Potential Due to a Continuous Charge Distribution — Question 8

Question 8

A graded solid sphere has a center voltage sensitive to inner charge weighting

A sphere of radius R has volume density $\rho(r)=\rho_0r/R$ with $\rho_0>0$. Use zero potential at infinity.

1. Find total charge Q and center potential by summing thin spherical shells.

2. Compare V(0) with its surface value and with a uniform solid sphere having the same Q and R; the uniform sphere has center potential $3kQ/(2R)$.

Question 8 - Solution

A graded solid sphere has a center voltage sensitive to inner charge weighting

(a) Method and calculation

$$Q=4\pi\int_0^R\rho_0\frac rR r^2dr=\pi\rho_0R^3.$$$$V(0)=k\int_0^R\frac{4\pi r^2\rho(r)dr}{r}=\frac{4\pi k\rho_0R^2}{3}=\frac{4kQ}{3R}.$$

(b) Interpretation and further calculation

$$V(R)=kQ/R,\quad V(0)/V(R)=4/3.$$$$\frac{V_{\mathrm{graded}}(0)}{V_{\mathrm{uniform}}(0)}=\frac{4/3}{3/2}=\frac89.$$

Physical check

Moving a larger fraction of fixed total charge outward lowers the center potential because each shell is weighted by inverse radius.