Applying Gauss' Law: Cylindrical Symmetry — Question 4

Question 4

A radial density increasing outward weakens the interior field at fixed line charge

An infinite insulating cylinder of radius R has density $\rho(r)=\rho_0r/R$ for $r<R$, with $\rho_0>0$. Compare it with a uniformly charged cylinder of the same radius and same total charge per length.

1. Derive the enclosed line charge, interior field and total line charge of the graded cylinder.

2. Find the uniform comparison density and the ratio of the two interior fields at the same nonzero r.

Question 4 - Solution

A radial density increasing outward weakens the interior field at fixed line charge

(a) Method and calculation

$$\lambda_{\mathrm{enc}}(r)=\int_0^r\rho_0\frac{s}{R}\,2\pi s\,ds=\frac{2\pi\rho_0r^3}{3R}.$$$$E_{\mathrm{graded}}(r)=\frac{\rho_0r^2}{3\epsilon_0R},\quad\lambda_{\mathrm{total}}=\frac{2\pi\rho_0R^2}{3}.$$

(b) Interpretation and further calculation

$$\rho_{\mathrm{uniform}}=\lambda_{\mathrm{total}}/(\pi R^2)=2\rho_0/3,\quad E_{\mathrm{uniform}}(r)=\frac{\rho_0r}{3\epsilon_0}.$$$$\frac{E_{\mathrm{graded}}}{E_{\mathrm{uniform}}}=\frac rR<1\quad(0<r<R).$$Moving more charge toward the outer region reduces the charge enclosed at an interior observation radius.

Physical check

Both distributions give the same exterior field because their total line charge is equal.