Gauss' Law and Coulomb's Law — Question 10

Question 10

A centered charge is not generally normal to an ellipsoidal surface

A positive point charge q is at the center of the ellipsoid $x^2/a^2+y^2/b^2+z^2/c^2=1$. A local normal is parallel to $(x/a^2,y/b^2,z/c^2)$. At $P=(a/\sqrt2,b/\sqrt2,0)$ take $a=2b$.

1. Calculate the cosine of the angle between the radial Coulomb field and the outward normal at P.

2. Find the flux through the whole ellipsoid, and state why Gauss law does not justify replacing the field by one constant scalar over its surface.

Question 10 - Solution

A centered charge is not generally normal to an ellipsoidal surface

(a) Method and calculation

$$\hat{\mathbf r}=\frac{(2,1,0)}{\sqrt5},\quad\hat{\mathbf n}=\frac{(1,2,0)}{\sqrt5}.$$$$\cos\gamma=\hat{\mathbf r}\cdot\hat{\mathbf n}=\frac45.$$

(b) Interpretation and further calculation

$$\Phi_{\mathrm{ellipsoid}}=q/\epsilon_0.$$$$d\Phi=\frac{kq}{r^2}\cos\gamma\,dA.$$Distance r and projection factor vary across the surface. The enclosed-charge result is exact even though the usual spherical shortcut is unavailable.

Physical check

For a sphere with $a=b=c$, the normal becomes radial everywhere and the angle reduces to zero.