Answer :
Answer:
To solve this problem, we will use the principles of electrostatics, particularly focusing on charge distribution and grounding.
**Part 1: Charge Flowed to Earth When Shell is Grounded**
1. Given:
- Radius of the metal shell, \( R = 5 \) cm = \( 0.05 \) meters
- Initial charge on the shell, \( Q = 10 \) mC = \( 10 \times 10^{-3} \) C
2. When the metal shell is grounded, it means it is connected to the Earth, which acts as a large reservoir of charge at zero potential. Grounding a charged conductor effectively neutralizes its charge by allowing charges to flow between the conductor and the Earth until they reach the same potential (zero potential).
3. Charge flow to the Earth:
- When the shell is grounded, excess charge \( Q \) on the shell will flow to the Earth until the shell's potential becomes zero.
- Therefore, the amount of charge that flows into the Earth is equal to the initial charge \( Q \) on the shell.
- So, the charge that flows into the Earth = \( Q = 10 \) mC = \( 10 \times 10^{-3} \) C
**Part 2: Charge Flowed to Shell When a Point Charge is Introduced**
1. Given:
- Additional point charge placed at the center of the shell, \( q = 30 \) mC = \( 30 \times 10^{-3} \) C
2. Now, the shell is still grounded after placing the point charge at its center.
3. Charge flow through the wire to the shell:
- The grounding wire provides a path for charge to flow between the Earth and the shell.
- The system (shell and point charge) will redistribute charges to reach a new equilibrium state.
- Due to the grounding, the shell remains at zero potential, and any excess charge on the shell is neutralized by the flow of charges through the wire.
- The net charge on the shell after grounding will be the sum of the initial charge on the shell and the point charge.
- New charge on the shell = Initial charge on the shell + Charge introduced by the point charge
- New charge on the shell = \( Q + q = 10 \times 10^{-3} \) C + \( 30 \times 10^{-3} \) C = \( 40 \times 10^{-3} \) C = \( 40 \) mC
4. Amount of charge that flows through the wire:
- The amount of charge that flows through the wire to the shell is the difference between the new charge on the shell and the initial charge on the shell.
- Charge flowed through the wire = New charge on the shell - Initial charge on the shell
- Charge flowed through the wire = \( 40 \) mC - \( 10 \) mC = \( 30 \) mC = \( 30 \times 10^{-3} \) C
Therefore:
- When the shell is initially grounded (Part 1), \( 10 \) mC of charge flows into the Earth.
- When a point charge of \( 30 \) mC is introduced at the center of the shell while keeping the shell grounded (Part 2), \( 30 \) mC of charge flows through the wire to the shell.