Question

A solid conducting sphere has net positive charge and radius R = 0.400 m. At a point 1.20 m from the center of the sphere, the electric potential due to the charge on the sphere is 24.0 V. Assume that V = 0 at an infinite distance from the sphere. What is the electric potential at the center of the sphere?

Short answer

The electric potential at the center of the sphere is 72.0V.

Step-by-step solution

Potential Energy

The equation gives the potential energy due to the sphere;

$V=\frac{1}{4\mathrm{\pi }{\pi }_0}\frac{q}{r}$

Here, r denotes the distance between the sphere and the point where the potential is measured; it can be inside or outside the sphere.

Electric potential at the center of the sphere

The potential is inversely proportional to the distance, as shown by the equation;

$V_{\infty }\frac{1}{r}$

The potential at the sphere's center is determined by the radius R, where the potential is constant throughout the sphere. Because the charge q is constant and the term $\frac{1}{4\mathrm{\pi }{€}_0}$is constant, a relationship between the states inside the sphere and the states outside the sphere can be established.

role="math" localid="1664260877266" $$\begin{gathered} \frac{V_1}{V_2}=\frac{r_2}{r_1} \\ \frac{V_{inside}}{V_{ouside}}=\frac{r}{R} \\ {V}_{inside}=\left(\frac{r}{R}\right)V_{ouside} \end{gathered}$$

Putting the values;

$V_{\mathrm{inside}}=\left(\frac{1.2\mathrm{m}}{0.40\mathrm{m}}\right)\left(24.0\mathrm{V}\right)$

Hence, the electric potential at the center of the sphere is 72.0V.