## Exploration 26.5: Capacitance of Concentric Cylinders

Wait for the calculation to finish. This animation shows a coaxial capacitor with cylindrical geometry: a very long cylinder (extending into and out of the page) in the center surrounded by a very long cylindrical shell (position is given in centimeters, electric field strength is given in newtons/coulomb, and electric potential is given in volts). The outside shell is grounded, while the inside shell is at 10 V. You can click-drag to measure the voltage at any position. Restart.

1. Use Gauss's law to show that the magnitude of the radial electric field between the two conductors for a cylindrical coaxial capacitor of length L is E = Q/2πrLε0 = 2kQ/(rL), where Q is the total charge on the inside (or outside) conductor and r is the distance from the center.
2. If L = 1 m, measure the electric field in the region between the two conductors and determine the charge on the inside (and outside) conductor.
3. Use V = -∫ E · dr to show that the potential at any point between the two conductors is V = (Q/2πLε0) ln(b/r)  = (2kQ/L) ln(b/r), where b is the radius of the outer conductor.
4. Given that the potential difference between the two cylinders is 10 V, verify your answer to (b) and find the charge on each conductor.
5. Given, then, that the potential difference between the two conductors is V = (Q/2πLε0) ln(b/a) = (2Qk/L) ln(b/a)–b is the radius of the outer shell and a is the radius of the inner cylinder—show that the capacitance of this capacitor is (2πLε0)/ln(b/a) = (L/2k)/ln(b/a).
6. What is the capacitance (numerical value) of this capacitor?

Exploration authored by Anne J. Cox.
Script authored by Mario Belloni, Wolfgang Christian and Anne J. Cox.

Physlets were developed at Davidson College and converted from Java to JavaScript using the SwingJS system developed at St. Olaf College.