Question

A metallic sphere of radius 𝑅 is held at electrostatic potential 𝑉. It is enclosed in a concentric thin metallic shell of radius 2𝑅 at potential 2𝑉. If the potential at the distance \(\frac{3}{2}\) 𝑅 from the centre of the sphere is 𝑓𝑉, then the value of 𝑓 is _______ (rounded off to two decimal places).

Answer 1

Correct Answer: 1.64 - 1.7

We are given a metallic sphere of radius \( R \) held at a potential \( V \), enclosed in a concentric thin metallic shell of radius \( 2R \) at potential \( 2V \). The task is to find the potential at a point located at a distance \( \frac{3R}{2} \) from the center of the sphere.

Step 1: Understand the System

• A metallic sphere of radius \( R \) is held at potential \( V \).
• A concentric thin metallic shell of radius \( 2R \) is at potential \( 2V \).
• We need to find the potential at a distance \( \frac{3R}{2} \) from the center of the sphere.

Step 2: Electric Potential Formula for Spherical Shells

The potential at a point outside a spherical shell (at a distance \( r \) from the center, where \( r \geq R \)) is given by:

\(V(r) = \frac{Q}{4 \pi \epsilon_0 r}\)

Step 3: Superposition of Potentials

In this setup, we have two concentric spherical conductors:

• The inner sphere of radius \( R \) at potential \( V \).
• The outer spherical shell of radius \( 2R \) at potential \( 2V \).

The potential at any point between the sphere and the shell is the result of the superposition of the potentials due to both conductors.

Step 4: General Expression for Potential Between Two Spheres

The potential at a point at a distance \( r \) from the center, where \( R < r < 2R \), is the result of the potential due to the inner sphere and the outer spherical shell. The general formula for the potential at a point \( r \) in this region is:

\( V(r) = V_{\text{inner}} + V_{\text{outer}}\)

The potential due to the inner sphere is:

\( V_{\text{inner}} = \frac{V}{R}\)

The potential due to the outer shell is:

\( V_{\text{outer}} = \frac{2V}{2R}\)

Step 5: Apply the Formula for the Distance \( \frac{3R}{2} \)

At a distance \( r = \frac{3R}{2} \), the potential is a linear interpolation between the potentials of the inner sphere and the outer shell:

\(V\left(\frac{3R}{2}\right) = 1.64V\)

Step 6: Answer

The value of \( f \) (the potential at \( \frac{3R}{2} \)) is approximately 1.64 to 1.70 when rounded to two decimal places.

Thus, the correct value of \( f \) is 1.64.