Question

Surface density of charge on a charged conducting sphere of radius \( R \) in terms of electric intensity \( E \) at a distance \( r \) in free space is
\textit{(where \( r>R, \, \epsilon_0 \) is permittivity of free space)}

• A. \( \epsilon_0 E \frac{R}{r} \)
• B. \( \epsilon_0 E \left( \frac{r}{R} \right)^2 \)
• C. \( \epsilon_0 E \frac{r}{R} \)
• D. \( \epsilon_0 E \left( \frac{R}{r} \right)^2 \)

Hint:

When dealing with spherical charge distributions, remember the relation between electric field and charge density at points outside the sphere.

Answer 1

The Correct Option is D

Step 1: Formula for surface charge density.
The surface charge density \( \sigma \) on a conducting sphere is related to the electric field at a distance \( r \) from the center of the sphere. The electric field \( E \) at a distance \( r \) from the center of a charged sphere is given by: \[ E = \frac{1}{4 \pi \epsilon_0} \frac{Q}{r^2} \] where \( Q \) is the total charge on the sphere. The surface charge density is related to the electric field by: \[ \sigma = \epsilon_0 E \left( \frac{R}{r} \right)^2 \]
Step 2: Conclusion.
Thus, the correct answer is (D) \( \epsilon_0 E \left( \frac{R}{r} \right)^2 \).