Question

Three rods $A,B$ and $C$ have the same dimensions. Their thermal conductivities are $k_A,k_B$ and $k_C$ respectively. $A$ and $B$ are placed end to end, with their free ends kept at a certain temperature difference.$C$ is placed separately, with its ends kept at the same temperature difference. The two arrangement conduct heat at the same rate. $K_C$ must be equal to

• A. $k_A+k_B$
• B. $\frac{k_A{k}_B}{k_A+k_B}$
• C. $\frac{1}{2}(k_A+k_B)$
• D. $2\left(\frac{k_A{k}_B}{k_A+k_B}\right)$

Answer 1

The correct option is B $\frac{k_A{k}_B}{k_A+k_B}$

We know,

Thermal Resistance$\left(R_{th}\right)=\frac{l}{kA}$

$R_{th\left(eq\right)}$ of A B combination$=R_{th\left(A\right)}+R_{th\left(B\right)}=\frac{l}{A}\frac{k_A+k_B}{k_A{k}_B}$

$R_{th}$ of C $=\frac{l}{k_{C}A}$

$\frac{dQ}{dt}=\frac{\Delta T}{R_{th}}$

For $\frac{dQ}{dt}$ of A and B to be same given that $\Delta T$ of both are same.

$R_{th\left(AC\right)}=R_{th\left(B\right)}$

$\frac{k_A+k_B}{k_A{k}_B}=\frac{1}{k_C}$

$k_C=\frac{k_A{k}_B}{k_A+k_B}$

Hence option $\text{B}$ is correct answer.