Question

Two rods \(A\) and \(B\) of different materials are welded together as shown in figure. Their thermal conductivities are \(K_1\) and \(K_2\). The thermal conductivity of the composite rod will be :

• A. $\frac{3(K_1 + K_2)}{2}$
• B. $K_1 + K_2$
• C. $2( K_1 + K_2)$
• D. $\frac{K_1 + K_2}{2} $

Answer 1

The Correct Option is D

To find the effective thermal conductivity of the composite rod, we need to consider the configuration of the rods. Here, rods \(A\) and \(B\) are in parallel configuration.

In a parallel configuration, the effective thermal conductivity \(K_{\text{eff}}\) is given by the area-weighted average of the individual conductivities:

K_{\text{eff}} = \frac{K_1 + K_2}{2}

The reasoning is based on the assumption that each rod has the same cross-sectional area and the heat flow is conducted in parallel. Therefore, the correct formula for the effective thermal conductivity of the composite rod is the arithmetic mean of the two thermal conductivities.

By using the given formula, we find the effective thermal conductivity of the composite system:

• Substitute the respective thermal conductivities \(K_1\) and \(K_2\) into the formula.
• The effective thermal conductivity is calculated as \frac{K_1 + K_2}{2}.

Therefore, the correct answer is:

• \frac{K_1 + K_2}{2}

This matches the given correct answer. Thus, the effective thermal conductivity of the composite rod is indeed \frac{K_1 + K_2}{2}.