Question

A shell of radius R has a charge Q spread uniformly on its surface. The potential at distance of R/2 from center is

• A. kQ/R
• B. 0
• C. 2KQ/R
• D. 3kQ/2R

Answer 1

Answer: (A)

kQ/R

Answer 2

The potential at a distance $r$ from the center of a uniformly charged spherical shell of radius $R$ and total charge $Q$ is given by:

$V(r) = \begin{cases} \frac{kQ}{R} & \text{for } r \le R \\ \frac{kQ}{r} & \text{for } r > R \end{cases}$

where $k = \frac{1}{4\pi\epsilon_0}$ is Coulomb's constant.

The question asks for the potential at a distance of $R/2$ from the center. Since $R/2 < R$, this point is inside the spherical shell. According to the formula, for points inside the shell ($r \le R$), the potential is constant and equal to the potential on the surface, which is $\frac{kQ}{R}$.

Therefore, the potential at a distance $R/2$ from the center is $V(R/2) = \frac{kQ}{R}$.

The electric field inside a uniformly charged spherical shell is zero. Since the electric field is the negative gradient of the potential ($E = -\nabla V$), a zero electric field implies that the potential is constant inside the shell. The value of this constant potential is equal to the potential on the surface of the shell, which is $kQ/R$. Thus, the potential at any point inside the shell, including at a distance $R/2$ from the center, is $kQ/R$.