Calculating the Potential from the Field — Question 2

Question 2

Signed areas under a piecewise field locate an interior voltage minimum

Along x, $E_x=200$ V/m for $0<x<.100$ m and $E_x=-100$ V/m for $.100<x<.300$ m. Set V(0)=0 and take V continuous at the interface.

1. Find the piecewise potential.

2. Find its minimum and V(.300 m), then determine the electric work on charge q over the full interval.

Question 2 - Solution

Signed areas under a piecewise field locate an interior voltage minimum

(a) Method and calculation

$$V(x)=-200x\quad(0\le x\le.100\,\mathrm m).$$$$V(x)=-20+100(x-.100)\quad(.100\le x\le.300\,\mathrm m),$$with x in meters and V in volts.

(b) Interpretation and further calculation

$$V_{\min}=V(.100\,\mathrm m)=-20.0\,\mathrm V,\quad V(.300\,\mathrm m)=0.$$$$W_e=q[V(0)-V(.300\,\mathrm m)]=0.$$

Physical check

A discontinuity of finite field changes the slope of V but produces no finite jump in V.