In an electrical cable there is a single wire of radius `9 mm` of copper. Its resistance is `5 Omega`. The cable is replaced by 6 different insulated copper, wires the radius of each wire is `3 mm`. Now the total resistance of the cable will be

Options

A. `7.5 Omega`
B. `45Omega`
C. `90Omega`
D. `270Omega`

Answer

To solve the problem, we need to find the total resistance of the cable when it is replaced by 6 different insulated copper wires, each with a radius of 3 mm.

### Step-by-step Solution:

1. **Identify the Resistance Formula**:
The resistance \( R \) of a wire is given by the formula:
\[ R = \frac{\rho L}{A} \]
where \( \rho \) is the resistivity of the material, \( L \) is the length of the wire, and \( A \) is the cross-sectional area of the wire.

2. **Calculate the Area of the Original Wire**:
The radius of the original wire is \( R_1 = 9 \, \text{mm} = 0.009 \, \text{m} \).
The cross-sectional area \( A_1 \) of the original wire is:
\[ A_1 = \pi R_1^2 = \pi (0.009)^2 = \pi \times 0.000081 = 0.000254469 \, \text{m}^2 \]

3. **Calculate the Resistance of the Original Wire**:
Given that the resistance \( R_1 = 5 \, \Omega \), we can express it as:
\[ R_1 = \frac{\rho L}{A_1} \]
Rearranging gives:
\[ \rho L = R_1 A_1 = 5 \times 0.000254469 = 0.001272345 \, \text{Ohm m}^2 \]

4. **Calculate the Area of Each New Wire**:
The radius of each new wire is \( r = 3 \, \text{mm} = 0.003 \, \text{m} \).
The cross-sectional area \( A_2 \) of each new wire is:
\[ A_2 = \pi r^2 = \pi (0.003)^2 = \pi \times 0.000009 = 0.000028274 \, \text{m}^2 \]

5. **Calculate the Total Area of 6 Wires**:
Since there are 6 wires, the total cross-sectional area \( A_{total} \) is:
\[ A_{total} = 6 \times A_2 = 6 \times 0.000028274 = 0.000169644 \, \text{m}^2 \]

6. **Calculate the Resistance of the New Configuration**:
The resistance \( R_2 \) of the new configuration can be expressed as:
\[ R_2 = \frac{\rho L}{A_{total}} \]
Substituting \( \rho L \) from step 3:
\[ R_2 = \frac{0.001272345}{0.000169644} \approx 7.5 \, \Omega \]

### Final Answer:
The total resistance of the cable when replaced by 6 different insulated copper wires is approximately \( 7.5 \, \Omega \).