Calculating the Potential from the Field — Question 3

Question 3

A linearly reversing field gives equal voltages at separated points

For $0\le x\le2L$, $E_x=E_0(1-x/L)$ with positive constants $E_0,L$. Set V(0)=0.

1. Integrate to find V(x) and its extremum.

2. Find all x in the interval with V=0 and the barrier energy for a negative charge $-q$ moving from 0 to 2L, where q is positive.

Question 3 - Solution

A linearly reversing field gives equal voltages at separated points

(a) Method and calculation

$$V(x)=-\int_0^x E_0(1-u/L)\,du=-E_0x+\frac{E_0x^2}{2L}.$$$$V\prime=0\Rightarrow x=L,\quad V_{\min}=-E_0L/2.$$

(b) Interpretation and further calculation

$$V=\frac{E_0}{2L}x(x-2L)=0\Rightarrow x=0,2L.$$$$U_{\max}-U(0)=(-q)(-E_0L/2)=qE_0L/2.$$

Physical check

The negative charge sees an energy barrier where the potential has a minimum.