Equipotential Surfaces — Question 3

Question 3

A curved level surface determines a local normal without a full field map

In a plane, $V(x,y)=A(x^2+4y^2)$ with $A>0$. Use $\mathbf E=-\nabla V$. Consider P=(2a,a), where a is positive.

1. Find the equipotential through P and a tangent direction there.

2. Find the electric field at P and verify its orthogonality to your tangent.

Question 3 - Solution

A curved level surface determines a local normal without a full field map

(a) Method and calculation

$$V_P=8Aa^2,\quad x^2+4y^2=8a^2.$$$$2x\,dx+8y\,dy=0\Rightarrow dy/dx=-\tfrac12\quad\text{at P};\quad\mathbf t=2\hat{\mathbf x}-\hat{\mathbf y}.$$

(b) Interpretation and further calculation

$$\mathbf E_P=-4Aa\hat{\mathbf x}-8Aa\hat{\mathbf y}.$$$$\mathbf E_P\cdot\mathbf t=-8Aa+8Aa=0,\qquad|\mathbf E_P|=4\sqrt5Aa.$$

Physical check

Equipotential curves are two-dimensional cross-sections; in three dimensions the field is normal to the entire level surface.