Transcript
Instructor: David Wood

David has taught Honors Physics, AP Physics, IB Physics and general science courses. He has a Masters in Education, and a Bachelors in Physics.

After watching this lesson, you will be able to explain what electric potential is, state and use the equation for the potential of a sphere, and plot a graph of electric potential with radius. A short quiz will follow.

What Is Potential?

Electric potential is the amount of electric potential energy that each unit charge would have at a particular point in space. It is measured in Joules per Coulomb or volts. Two points in space have different electric potentials due to their positions inside a field.

Since gravity is more in our everyday experiences, sometimes it's easier to compare this to the situation for gravity. The gravitational potential is the amount of gravitational potential energy that each unit mass would have at a particular point in space. If you raise a ball to a height, h, that point in space has a larger gravitational potential than it did when it was at ground level, and if you release a point mass at that height, it will start to move because it has gravitational potential energy. If you drew field lines, they would point from the raised ball towards the ground. So, as you follow field lines, the amount of potential changes - or to be exact, it decreases. This is the same for electric potential.

Gauss's Law

In another lesson, we discussed Gauss's Law and how it can be used to derive an equation for the electric field around a uniform object, like a conducting sphere. When you do that, you get this equation, where Q is the charge on the sphere, epsilon-zero is a constant that is always equal to 8.85 * 10^-12 and r is the distance you are from the center of the sphere:

Electric field around a uniform object

This is the same as the equation for the electric field created by a point charge, so in other words, the field created by a conducting sphere is the same as that of a point charge.

Electric field is defined as the force that a +1 coulomb test charge would feel at a particular location. But electric field is also the rate of change of potential. So, if the electric field of a sphere is the same as a point charge, it follows that the potential will also be the same as a point charge.

The equation for the potential of a point charge looks like this:

Potential of a point charge
equation for potential of point charge

Exactly the same as the electric field equation but with a single radius r instead of r-squared. To use the equation, all you have to do is plug in some numbers and solve.

Potential as a Graph

If we take that equation for potential and plot it as a graph, we find that the electric potential outside of the surface of the sphere looks like this:

Graph of equation for potential
graph of equation for potential

It starts off at some maximum value at the surface and then decreases quickly as you move further away. But what about inside the sphere? What about this region of the graph?

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