Calculating the Potential from the Field — Question 9

Question 9

A sinusoidal field has repeated equal-potential points

Along x, $E_x=E_0\cos(\pi x/L)$ with $E_0,L>0$. Set V(0)=0 and consider $0\le x\le2L$.

1. Find V(x) and its largest and smallest values.

2. Find the voltage from L/4 to 3L/4 and explain whether the field vanishes at these endpoints.

Question 9 - Solution

A sinusoidal field has repeated equal-potential points

(a) Method and calculation

$$V(x)=-\frac{E_0L}{\pi}\sin(\pi x/L).$$$$V_{\min}=-E_0L/\pi\ (x=L/2),\quad V_{\max}=+E_0L/\pi\ (x=3L/2).$$

(b) Interpretation and further calculation

$$V(3L/4)-V(L/4)=0.$$$$E_x(L/4)=E_0/\sqrt2,\quad E_x(3L/4)=-E_0/\sqrt2.$$

Physical check

Cancellation in a definite integral does not require a zero integrand at either endpoint.