A conducting wire has a resistance of `5 Omega "at" 0^@C and 5.6 Omega "at" 100 ^@C`. When the wire is inserted in a hot bath with temperature `T^@C` , its resistance is `5.9 Omega` . What is the value of T ?

Answer

To find the temperature \( T \) at which the resistance of the wire is \( 5.9 \, \Omega \), we can use the formula for the temperature dependence of resistance:

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

Where:
- \( R_T \) is the resistance at temperature \( T \)
- \( R_0 \) is the resistance at a reference temperature (in this case, \( 0^\circ C \))
- \( \alpha \) is the temperature coefficient of resistance
- \( \Delta T \) is the change in temperature from the reference temperature

### Step 1: Calculate the temperature coefficient \( \alpha \)

We know:
- \( R_0 = 5 \, \Omega \) at \( 0^\circ C \)
- \( R_{100} = 5.6 \, \Omega \) at \( 100^\circ C \)

Using the formula:

\[ R_{100} = R_0 (1 + \alpha \Delta T) \]

Substituting the known values:

\[ 5.6 = 5 (1 + \alpha (100 - 0)) \]

This simplifies to:

\[ 5.6 = 5 (1 + 100\alpha) \]

Dividing both sides by 5:

\[ 1.12 = 1 + 100\alpha \]

Subtracting 1 from both sides:

\[ 0.12 = 100\alpha \]

Now, solving for \( \alpha \):

\[ \alpha = \frac{0.12}{100} = 0.0012 \, \text{per degree Celsius} \]

### Step 2: Use \( \alpha \) to find the temperature \( T \)

Now we know the resistance at temperature \( T \) is \( R_T = 5.9 \, \Omega \). We can use the same formula:

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

Substituting the known values:

\[ 5.9 = 5 (1 + 0.0012 (T - 0)) \]

This simplifies to:

\[ 5.9 = 5 (1 + 0.0012 T) \]

Dividing both sides by 5:

\[ 1.18 = 1 + 0.0012 T \]

Subtracting 1 from both sides:

\[ 0.18 = 0.0012 T \]

Now, solving for \( T \):

\[ T = \frac{0.18}{0.0012} = 150 \, \text{degrees Celsius} \]

### Final Answer:
The value of \( T \) is \( 150^\circ C \).

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