Calculating the Potential from the Field — Question 8

Question 8

An infinite charged sheet gives a continuous cusp in potential

An infinite sheet at x=0 produces $E_x=E_s\operatorname{sgn}(x)$ with $E_s>0$. Set V(0)=0; there is no ideal dipole layer.

1. Find V(x) on both sides and check continuity.

2. Find the voltage between x=-a and x=2a and the work on positive charge q between these points.

Question 8 - Solution

An infinite charged sheet gives a continuous cusp in potential

(a) Method and calculation

$$x>0:\ V=-E_sx;\qquad x<0:\ V=+E_sx.$$$$V(x)=-E_s|x|,\quad V(0^-)=V(0^+)=0.$$

(b) Interpretation and further calculation

$$V(2a)-V(-a)=-2E_sa+E_sa=-E_sa.$$$$W_e=-q\Delta V=qE_sa.$$

Physical check

The cusp means the normal field jumps; it does not mean the potential jumps.