Two charged spheres of radii `R_(1) and R_(2)` having equal surface charge density. The ratio of their potential is

Options

A. `R_(1)//R_(2)`
B. `R_(2)//R_(1)`
C. `(R_(1)//R_(2))^(2)`
D. `(R_(2)//R_(1))^(2)`

Answer

To solve the problem of finding the ratio of the potentials of two charged spheres with equal surface charge densities, we can follow these steps:

### Step 1: Understand the relationship between surface charge density and charge
The surface charge density (σ) is defined as charge per unit area. For a sphere, the surface area \( A \) is given by:
\[ A = 4\pi R^2 \]
Thus, the surface charge density for each sphere can be expressed as:
\[ \sigma = \frac{q_1}{4\pi R_1^2} \quad \text{and} \quad \sigma = \frac{q_2}{4\pi R_2^2} \]
Since the surface charge densities are equal, we have:
\[ \frac{q_1}{4\pi R_1^2} = \frac{q_2}{4\pi R_2^2} \]

### Step 2: Derive the ratio of charges
From the equality of the surface charge densities, we can derive the ratio of the charges \( q_1 \) and \( q_2 \):
\[ q_1 R_2^2 = q_2 R_1^2 \implies \frac{q_1}{q_2} = \frac{R_1^2}{R_2^2} \]

### Step 3: Write the expression for the potential of each sphere
The electric potential \( V \) at the surface of a charged sphere is given by:
\[ V = k \frac{q}{R} \]
where \( k \) is Coulomb's constant. Therefore, the potentials for the two spheres are:
\[ V_1 = k \frac{q_1}{R_1} \quad \text{and} \quad V_2 = k \frac{q_2}{R_2} \]

### Step 4: Find the ratio of the potentials
We need to find the ratio \( \frac{V_1}{V_2} \):
\[ \frac{V_1}{V_2} = \frac{k \frac{q_1}{R_1}}{k \frac{q_2}{R_2}} = \frac{q_1}{q_2} \cdot \frac{R_2}{R_1} \]
Substituting the ratio of charges from Step 2:
\[ \frac{V_1}{V_2} = \frac{R_1^2}{R_2^2} \cdot \frac{R_2}{R_1} = \frac{R_1^2 \cdot R_2}{R_2^2 \cdot R_1} = \frac{R_1}{R_2} \]

### Conclusion
Thus, the ratio of the potentials of the two spheres is:
\[ \frac{V_1}{V_2} = \frac{R_1}{R_2} \]