Two conducting spheres of radii `5 cm` and `10 cm` are given a charge of `15mu F` each. After the two spheres are joined by a conducting wire, the charge on the smaller sphere is

Options

A. `5 muC`
B. `10 muC`
C. `15 muC`
D. `20 muC`

Explanation

To solve the problem step by step, we will follow these instructions:

### Step 1: Understand the given information
We have two conducting spheres:
- Sphere 1 (smaller) has a radius of \( r_1 = 5 \, \text{cm} \)
- Sphere 2 (larger) has a radius of \( r_2 = 10 \, \text{cm} \)

Each sphere is given a charge of \( Q_1 = Q_2 = 15 \, \mu C \).

### Step 2: Calculate the total charge
When the two spheres are connected by a conducting wire, the total charge \( Q_{total} \) is the sum of the charges on both spheres:
\[ Q_{total} = Q_1 + Q_2 = 15 \, \mu C + 15 \, \mu C = 30 \, \mu C \]

### Step 3: Understand the concept of potential
When the spheres are connected by a wire, they will reach the same electric potential \( V \). The potential \( V \) of a charged sphere 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 sphere.

### Step 4: Set up the equation for equal potentials
Let \( Q_1' \) be the final charge on the smaller sphere and \( Q_2' \) be the final charge on the larger sphere after they are connected. Since the total charge is conserved, we have:
\[ Q_1' + Q_2' = 30 \, \mu C \]

The potentials of both spheres must be equal:
\[ \frac{k \cdot Q_1'}{r_1} = \frac{k \cdot Q_2'}{r_2} \]

### Step 5: Simplify the equation
Since \( k \) is a constant, we can cancel it out:
\[ \frac{Q_1'}{r_1} = \frac{Q_2'}{r_2} \]
Substituting \( r_1 = 0.05 \, \text{m} \) and \( r_2 = 0.10 \, \text{m} \):
\[ \frac{Q_1'}{0.05} = \frac{Q_2'}{0.10} \]
This simplifies to:
\[ 2Q_1' = Q_2' \]

### Step 6: Substitute \( Q_2' \) in terms of \( Q_1' \)
From the equation \( Q_2' = 2Q_1' \), we can substitute into the total charge equation:
\[ Q_1' + 2Q_1' = 30 \, \mu C \]
\[ 3Q_1' = 30 \, \mu C \]
\[ Q_1' = \frac{30 \, \mu C}{3} = 10 \, \mu C \]

### Step 7: Conclusion
The charge on the smaller sphere after they are connected is:
\[ Q_1' = 10 \, \mu C \]

### Final Answer
The charge on the smaller sphere is \( 10 \, \mu C \).

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