Question

Wires $A$ and $B$ have identical lengths and have circular cross-sections. The radius of $A$ is twice the radius of $B$ i.e. $R_{A}=2 R_{B}$. For a given temperature difference between the two ends, both wires conduct heat at the same rate. The relation between the thermal conductivities is given by :

• A. $K_{A}=4 K_{B}$
• B. $K_{A}=2 K_{B}$
• C. $K_{A}=K_{B} / 2$
• D. $K_{A}=K_{B} / 4$

Answer 1

Answer: (D)

We use $r=\frac{Q}{t}=\frac{k A \Delta T}{L}$
As $r_{A}=r_{B}$
$\Rightarrow K_{A} A_{A}=K_{B} A_{B}$
$\Rightarrow K_{A} \pi R_{A}{ }^{2}=K_{B} \pi R_{B}^{2}$
$\Rightarrow K_{A}\left(2 R_{B}\right)^{2}=K_{B} R_{B}{ }^{2}$
$\Rightarrow K_{A}=\frac{K_{B}}{4}$