Two spherical conductors of radii 4 m and 5 m are charged to the same potential. If `sigma_(1) and sigma_(2)` be the respective value of the surface density of charge on the two conductors, then the ratio `(sigma_(1))/(sigma_(2))`is

Options

A. `(25)/(16)`
B. `(16)/(25)`
C. `(5)/(4)`
D. `(4)/(5)`

Answer

To solve the problem, we need to find the ratio of the surface charge densities (\( \sigma_1 \) and \( \sigma_2 \)) of two spherical conductors that are charged to the same potential. The radii of the conductors are given as \( R_1 = 4 \, \text{m} \) and \( R_2 = 5 \, \text{m} \).

### Step-by-Step Solution:

1. **Understanding the Potential of Spherical Conductors**:
The potential \( V \) of a spherical conductor is given by the formula:
\[ V = \frac{KQ}{R} \]
where \( K \) is the electrostatic constant, \( Q \) is the charge, and \( R \) is the radius of the sphere.

2. **Setting Up the Equation for Both Conductors**:
Since both conductors are charged to the same potential \( V \), we can write:
\[ V_1 = \frac{KQ_1}{R_1} \quad \text{and} \quad V_2 = \frac{KQ_2}{R_2} \]
Since \( V_1 = V_2 \), we have:
\[ \frac{KQ_1}{R_1} = \frac{KQ_2}{R_2} \]

3. **Simplifying the Equation**:
The constant \( K \) cancels out, leading to:
\[ \frac{Q_1}{R_1} = \frac{Q_2}{R_2} \]
Rearranging gives us:
\[ \frac{Q_1}{Q_2} = \frac{R_1}{R_2} \]

4. **Calculating Surface Charge Densities**:
The surface charge density \( \sigma \) is defined as:
\[ \sigma = \frac{Q}{A} \]
For a spherical conductor, the surface area \( A \) is \( 4\pi R^2 \). Therefore, we can express \( \sigma_1 \) and \( \sigma_2 \) as:
\[ \sigma_1 = \frac{Q_1}{4\pi R_1^2} \quad \text{and} \quad \sigma_2 = \frac{Q_2}{4\pi R_2^2} \]

5. **Finding the Ratio of Surface Charge Densities**:
The ratio \( \frac{\sigma_1}{\sigma_2} \) can be expressed as:
\[ \frac{\sigma_1}{\sigma_2} = \frac{Q_1}{Q_2} \cdot \frac{R_2^2}{R_1^2} \]
Substituting \( \frac{Q_1}{Q_2} = \frac{R_1}{R_2} \) into the equation gives:
\[ \frac{\sigma_1}{\sigma_2} = \frac{R_1}{R_2} \cdot \frac{R_2^2}{R_1^2} \]
This simplifies to:
\[ \frac{\sigma_1}{\sigma_2} = \frac{R_2}{R_1} \]

6. **Substituting the Values**:
Now, substituting the values of the radii:
\[ R_1 = 4 \, \text{m} \quad \text{and} \quad R_2 = 5 \, \text{m} \]
We find:
\[ \frac{\sigma_1}{\sigma_2} = \frac{5}{4} \]

### Final Answer:
Thus, the ratio of the surface charge densities is:
\[ \frac{\sigma_1}{\sigma_2} = \frac{5}{4} \]