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

Options

A. `3 xx 10^(-2)` C
B. `2 xx 10^(-2)` C
C. `1 xx 10^(-2)`C
D. `6 xx 10^(-2)` C

Answer

To find the final charge on the smaller sphere after connecting it with the larger sphere via a conducting wire, we can follow these steps:

### Step 1: Understand the system
We have two metallic spheres:
- Sphere 1 (smaller) with radius \( R_1 = 1 \, \text{cm} \) and initial charge \( Q_1 = 10^{-2} \, \text{C} \).
- Sphere 2 (larger) with radius \( R_2 = 2 \, \text{cm} \) and initial charge \( Q_2 = 5 \times 10^{-2} \, \text{C} \).

### Step 2: Apply the principle of conservation of charge
When the two spheres are connected by a conducting wire, the total charge is conserved. Thus, the total initial charge \( Q_{total} \) is given by:
\[ Q_{total} = Q_1 + Q_2 = 10^{-2} + 5 \times 10^{-2} = 6 \times 10^{-2} \, \text{C} \]

### Step 3: Set up the equation for equal potentials
The potential \( V \) on the surface of a sphere is given by:
\[ V = \frac{kQ}{R} \]
where \( k \) is Coulomb's constant, \( Q \) is the charge, and \( R \) is the radius of the sphere.

Since the spheres are connected, their potentials will be equal:
\[ \frac{kQ_1'}{R_1} = \frac{kQ_2'}{R_2} \]
where \( Q_1' \) and \( Q_2' \) are the final charges on the smaller and larger spheres, respectively.

### Step 4: Substitute the values
Substituting the radii:
\[ \frac{Q_1'}{1} = \frac{Q_2'}{2} \]
This simplifies to:
\[ Q_2' = 2Q_1' \]

### Step 5: Use the conservation of charge equation
From the conservation of charge, we have:
\[ Q_1' + Q_2' = Q_{total} \]
Substituting \( Q_2' \) from the previous step:
\[ Q_1' + 2Q_1' = 6 \times 10^{-2} \]
This simplifies to:
\[ 3Q_1' = 6 \times 10^{-2} \]

### Step 6: Solve for \( Q_1' \)
Dividing both sides by 3:
\[ Q_1' = \frac{6 \times 10^{-2}}{3} = 2 \times 10^{-2} \, \text{C} \]

### Conclusion
The final charge on the smaller sphere is:
\[ \boxed{2 \times 10^{-2} \, \text{C}} \]