The resistance of a copper wire and an iron at `20^@C` are `4.1 Omega` and `3.9Omega` respectively. Neglecting any thermal expansion, find the temperature at which resistane of both are equal.
`alpha_(Cu)=4.0xx10^-3K^-1` an `alpha_(Fe)=5.0xx10^-3K^-1`

Explanation

To find the temperature at which the resistance of copper and iron wires are equal, we will use the formula for the resistance change with temperature:

\[ R = R_0 (1 + \alpha \Delta T) \]

Where:
- \( R \) is the final resistance,
- \( R_0 \) is the initial resistance,
- \( \alpha \) is the temperature coefficient of resistance,
- \( \Delta T \) is the change in temperature from the initial temperature.

### Step 1: Write the resistance equations for both materials.

For copper:
\[ R_{Cu} = R_{0,Cu} (1 + \alpha_{Cu} \Delta T) \]
\[ R_{Cu} = 4.1 \, \Omega (1 + 4.0 \times 10^{-3} \Delta T) \]

For iron:
\[ R_{Fe} = R_{0,Fe} (1 + \alpha_{Fe} \Delta T) \]
\[ R_{Fe} = 3.9 \, \Omega (1 + 5.0 \times 10^{-3} \Delta T) \]

### Step 2: Set the equations equal to each other.

Since we want the resistances to be equal:
\[ 4.1 (1 + 4.0 \times 10^{-3} \Delta T) = 3.9 (1 + 5.0 \times 10^{-3} \Delta T) \]

### Step 3: Expand both sides.

Expanding both sides gives:
\[ 4.1 + 4.1 \times 4.0 \times 10^{-3} \Delta T = 3.9 + 3.9 \times 5.0 \times 10^{-3} \Delta T \]

### Step 4: Calculate the coefficients.

Calculating the coefficients:
- For copper: \( 4.1 \times 4.0 \times 10^{-3} = 0.0164 \)
- For iron: \( 3.9 \times 5.0 \times 10^{-3} = 0.0195 \)

So the equation becomes:
\[ 4.1 + 0.0164 \Delta T = 3.9 + 0.0195 \Delta T \]

### Step 5: Rearrange the equation.

Rearranging gives:
\[ 0.0164 \Delta T - 0.0195 \Delta T = 3.9 - 4.1 \]
\[ -0.0031 \Delta T = -0.2 \]

### Step 6: Solve for \( \Delta T \).

Dividing both sides by -0.0031:
\[ \Delta T = \frac{0.2}{0.0031} \approx 64.52 \, \text{°C} \]

### Step 7: Find the final temperature.

Adding the initial temperature (20 °C) to the change in temperature:
\[ T = 20 + 64.52 \approx 84.52 \, \text{°C} \]

### Final Answer:
The temperature at which the resistance of both copper and iron wires are equal is approximately **84.52 °C**.