Potential Due to a Continuous Charge Distribution — Question 10

Question 10

A radially graded disk changes potential even when its total charge is fixed

A disk of radius R has $\sigma(s)=Cs$ with C>0. Compare it with a uniform disk of the same total charge Q and radius R. Use infinity reference.

1. Normalize C in terms of Q and find the graded disk center potential.

2. Find the ratio to the uniform disk center potential and explain the direction of the change through the radial weighting.

Question 10 - Solution

A radially graded disk changes potential even when its total charge is fixed

(a) Method and calculation

$$Q=2\pi C\int_0^R s^2ds=\frac{2\pi CR^3}{3}\Rightarrow C=\frac{3Q}{2\pi R^3}.$$$$V_{\mathrm{graded}}(0)=k\int_0^R\frac{2\pi Cs^2ds}{s}=\pi kCR^2=\frac{3kQ}{2R}.$$

(b) Interpretation and further calculation

$$V_{\mathrm{uniform}}(0)=2\pi k\frac{Q}{\pi R^2}R=\frac{2kQ}{R}.$$$$V_{\mathrm{graded}}(0)/V_{\mathrm{uniform}}(0)=3/4.$$

Physical check

The graded disk weights outer radii more strongly; at the center those charges contribute less voltage per unit charge.