[ JEE ADVANCED 2024 ] DIELECTRIC SHELL AND CONDUCTING SPHERE [INSP ORIGINALS ]

Step-by-Step Tutorial: Solving the Dielectric Shell and Conducting Sphere Problem

Problem Statement:

1. The problem involves a conducting sphere with inner radius R and outer radius 2R.
2. Outside the sphere, from 2R to 3R, there is a dielectric with a dielectric constant of 2.
3. A point charge is placed inside the shell.
4. The goal is to calculate the work done by an agent in two scenarios:
   • W1: Removing the dielectric coat from the system.
   • W2: Moving the point charge to a faraway point (Infinity) after removing the dielectric coat.

Method 1: Using Capacitor Logic

1. Calculate the energy U1 stored in the system with the dielectric coat using the formula: (U1 = \frac{Q^2}{48 \pi \epsilon_0 R}).
2. Calculate the energy U2 stored in the system with air instead of the dielectric using the formula: (U2 = \frac{Q^2}{96 \pi \epsilon_0 R}).
3. Find the work done W1 by subtracting U1 from U2: (W1 = U2 - U1).

Method 2: Using Energy Density

1. Calculate the energy density formula for air and dielectric: (U = \frac{1}{2} \epsilon E^2) and (U = \frac{1}{2} \epsilon_0 K E^2).
2. Integrate the energy density from 2R to 3R to find U1 and U2 for the dielectric and air regions.
3. Calculate U1 and U2 values and find W1 by subtracting U1 from U2.

Calculating W2:

1. For W2, consider the self-energy of the charge Q at the center.
2. Calculate the energy of the system with the charge at the center and without induction on the inner and outer surfaces of the conductor.
3. Find W2 by subtracting the initial energy from the final energy.

Final Steps:

1. Simplify the expressions to find the final values of W1 and W2.
2. Calculate the ratio of W1 to W2, which is the answer to the problem.
3. Verify the calculations using both methods to ensure accuracy.
4. Understand the concept of self-energy and mutual potential energy in the system.

By following these steps and understanding the concepts of energy storage in capacitors and energy density in dielectric media, you can successfully solve the given problem related to the dielectric shell and conducting sphere scenario. Good luck with your problem-solving and preparation for JEE Advanced 2024!