Write the dimensional formula for thermal conductivity.

Options

A. `MLT^(-2)K`
B. `MLT^(-2)K^(-2)`
C. `MLT^(-3)K^(-1)`
D. `MLT^(-3)K`

Answer

To find the dimensional formula for thermal conductivity, we start with the formula for thermal conductivity \( k \):

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

Where:
- \( Q \) is the heat transferred,
- \( \Delta x \) is the change in length,
- \( A \) is the area,
- \( \Delta T \) is the change in temperature,
- \( t \) is the time.

### Step 1: Identify the dimensional formulas of each quantity.

1. **Heat (Q)**:
- Heat is equivalent to work done, which has the dimensional formula:
\[ [Q] = [M L^2 T^{-2}] \]

2. **Change in length (\( \Delta x \))**:
- The dimensional formula for length is:
\[ [\Delta x] = [L] \]

3. **Area (A)**:
- The dimensional formula for area is:
\[ [A] = [L^2] \]

4. **Change in temperature (\( \Delta T \))**:
- The dimensional formula for temperature is:
\[ [\Delta T] = [K] \]

5. **Time (t)**:
- The dimensional formula for time is:
\[ [t] = [T] \]

### Step 2: Substitute the dimensional formulas into the thermal conductivity formula.

Now, substituting these into the formula for thermal conductivity:

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

Substituting the dimensional formulas we derived:

\[ [k] = \frac{[M L^2 T^{-2}] \cdot [L]}{[L^2] \cdot [K] \cdot [T]} \]

### Step 3: Simplify the expression.

This simplifies to:

\[ [k] = \frac{[M L^3 T^{-2}]}{[L^2 K T]} \]

Breaking it down further:

\[ [k] = [M] \cdot [L^{3-2}] \cdot [T^{-2-1}] \cdot [K^{-1}] \]

This results in:

\[ [k] = [M L^1 T^{-3} K^{-1}] \]

### Final Answer

Thus, the dimensional formula for thermal conductivity \( k \) is:

\[ [k] = [M L^1 T^{-3} K^{-1}] \]