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.
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
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
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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For that, we need to think about the fact that electric field is the rate of change of potential. We know from Gauss's Law that the electric field inside a conducting sphere is zero. This is because the charge on a sphere spreads out on the surface. This means that once you enter the sphere, the charge enclosed by a radius, r immediately becomes zero.
If the electric field is zero all the way through the inside of the sphere, then that means that the rate of change of potential is zero. Or in other words, the potential inside the sphere doesn't change. So, the graph must be flat, like this:
Electric field of zero
From the surface, all the way to the center, the electric potential stays constant.
Electric potential is the amount of electric potential energy that each unit charge would have at a particular point in space. Two points in space have different electric potentials due to their different positions inside a field.
From Gauss's Law, we know that the equation for the electric field of a conducting, charged sphere is the same as the equation for the electric field of a point charge. Electric field is also the change in potential. So, it follows that the equation for the potential of a charged sphere is also the same as the equation for the potential of a point charge. It looks like the equation in the section above, 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. We can plot this equation with respect to the radius from the center of the sphere, creating a graph that looks like the one in the section above.
Gauss's Law also tells us that the electric field inside a conducting sphere is zero. Since electric field is the rate of change of potential, this means that the potential must not change inside the sphere. So, for the inside of the sphere, the graph remains flat, as also shown above.
Learn from this lesson as you prepare to:
Provide the definition of electric potential
Illustrate the equations for potential derived from Gauss's Law
Understand the effect of an electric field at zero inside a conducting sphere
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