Two thin spherical shells made of metal are at a large distance apart. One of radius 10cm carries a charge of `+0.5muC` and the other of radius 20cm carries a charge of `+0.7muC`. The charge on each, when they are connected by a suitable conducting wire is respectively

Options

A. `0.4 and 0.8 muC`
B. `+0.425 and +0.85 muC`
C. `+0.5 and +0.7 muC`
D. `+0.6 and +0.6 muC`

Answer

To solve the problem, we need to determine the final charges on two spherical shells when they are connected by a conducting wire. The steps are as follows:

### Step 1: Understand the Initial Charges
- We have two spherical shells:
- Shell 1 (radius \( R_1 = 10 \, \text{cm} \)) with charge \( Q_1 = +0.5 \, \mu\text{C} \)
- Shell 2 (radius \( R_2 = 20 \, \text{cm} \)) with charge \( Q_2 = +0.7 \, \mu\text{C} \)

### Step 2: Calculate Total Charge
- The total charge before connecting the shells is given by:
\[ Q_{\text{net}} = Q_1 + Q_2 = 0.5 \, \mu\text{C} + 0.7 \, \mu\text{C} = 1.2 \, \mu\text{C} \]

### Step 3: Understand Charge Distribution
- When the shells are connected by a conducting wire, charge will redistribute until both shells reach the same electric potential.
- The potential \( V \) of a charged spherical shell is given by:
\[ V = \frac{k \cdot Q}{R} \]
where \( k \) is Coulomb's constant, \( Q \) is the charge, and \( R \) is the radius of the shell.

### Step 4: Set Up the Equation for Equal Potentials
- After connecting the shells, let the final charges be \( Q_1' \) and \( Q_2' \).
- The potentials of both shells must be equal:
\[ \frac{k \cdot Q_1'}{R_1} = \frac{k \cdot Q_2'}{R_2} \]
Simplifying this gives:
\[ \frac{Q_1'}{10} = \frac{Q_2'}{20} \]
or
\[ Q_2' = 2 \cdot Q_1' \]

### Step 5: Use Charge Conservation
- The total charge after redistribution must equal the initial total charge:
\[ Q_1' + Q_2' = Q_{\text{net}} = 1.2 \, \mu\text{C} \]
Substituting \( Q_2' = 2 \cdot Q_1' \) into this equation gives:
\[ Q_1' + 2 \cdot Q_1' = 1.2 \, \mu\text{C} \]
\[ 3 \cdot Q_1' = 1.2 \, \mu\text{C} \]
\[ Q_1' = \frac{1.2 \, \mu\text{C}}{3} = 0.4 \, \mu\text{C} \]

### Step 6: Calculate \( Q_2' \)
- Now, using \( Q_2' = 2 \cdot Q_1' \):
\[ Q_2' = 2 \cdot 0.4 \, \mu\text{C} = 0.8 \, \mu\text{C} \]

### Final Answer
- The final charges on the shells are:
- Charge on shell 1: \( Q_1' = 0.4 \, \mu\text{C} \)
- Charge on shell 2: \( Q_2' = 0.8 \, \mu\text{C} \)