Exploration 25.5: Spherical Conductor and Insulator

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How does the electric potential around a charged solid insulating sphere (with charge distributed throughout the volume of the sphere) compare with the electric potential around a charged conducting sphere?  Move the test charge to map out the electric potential as a function of distance from the center (position is given in centimeters and electric potential is given in volts).  Restart.

1. Why is the voltage constant inside the conductor? 
2. Why is there no electric field and no force on the test charge inside the conductor? 
3. Looking at the plots you make of voltage as a function of radial distance (as you move the test charge), what is the same and what is different between the two cases?  Given that both spheres have the same total charge, explain the similarities and differences in the plots.
4. The electric field outside both spheres is Q/4πε₀r².  Using this and the reference point of V = 0 volts at infinity, find an expression for the electric potential at a point outside the sphere and a distance r from the center of the sphere.  V = - ∫ E • dr and integrate from r = infinity (where V = 0 volts) to a point r. 
5. Measure the voltage at some point outside the sphere and find the charge on both spheres.  Verify that the total charge is the same.

Now for the voltage inside the uniformly charged insulator.  Here the electric field is Qr/(4πε₀R³), where R is the radius of the sphere itself.  In this case, to find the potential as a function of r, you again need to integrate V = - ∫ E • dr, but this time you must break up the integral and integrate from infinity to R using E = Q/4πε₀r² (to find the electric potential associated with getting all the charges to the surface of the sphere) and then integrate from R to r (an arbitrary point inside the sphere) using the expression for the electric field inside the insulating sphere. 

6. Verify that your calculation gives the same results as shown on the graph.

Exploration authored by Anne J. Cox.
Script authored by Mario Belloni and Anne J. Cox.
© 2004 by Prentice-Hall, Inc. A Pearson Company