Two metallic spheres of radii 1 cm and 3 cm are given charges of `-1 xx 10^(-2) C ` and `5 xx 10^(-2) C` , respectively . If these are connected by a conducting wire , the final charge on the bigger sphere is

Options

A. ` 2 xx 10^(-7) C `
B. `3 xx 10^(-2) C`
C. `4 xx 10^(-2) C `
D. `1 xx 10^(-2) C `

Explanation

To solve the problem, we need to find the final charge on the bigger sphere after connecting two metallic spheres with different charges and radii. Let's break it down step by step.

### Step 1: Identify Given Data
- Radius of Sphere 1 (R1) = 1 cm = 0.01 m
- Radius of Sphere 2 (R2) = 3 cm = 0.03 m
- Charge on Sphere 1 (Q1) = -1 × 10^(-2) C
- Charge on Sphere 2 (Q2) = 5 × 10^(-2) C

### Step 2: Calculate Initial Charges
The total initial charge (Q_total) before connecting the spheres is:
\[ Q_{\text{total}} = Q1 + Q2 = (-1 \times 10^{-2}) + (5 \times 10^{-2}) = 4 \times 10^{-2} \, \text{C} \]

### Step 3: Calculate Capacitance of Each Sphere
The capacitance (C) of a spherical conductor is given by:
\[ C = 4 \pi \epsilon_0 R \]
Where \( \epsilon_0 \) (the permittivity of free space) is approximately \( 8.85 \times 10^{-12} \, \text{F/m} \).

- Capacitance of Sphere 1 (C1):
\[ C1 = 4 \pi \epsilon_0 R1 = 4 \pi (8.85 \times 10^{-12}) (0.01) \]
\[ C1 \approx 1.11 \times 10^{-13} \, \text{F} \]

- Capacitance of Sphere 2 (C2):
\[ C2 = 4 \pi \epsilon_0 R2 = 4 \pi (8.85 \times 10^{-12}) (0.03) \]
\[ C2 \approx 3.33 \times 10^{-12} \, \text{F} \]

### Step 4: Calculate Total Capacitance
Since the spheres are connected in parallel, the total capacitance (C_total) is:
\[ C_{\text{total}} = C1 + C2 \]
\[ C_{\text{total}} \approx 1.11 \times 10^{-13} + 3.33 \times 10^{-12} \]
\[ C_{\text{total}} \approx 3.44 \times 10^{-12} \, \text{F} \]

### Step 5: Calculate Common Potential (V)
The common potential (V) when the spheres are connected is given by:
\[ V = \frac{Q_{\text{total}}}{C_{\text{total}}} \]
\[ V = \frac{4 \times 10^{-2}}{3.44 \times 10^{-12}} \]
\[ V \approx 1.16 \times 10^{10} \, \text{V} \]

### Step 6: Calculate Final Charge on the Bigger Sphere
The final charge on the bigger sphere (Q_final2) can be calculated using its capacitance and the common potential:
\[ Q_{\text{final2}} = C2 \times V \]
\[ Q_{\text{final2}} = 3.33 \times 10^{-12} \times 1.16 \times 10^{10} \]
\[ Q_{\text{final2}} \approx 3.86 \times 10^{-2} \, \text{C} \]

### Conclusion
The final charge on the bigger sphere is approximately \( 3.86 \times 10^{-2} \, \text{C} \).

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