Resistance of two wires A and B at `10^@C` are `3.8 Omega and 4.0 Omega` respectively . If temperature coefficient of resistance for wire A and B are `4.8 xx10^(-3) K^(-1) and 3.8 xx10^(-3)K^(-1)` respectively, calculate the temperature at which resistance of both the wires are equal. Ignore thermal expansion.

Answer

To find the temperature at which the resistances of two wires A and B are equal, we can use the formula for resistance as a function of temperature:

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

Where:
- \( R(T) \) is the resistance at temperature \( T \)
- \( R_0 \) is the resistance at the reference temperature \( T_0 \)
- \( \alpha \) is the temperature coefficient of resistance
- \( T \) is the final temperature we want to find
- \( T_0 \) is the initial reference temperature

### Step-by-Step Solution

1. **Identify Given Values:**
- For wire A:
- \( R_{A0} = 3.8 \, \Omega \) (at \( T_0 = 10^\circ C \))
- \( \alpha_A = 4.8 \times 10^{-3} \, K^{-1} \)
- For wire B:
- \( R_{B0} = 4.0 \, \Omega \) (at \( T_0 = 10^\circ C \))
- \( \alpha_B = 3.8 \times 10^{-3} \, K^{-1} \)

2. **Write the Resistance Equations:**
- For wire A at temperature \( T \):
\[ R_A(T) = 3.8 \left(1 + 4.8 \times 10^{-3} (T - 10)\right) \]
- For wire B at temperature \( T \):
\[ R_B(T) = 4.0 \left(1 + 3.8 \times 10^{-3} (T - 10)\right) \]

3. **Set the Resistances Equal:**
\[ 3.8 \left(1 + 4.8 \times 10^{-3} (T - 10)\right) = 4.0 \left(1 + 3.8 \times 10^{-3} (T - 10)\right) \]

4. **Expand Both Sides:**
- Left side:
\[ 3.8 + 3.8 \times 4.8 \times 10^{-3} (T - 10) \]
- Right side:
\[ 4.0 + 4.0 \times 3.8 \times 10^{-3} (T - 10) \]

5. **Combine Like Terms:**
\[ 3.8 + 3.8 \times 4.8 \times 10^{-3} (T - 10) = 4.0 + 4.0 \times 3.8 \times 10^{-3} (T - 10) \]

6. **Rearranging the Equation:**
\[ 3.8 - 4.0 = 4.0 \times 3.8 \times 10^{-3} (T - 10) - 3.8 \times 4.8 \times 10^{-3} (T - 10) \]
\[ -0.2 = (4.0 \times 3.8 - 3.8 \times 4.8) \times 10^{-3} (T - 10) \]

7. **Calculate the Coefficient:**
\[ 4.0 \times 3.8 - 3.8 \times 4.8 = 15.2 - 18.24 = -3.04 \]

8. **Substituting Back:**
\[ -0.2 = -3.04 \times 10^{-3} (T - 10) \]
\[ 0.2 = 3.04 \times 10^{-3} (T - 10) \]

9. **Solve for \( T \):**
\[ T - 10 = \frac{0.2}{3.04 \times 10^{-3}} \approx 65.79 \]
\[ T \approx 75.79^\circ C \]

### Final Answer:
The temperature at which the resistance of both wires is equal is approximately \( 75.79^\circ C \).