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 are equal, we can use the formula for the resistance of a conductor as a function of temperature:

\[ R(T) = R_0 (1 + \alpha (T - T_0)) \]

Where:
- \( R(T) \) = resistance at temperature \( T \)
- \( R_0 \) = initial resistance at temperature \( T_0 \)
- \( \alpha \) = temperature coefficient of resistance
- \( T \) = final temperature
- \( T_0 \) = initial temperature

Given:
- Resistance of copper at \( 20^\circ C \), \( R_{Cu0} = 4.1 \, \Omega \)
- Resistance of iron at \( 20^\circ C \), \( R_{Fe0} = 3.9 \, \Omega \)
- Temperature coefficient for copper, \( \alpha_{Cu} = 4.0 \times 10^{-3} \, K^{-1} \)
- Temperature coefficient for iron, \( \alpha_{Fe} = 5.0 \times 10^{-3} \, K^{-1} \)

We need to find the temperature \( T \) at which \( R_{Cu}(T) = R_{Fe}(T) \).

### Step 1: Write the equations for resistance at temperature \( T \)

For copper:
\[ R_{Cu}(T) = R_{Cu0} (1 + \alpha_{Cu} (T - 20)) \]
\[ R_{Cu}(T) = 4.1 (1 + 4.0 \times 10^{-3} (T - 20)) \]

For iron:
\[ R_{Fe}(T) = R_{Fe0} (1 + \alpha_{Fe} (T - 20)) \]
\[ R_{Fe}(T) = 3.9 (1 + 5.0 \times 10^{-3} (T - 20)) \]

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

Set the resistance of copper equal to the resistance of iron:
\[ 4.1 (1 + 4.0 \times 10^{-3} (T - 20)) = 3.9 (1 + 5.0 \times 10^{-3} (T - 20)) \]

### Step 3: Expand both sides

Expanding both sides gives:
\[ 4.1 + 4.1 \cdot 4.0 \times 10^{-3} (T - 20) = 3.9 + 3.9 \cdot 5.0 \times 10^{-3} (T - 20) \]

### Step 4: Simplify the equation

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

So the equation becomes:
\[ 4.1 + 0.0164 (T - 20) = 3.9 + 0.0195 (T - 20) \]

### Step 5: Rearranging the equation

Rearranging gives:
\[ 4.1 - 3.9 = 0.0195 (T - 20) - 0.0164 (T - 20) \]
\[ 0.2 = (0.0195 - 0.0164)(T - 20) \]
\[ 0.2 = 0.0031 (T - 20) \]

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

Dividing both sides by \( 0.0031 \):
\[ T - 20 = \frac{0.2}{0.0031} \]
\[ T - 20 \approx 64.51 \]
\[ T \approx 84.51 \]

### Final Answer
The temperature at which the resistance of both wires is equal is approximately \( 84.51^\circ C \).

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