A copper wire of length 3 m and radius r is nickel plated till its radius becomes 2r. What would be the effective resistance of the wire, if specific resistance of copper and nickel are ` rho _c and rho_n` respectively.

Explanation

To find the effective resistance of a copper wire that has been nickel plated, we will follow these steps:

### Step 1: Identify the given parameters
- Length of the copper wire, \( L = 3 \, \text{m} \)
- Radius of the copper wire, \( r \)
- Radius of the nickel-plated wire, \( R = 2r \)
- Specific resistance of copper, \( \rho_c \)
- Specific resistance of nickel, \( \rho_n \)

### Step 2: Calculate the resistance of the copper wire
The resistance \( R_c \) of the copper wire can be calculated using the formula:
\[ R_c = \frac{\rho_c L}{A_c} \]
where \( A_c \) is the cross-sectional area of the copper wire.
\[ A_c = \pi r^2 \]
Thus, the resistance of the copper wire becomes:
\[ R_c = \frac{\rho_c \cdot 3}{\pi r^2} \]

### Step 3: Calculate the resistance of the nickel layer
The resistance \( R_n \) of the nickel layer can be calculated using the formula:
\[ R_n = \frac{\rho_n L}{A_n} \]
where \( A_n \) is the cross-sectional area of the nickel layer. The area of the nickel layer is the area of the larger circle minus the area of the smaller circle:
\[ A_n = \pi (2r)^2 - \pi r^2 = \pi (4r^2 - r^2) = \pi (3r^2) \]
Thus, the resistance of the nickel layer becomes:
\[ R_n = \frac{\rho_n \cdot 3}{\pi (3r^2)} = \frac{\rho_n}{\pi r^2} \]

### Step 4: Combine the resistances
Since the copper wire and the nickel layer are in parallel, we can use the formula for resistances in parallel:
\[ \frac{1}{R_{\text{effective}}} = \frac{1}{R_c} + \frac{1}{R_n} \]
Substituting the values of \( R_c \) and \( R_n \):
\[ \frac{1}{R_{\text{effective}}} = \frac{\pi r^2}{3 \rho_c} + \frac{\pi r^2}{\rho_n} \]

### Step 5: Simplify the equation
Finding a common denominator:
\[ \frac{1}{R_{\text{effective}}} = \frac{\pi r^2 \cdot \rho_n + 3 \pi r^2 \cdot \rho_c}{3 \rho_c \cdot \rho_n} \]
Thus,
\[ R_{\text{effective}} = \frac{3 \rho_c \cdot \rho_n}{\pi r^2 (\rho_n + 3 \rho_c)} \]

### Final Result
The effective resistance of the nickel-plated copper wire is:
\[ R_{\text{effective}} = \frac{3 \rho_c \cdot \rho_n}{\pi r^2 (\rho_n + 3 \rho_c)} \]

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