Potential Due to a Continuous Charge Distribution — Question 2

Question 2

An angular charge ripple is invisible to every axial potential measurement

A ring of radius R in the xy plane has line density $\lambda(\phi)=\lambda_0[1+\eta\cos\phi]$ with $\lambda_0>0$ and $|\eta|<1$. Use infinity reference.

1. Find its total charge and V at every point on the z axis.

2. Find the electric field at the center by direct vector integration and explain what axial potential data fail to determine.

Question 2 - Solution

An angular charge ripple is invisible to every axial potential measurement

(a) Method and calculation

$$Q=\int_0^{2\pi}\lambda(\phi)R\,d\phi=2\pi R\lambda_0.$$$$V(0,0,z)=\frac{2\pi kR\lambda_0}{\sqrt{R^2+z^2}},$$independent of $\eta$.

(b) Interpretation and further calculation

$$\mathbf E(0)=-\frac{k\lambda_0}{R}\int_0^{2\pi}(1+\eta\cos\phi)(\cos\phi\hat{\mathbf x}+\sin\phi\hat{\mathbf y})d\phi.$$$$\mathbf E(0)=-\frac{\pi k\lambda_0\eta}{R}\hat{\mathbf x}.$$

Physical check

Values of V on one axis determine its axial derivative but do not determine transverse derivatives off that axis.