Two spheres of radii 2 cm and 3 cm are charged to the same potential. If `sigma` and `sigma_2` be respectively the values of surface charge density on the conductors, then the ratio `(sigma_1)/(sigma_2)` will be

Options

A. (a)`(4)/(9)`
B. (b)`(2)/(3)`
C. (c)`(3)/(2)`
D. (d)`(9)/(4)`

Answer

To solve the problem, we need to find the ratio of surface charge densities (\( \sigma_1 \) and \( \sigma_2 \)) on two spheres of different radii that are charged to the same potential.

### Step-by-step Solution:

1. **Understand the relationship between charge density and potential**:
The surface charge density (\( \sigma \)) on a sphere is related to its potential (\( V \)) and radius (\( r \)) by the formula:
\[ \sigma = \frac{Q}{A} = \frac{V \cdot r}{k \cdot A} \]
where \( A \) is the surface area of the sphere, \( A = 4\pi r^2 \).

2. **Express charge density in terms of potential and radius**:
The potential \( V \) of a charged sphere is given by:
\[ V = k \frac{Q}{r} \]
Rearranging gives:
\[ Q = \frac{V \cdot r}{k} \]
The surface area \( A \) of a sphere is \( 4\pi r^2 \), so the surface charge density can be expressed as:
\[ \sigma = \frac{Q}{A} = \frac{V \cdot r}{k \cdot 4\pi r^2} = \frac{V}{4\pi k r} \]

3. **Relate the surface charge densities of the two spheres**:
Let \( \sigma_1 \) be the surface charge density of the sphere with radius \( r_1 = 2 \, \text{cm} \) and \( \sigma_2 \) be that of the sphere with radius \( r_2 = 3 \, \text{cm} \). Since both spheres are at the same potential \( V \):
\[ \sigma_1 = \frac{V}{4\pi k r_1} \quad \text{and} \quad \sigma_2 = \frac{V}{4\pi k r_2} \]

4. **Calculate the ratio of surface charge densities**:
Now, we can find the ratio \( \frac{\sigma_1}{\sigma_2} \):
\[ \frac{\sigma_1}{\sigma_2} = \frac{\frac{V}{4\pi k r_1}}{\frac{V}{4\pi k r_2}} = \frac{r_2}{r_1} \]

5. **Substitute the values of the radii**:
Substitute \( r_1 = 2 \, \text{cm} \) and \( r_2 = 3 \, \text{cm} \):
\[ \frac{\sigma_1}{\sigma_2} = \frac{3}{2} \]

### Final Answer:
The ratio of the surface charge densities \( \frac{\sigma_1}{\sigma_2} \) is \( \frac{3}{2} \).