Two spheres `A` and `B` of radius 'a' and 'b' respectively are at same electric potential. The ratio of the surface charge densities of `A` and `B` is

Options

A. `(a)/(b)`
B. `(b)/(a)`
C. `(a^(2))/(b^(2))`
D. `(b^(2))/(a^(2))`

Explanation

To solve the problem of finding the ratio of surface charge densities of two spheres `A` and `B` that are at the same electric potential, we can follow these steps:

### Step 1: Understanding Electric Potential
The electric potential \( V \) of a charged sphere is given by the formula:
\[ V = \frac{kQ}{R} \]
where \( k \) is Coulomb's constant, \( Q \) is the charge on the sphere, and \( R \) is the radius of the sphere.

### Step 2: Setting up the Equations
Let the charge on sphere A be \( Q_1 \) and its radius be \( a \). Similarly, let the charge on sphere B be \( Q_2 \) and its radius be \( b \). Since both spheres are at the same electric potential, we can write:
\[ V_A = V_B \]
This gives us:
\[ \frac{kQ_1}{a} = \frac{kQ_2}{b} \]
We can simplify this equation by canceling \( k \):
\[ \frac{Q_1}{a} = \frac{Q_2}{b} \]
From this, we can derive the relationship between the charges:
\[ \frac{Q_1}{Q_2} = \frac{a}{b} \quad \text{(Equation 1)} \]

### Step 3: Surface Charge Density
The surface charge density \( \sigma \) is defined as the charge per unit area. For a sphere, the surface area \( A \) is given by \( 4\pi R^2 \). Therefore, the surface charge densities for spheres A and B are:
\[ \sigma_1 = \frac{Q_1}{4\pi a^2} \]
\[ \sigma_2 = \frac{Q_2}{4\pi b^2} \]

### Step 4: Finding the Ratio of Surface Charge Densities
To find the ratio of the surface charge densities \( \sigma_1 \) and \( \sigma_2 \):
\[ \frac{\sigma_1}{\sigma_2} = \frac{Q_1}{4\pi a^2} \cdot \frac{4\pi b^2}{Q_2} = \frac{Q_1}{Q_2} \cdot \frac{b^2}{a^2} \]
Substituting Equation 1 into this expression:
\[ \frac{\sigma_1}{\sigma_2} = \frac{a}{b} \cdot \frac{b^2}{a^2} = \frac{b}{a} \]

### Final Result
Thus, the ratio of the surface charge densities of spheres A and B is:
\[ \frac{\sigma_1}{\sigma_2} = \frac{b}{a} \]