The temperature gradient in a rod of `0.5 m` length is `80^(@)C//m`. It the temperature of hotter end of the rod is `30^(@)C`, then the temperature of the cooler end is

Options

A. `40^(@)C`
B. `-10^(@)C`
C. `10^(@)C`
D. `0^(@)C`

Explanation

To find the temperature of the cooler end of the rod, we can use the concept of temperature gradient. The temperature gradient is defined as the change in temperature per unit length of the rod.

### Step-by-Step Solution:

1. **Identify the given values:**
- Length of the rod, \( L = 0.5 \, \text{m} \)
- Temperature gradient, \( \frac{dT}{dx} = 80 \, \text{°C/m} \)
- Temperature at the hotter end, \( T_h = 30 \, \text{°C} \)

2. **Set up the equation for temperature gradient:**
The temperature gradient can be expressed as:
\[ \frac{T_h - T_c}{L} = \frac{dT}{dx} \]
where \( T_c \) is the temperature at the cooler end.

3. **Substitute the known values into the equation:**
\[ \frac{30 - T_c}{0.5} = 80 \]

4. **Multiply both sides by \( 0.5 \) to eliminate the denominator:**
\[ 30 - T_c = 40 \]

5. **Rearrange the equation to solve for \( T_c \):**
\[ T_c = 30 - 40 \]

6. **Calculate \( T_c \):**
\[ T_c = -10 \, \text{°C} \]

### Final Answer:
The temperature of the cooler end of the rod is \( -10 \, \text{°C} \).

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