A heat flux of 4000 J/s is to be passed through a copper rod of length 10 cm and area of cross section `100cm^2`. The thermal conductivity of copper is `400W//m//^(@)C` The two ends of this rod must be kept at a temperature difference of

Options

A. `1^(@)C`
B. `10^(@)C`
C. `100^(@)C`
D. `1000^(@)C`

Explanation

To solve the problem, we will use the formula for heat transfer through a rod, which is given by:

\[ Q = \frac{k \cdot A \cdot \Delta T}{L} \]

Where:
- \( Q \) is the heat transfer rate (in Watts, which is equivalent to J/s),
- \( k \) is the thermal conductivity of the material (in W/m°C),
- \( A \) is the area of cross-section (in m²),
- \( \Delta T \) is the temperature difference across the length of the rod (in °C),
- \( L \) is the length of the rod (in meters).

### Step 1: Convert given values to SI units
- Length of the rod, \( L = 10 \, \text{cm} = 0.1 \, \text{m} \)
- Area of cross-section, \( A = 100 \, \text{cm}^2 = 0.01 \, \text{m}^2 \)
- Heat flux, \( Q = 4000 \, \text{J/s} = 4000 \, \text{W} \)
- Thermal conductivity of copper, \( k = 400 \, \text{W/m°C} \)

### Step 2: Rearrange the formula to find the temperature difference \( \Delta T \)
We can rearrange the formula to solve for \( \Delta T \):

\[ \Delta T = \frac{Q \cdot L}{k \cdot A} \]

### Step 3: Substitute the known values into the equation
Now, substitute the known values into the equation:

\[ \Delta T = \frac{4000 \, \text{W} \cdot 0.1 \, \text{m}}{400 \, \text{W/m°C} \cdot 0.01 \, \text{m}^2} \]

### Step 4: Calculate \( \Delta T \)
Calculating the numerator and denominator separately:

Numerator:
\[ 4000 \cdot 0.1 = 400 \, \text{W·m} \]

Denominator:
\[ 400 \cdot 0.01 = 4 \, \text{W·m/°C} \]

Now, divide the numerator by the denominator:

\[ \Delta T = \frac{400 \, \text{W·m}}{4 \, \text{W·m/°C}} = 100 \, °C \]

### Final Answer
The temperature difference that must be maintained across the ends of the copper rod is:

\[ \Delta T = 100 \, °C \]