In a platinum resistance thermometer resitance of the platinum wire at `0^(@)C, 100^(@) C ` and in hot bath are `8 Omega , 12 Omega ` and `11 Omega ` respectively. Temperature of hot bath is

Options

A. `15^(@)C`
B. ` 175^(@)C`
C. ` 60^(@)C`
D. ` 75^(@)C`

Answer

To solve the problem of finding the temperature of the hot bath using the resistance values of a platinum resistance thermometer, we can follow these steps:

### Step-by-Step Solution:

1. **Identify the Given Values:**
- Resistance at 0°C, \( R_0 = 8 \, \Omega \)
- Resistance at 100°C, \( R_{100} = 12 \, \Omega \)
- Resistance in the hot bath, \( R_T = 11 \, \Omega \)

2. **Calculate the Temperature Coefficient of Resistance (\( \alpha \)):**
The formula for the temperature coefficient of resistance is given by:
\[ \alpha = \frac{R_{100} - R_0}{R_0 \times 100} \]
Substituting the known values:
\[ \alpha = \frac{12 \, \Omega - 8 \, \Omega}{8 \, \Omega \times 100} = \frac{4 \, \Omega}{800 \, \Omega} = \frac{1}{200} \]

3. **Use the Resistance Formula to Find Temperature \( T \):**
The formula relating resistance at temperature \( T \) is:
\[ R_T = R_0 + \alpha \times R_0 \times T \]
Rearranging this to find \( T \):
\[ T = \frac{R_T - R_0}{\alpha \times R_0} \]
Substituting the values we have:
\[ T = \frac{11 \, \Omega - 8 \, \Omega}{\left(\frac{1}{200}\right) \times 8 \, \Omega} \]
Simplifying the numerator:
\[ T = \frac{3 \, \Omega}{\left(\frac{1}{200}\right) \times 8 \, \Omega} \]
\[ T = \frac{3 \, \Omega}{\frac{8 \, \Omega}{200}} = \frac{3 \times 200}{8} = \frac{600}{8} = 75 \, °C \]

4. **Conclusion:**
The temperature of the hot bath is \( 75 \, °C \).