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

Options

A. `(16)/(25)`
B. `(15)/(10)`
C. `(4)/(5)`
D. `(5)/(4)`

Answer

To solve the problem of finding the ratio of surface charge densities \(\sigma_1\) and \(\sigma_2\) on two spherical conductors of radii \(R_1 = 4 \, \text{cm}\) and \(R_2 = 5 \, \text{cm}\) that are charged to the same potential, we can follow these steps:

### Step 1: Understanding the relationship between charge, potential, and surface charge density
The electric potential \(V\) of a charged conductor is given by the formula:
\[ V = \frac{KQ}{R} \]
where \(K\) is a constant, \(Q\) is the charge, and \(R\) is the radius of the conductor.

### Step 2: Expressing charge in terms of surface charge density
The charge \(Q\) on the surface of a spherical conductor can be expressed in terms of surface charge density \(\sigma\) as:
\[ Q = \sigma \cdot A \]
where \(A\) is the surface area of the sphere. The surface area \(A\) of a sphere is given by:
\[ A = 4\pi R^2 \]
Thus, we can write:
\[ Q = \sigma \cdot 4\pi R^2 \]

### Step 3: Substituting charge into the potential formula
Substituting the expression for \(Q\) into the potential formula gives:
\[ V = \frac{K(\sigma \cdot 4\pi R^2)}{R} = K\sigma \cdot 4\pi R \]

### Step 4: Relating the potentials of both conductors
Since both conductors are charged to the same potential, we can write:
\[ V_1 = V_2 \]
This leads to:
\[ K\sigma_1 \cdot 4\pi R_1 = K\sigma_2 \cdot 4\pi R_2 \]
The \(K\) and \(4\pi\) terms cancel out, simplifying to:
\[ \sigma_1 R_1 = \sigma_2 R_2 \]

### Step 5: Finding the ratio of surface charge densities
Rearranging the above equation gives:
\[ \frac{\sigma_1}{\sigma_2} = \frac{R_2}{R_1} \]
Substituting the values of \(R_1\) and \(R_2\):
\[ \frac{\sigma_1}{\sigma_2} = \frac{5 \, \text{cm}}{4 \, \text{cm}} = \frac{5}{4} \]

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

---