Step 1: Understanding the Concept:
Heat transfer through a solid material by conduction is governed by Fourier's law of heat conduction.
In a steady state, the rate of heat flow is constant and uniform throughout the cross-section of the rod.
Step 2: Key Formula or Approach:
Fourier's law for one-dimensional heat conduction states that the rate of heat transfer \( H \) (which is \( Q/t \)) is proportional to the cross-sectional area \( A \) and the temperature gradient.
The formula is:
\[ H = \frac{Q}{t} = \frac{K \cdot A \cdot \Delta T}{L} \]
where \( K \) is the coefficient of thermal conductivity, \( A \) is the area, \( \Delta T \) is the temperature difference, and \( L \) is the length of the conductor.
Step 3: Detailed Explanation:
From the problem description, we have the following parameters:
- Rate of heat transfer \( = \frac{Q}{t} \)
- Length of the rod \( = x \)
- Cross-sectional area \( = A \)
- Temperature difference \( \Delta T = T_1 - T_2 \) (since \( T_1>T_2 \))
Substitute these specific variables into the standard heat conduction formula:
\[ \frac{Q}{t} = \frac{K \cdot A \cdot (T_1 - T_2)}{x} \]
We need to find the expression for the coefficient of thermal conductivity '\( K \)'. Rearrange the equation to solve for \( K \).
Multiply both sides by \( x \):
\[ \left(\frac{Q}{t}\right) \cdot x = K \cdot A \cdot (T_1 - T_2) \]
Now, divide both sides by \( A(T_1 - T_2) \):
\[ K = \frac{\left(\frac{Q}{t}\right) \cdot x}{A \cdot (T_1 - T_2)} \]
This can be rewritten more neatly as:
\[ K = \frac{x Q}{t A (T_1 - T_2)} \]
This expression matches option (B).
Step 4: Final Answer:
The coefficient of thermal conductivity is \( \frac{x Q}{t A(T_1-T_2)} \).