Answer to Question #98902 in Electricity and Magnetism for Nayab

Question

Question- Check which of the two expressions below for electrical potential is applicable for a charged region. Correspondingly find the charge density:

(i) 3x^2 + y^2 + 2z^2
(ii) x^2 - y^2 + 8z

Answer 1

The first equation, since it have x, y, and z squared, may represent a closed region of space (ellipsoid) and it is applicable for electrical potential. The second one represents an opened region of space, that is why it is not applicable for the electric potential.

If we write equation i as

$$3x^2+y^2+2z^2=1,$$

we can see that the semi-axes of that ellipse are

$$\sqrt{3},1,\sqrt{2}.$$

Hence, the volume is

$$V=\frac{4}{3}\pi \sqrt{3}\cdot1\cdot\sqrt{2}=\frac{4\pi\sqrt{6}}{3}.$$

The charge density:

$$\rho=Q/V=\frac{3Q}{4\pi\sqrt{6}}.$$