The dimensions of coefficient of thermal conductivity is

Answer

To find the dimensions of the coefficient of thermal conductivity, we start with the formula for thermal conductivity (K):

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

Where:
- \( K \) = Thermal conductivity
- \( Q \) = Heat transferred
- \( \Delta x \) = Thickness (change in length)
- \( A \) = Area
- \( \Delta T \) = Change in temperature

### Step 1: Identify the dimensions of each quantity
1. **Heat (Q)**: The dimension of heat is given by the formula for energy. The dimension of energy is:
\[ [Q] = [M][L^2][T^{-2}] \quad \text{(Mass × Velocity²)} \]
So, \( [Q] = M L^2 T^{-2} \).

2. **Thickness (\(\Delta x\))**: This is a length dimension:
\[ [\Delta x] = [L] \]

3. **Area (A)**: The dimension of area is:
\[ [A] = [L^2] \]

4. **Change in temperature (\(\Delta T\))**: The dimension of temperature is:
\[ [\Delta T] = [\Theta] \quad \text{(Temperature)} \]
However, for dimensional analysis, we can consider it as a separate dimension.

### Step 2: Substitute dimensions into the formula
Now, substituting these dimensions into the formula for thermal conductivity:

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

Substituting the dimensions we found:

\[ K = \frac{(M L^2 T^{-2}) \cdot (L)}{(L^2) \cdot [\Delta T]} \]

### Step 3: Simplify the expression
Now, simplifying the right-hand side:

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

### Step 4: Final expression for dimensions
Thus, we can express the dimensions of thermal conductivity as:

\[ [K] = M L T^{-2} [\Delta T]^{-1} \]

If we denote the dimension of temperature as \( K^{-1} \), we can write:

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

### Final Answer
The dimensions of the coefficient of thermal conductivity are:

\[ \boxed{M L T^{-3} \Theta^{-1}} \]