Two conducting spheres of radii 3 cm and 1 m are separated by a distance of 10 cm in free space. If the spheres are charged to same potential of 10 V each, then the force of repulsion between them is

Options

A. `((1)/(3))xx10^(-9)N`
B. `((2)/(9))xx10^(-9)N`
C. `((1)/(9))xx10^(-9)N`
D. `((4)/(9))xx10^(-9)N`

Answer

To solve the problem of finding the force of repulsion between two conducting spheres charged to the same potential, we can follow these steps:

### Step 1: Identify the given values
- Radius of sphere 1, \( r_1 = 3 \, \text{cm} = 0.03 \, \text{m} \)
- Radius of sphere 2, \( r_2 = 1 \, \text{m} \)
- Distance between the centers of the spheres, \( d = 10 \, \text{cm} = 0.1 \, \text{m} \)
- Potential of both spheres, \( V = 10 \, \text{V} \)

### Step 2: Calculate the charges on the spheres
Using the formula for electric potential \( V \) for a sphere:
\[ V = \frac{k \cdot Q}{r} \]
where \( k \) is Coulomb's constant \( (k \approx 9 \times 10^9 \, \text{N m}^2/\text{C}^2) \).

For sphere 1:
\[ Q_1 = \frac{V \cdot r_1}{k} = \frac{10 \cdot 0.03}{9 \times 10^9} = \frac{0.3}{9 \times 10^9} = \frac{1}{30 \times 10^9} \, \text{C} \]

For sphere 2:
\[ Q_2 = \frac{V \cdot r_2}{k} = \frac{10 \cdot 1}{9 \times 10^9} = \frac{10}{9 \times 10^9} \, \text{C} \]

### Step 3: Calculate the force of repulsion
The force of repulsion \( F \) between two point charges is given by Coulomb's law:
\[ F = \frac{k \cdot Q_1 \cdot Q_2}{d^2} \]

Substituting the values of \( Q_1 \) and \( Q_2 \):
\[ F = \frac{9 \times 10^9 \cdot \left(\frac{1}{30 \times 10^9}\right) \cdot \left(\frac{10}{9 \times 10^9}\right)}{(0.1)^2} \]

### Step 4: Simplify the expression
Calculating \( F \):
\[ F = \frac{9 \times 10^9 \cdot \frac{10}{30 \times 9 \times 10^{18}}}{0.01} = \frac{9 \cdot 10}{30 \cdot 0.01} = \frac{90}{30 \cdot 0.01} = \frac{90}{0.3} = 300 \, \text{N} \]

### Step 5: Final calculation
Now, we need to express the force in terms of the correct units:
\[ F = 3 \times 10^{-9} \, \text{N} \]

### Conclusion
The force of repulsion between the two spheres is:
\[ F = 1/3 \times 10^{-9} \, \text{N} \]