Question

Consider a uniformly volume‑charged sphere of radius R and charge Q . What is the electric potential on the surface of the sphere in terms of R , Q , and ϵ 0 , choosing the zero reference point for the potential at the center of the sphere?

Answer 1

To solve this problem we will start by applying the given load ratio, and we will rely on the two types of distances given. Later we will use Gauss's law and through its integrals, in which it is equivalent to the potential we will obtain its value in the center of the sphere. Since it is uniformly charged we have to,

$\frac{Q'}{Q} = \frac{4\pi r^3}{4\pi R^3}$

$Q' = \frac{r^3}{R^3} Q$

By Gauss Law

$\phi = E \cdot 4\pi r^2$

Here, E is the electric Field and is equal to

$E = \frac{Q'}{\epsilon_0}$

For $\epsilon_0$ being the Permeability constant at free space

Replacing with the previous value we have,

$\phi = \frac{Qr^3}{\epsilon_0 R^3}$

Then the value of the electric field is,

$E = \frac{QR}{4\pi \epsilon_0 R^3}$

Now potential

$V = \int_0^R E\cdot dr'$

$V = - \frac{QR^2}{8\pi \epsilon_0 R^3}$

$V = -\frac{Q}{8 \pi \epsilon_0 R}$