The Electric Field Due to a Line of Charge — Question 9

Question 9

A nonuniform ring has a center field controlled by its first angular harmonic

A circular ring of radius R has density $\lambda(\phi)=\lambda_0(1+\beta\cos\phi)$, with $\lambda_0>0$ and $|\beta|\le1$. Angle is measured from $+x$.

1. Find its total charge and center-field vector.

2. Find its dipole moment about the center and explain how the same total charge can coexist with different center fields.

Question 9 - Solution

A nonuniform ring has a center field controlled by its first angular harmonic

(a) Method and calculation

$$Q=R\int_0^{2\pi}\lambda(\phi)\,d\phi=2\pi R\lambda_0.$$$$E_x=-\frac{k\lambda_0}{R}\int_0^{2\pi}(1+\beta\cos\phi)\cos\phi\,d\phi=-\frac{\pi k\lambda_0\beta}{R},\quad E_y=0.$$

(b) Interpretation and further calculation

$$p_x=R^2\int_0^{2\pi}\lambda(\phi)\cos\phi\,d\phi=\pi\lambda_0\beta R^2,\quad p_y=0.$$$$\mathbf E(0)=-k\mathbf p/R^3.$$Varying $\beta$ redistributes charge without changing Q; the stronger positive-charge side pushes the center field toward the opposite side.

Physical check

The bound on beta keeps the density nonnegative everywhere, including the least populated side.