Question

Two rods of same length and cross-sectional area are joined in series. Thermal conductivity of the rods are in ratio of $2:1$.


The ends are maintained at temperatures ${\theta }_A$ and ${\theta }_B$ as shown with ${\theta }_A>{\theta }_B$ and sides are thermally insulated.
Which of the following graph represents temperature gradient $\left(\frac{dT}{dx}\right)$ against $x$ in steady state.

• A.
• B.
• C.
• D.

Answer 1

The correct option is D

Sides are thermally insulated that means no heat transfer is occuring through sides to ambience.
For 1 - dimensional steady state heat transfer.
$\frac{dQ}{dt}=-KA\left(\frac{dT}{dx}\right)$
In steady state $\frac{dQ}{dt}=$ constant at every point
${\left(\frac{dT}{dx}\right)}_1=-\left(\frac{dQ}{dt}\right)\frac{1}{2KA}=\text{constant}=-C{\left(\frac{dT}{dx}\right)}_2=-\left(\frac{dQ}{dt}\right)\frac{1}{KA}=\text{constant}=-2C$
Hence, option (D) is correct.