Equipotential Surfaces — Question 9

Question 9

A sampled map constrains a directional derivative but not the full field

Near P a potential is locally linear: $V=V_P+ax+by$, with coordinates measured from P. Measurements along the x direction give a rise of 6.00 V over .0200 m. No y-direction measurements are available.

1. Find $E_x$ and the smallest possible field magnitude.

2. If a second measurement gives a drop of 8.00 V over .0200 m in the positive y direction, find the full field and an equipotential tangent.

Question 9 - Solution

A sampled map constrains a directional derivative but not the full field

(a) Method and calculation

$$a=6.00/.0200=300\,\mathrm{V/m},\quad E_x=-300\,\mathrm{V/m}.$$$$|\mathbf E|=\sqrt{300^2+b^2}\ge300\,\mathrm{V/m}.$$

(b) Interpretation and further calculation

$$b=-400\,\mathrm{V/m},\quad\mathbf E=(-300\hat{\mathbf x}+400\hat{\mathbf y})\,\mathrm{V/m},\quad|\mathbf E|=500\,\mathrm{V/m}.$$$$300\,dx-400\,dy=0\Rightarrow dy/dx=3/4.$$

Physical check

One directional voltage measurement determines one component; assuming it determines the magnitude silently assumes the transverse components vanish.