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 of finding the temperature difference required to pass a heat flux of 4000 J/s through a copper rod, we can use the formula for heat conduction, which is derived from Fourier's law:

\[ \frac{dQ}{dt} = k \cdot A \cdot \frac{\Delta T}{\Delta x} \]

Where:
- \(\frac{dQ}{dt}\) is the heat flux (in watts, which is equivalent to J/s),
- \(k\) is the thermal conductivity of the material (in W/m·°C),
- \(A\) is the cross-sectional area (in m²),
- \(\Delta T\) is the temperature difference (in °C),
- \(\Delta x\) is the length of the rod (in meters).

### Step-by-step Solution:

1. **Identify the given values:**
- Heat flux, \(\frac{dQ}{dt} = 4000 \, \text{J/s}\)
- Length of the rod, \(\Delta x = 10 \, \text{cm} = 0.1 \, \text{m}\)
- Area of cross-section, \(A = 100 \, \text{cm}^2 = 100 \times 10^{-4} \, \text{m}^2 = 0.01 \, \text{m}^2\)
- Thermal conductivity of copper, \(k = 400 \, \text{W/m·°C}\)

2. **Rearrange the formula to solve for \(\Delta T\):**
\[ \Delta T = \frac{\frac{dQ}{dt} \cdot \Delta x}{k \cdot A} \]

3. **Substitute the values into the equation:**
\[ \Delta T = \frac{4000 \, \text{J/s} \cdot 0.1 \, \text{m}}{400 \, \text{W/m·°C} \cdot 0.01 \, \text{m}^2} \]

4. **Calculate the denominator:**
\[ k \cdot A = 400 \cdot 0.01 = 4 \, \text{W/°C} \]

5. **Now substitute this back into the equation:**
\[ \Delta T = \frac{4000 \cdot 0.1}{4} \]

6. **Perform the calculations:**
\[ \Delta T = \frac{400}{4} = 100 \, \text{°C} \]

### Final Answer:
The temperature difference required across the ends of the copper rod is \(100 \, \text{°C}\).

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