The Electric Field Due to a Charged Disk — Question 9

Question 9

A density that vanishes at the disk center changes the near-axis limit

A disk of radius R has prescribed density $\sigma(\rho)=\sigma_0\rho^2/R^2$, with $\sigma_0>0$. Use ring integration and k for the Coulomb constant.

1. Derive its axial field for $z>0$ and total charge.

2. Find the leading behavior for $z\ll R$ and the distant point-charge limit.

Question 9 - Solution

A density that vanishes at the disk center changes the near-axis limit

(a) Method and calculation

$$E_z=\frac{2\pi k\sigma_0z}{R^2}\int_0^R\frac{\rho^3\,d\rho}{(z^2+\rho^2)^{3/2}}.$$$$E_z=\frac{2\pi k\sigma_0z}{R^2}\left[\frac{R^2+2z^2}{\sqrt{R^2+z^2}}-2z\right],\quad Q=\frac{\pi\sigma_0R^2}{2}.$$

(b) Interpretation and further calculation

$$z\ll R:\quad E_z\simeq2\pi k\sigma_0z/R\to0.$$$$z\gg R:\quad E_z\simeq kQ/z^2=\frac{\pi k\sigma_0R^2}{2z^2}.$$The local density is zero at the center, so the uniform-disk surface limit does not apply there.

Physical check

Total charge must be found by area integration; multiplying the edge density by the full area would overcount by a factor of two.