Question

$A$ and $B$ are two points on a uniform metal ring whose center is $C$. The angle $ACB$ = $\theta$. $A$ and $B$ are maintained at two different constant temperatures. When $\theta ={180}^o$, the rate of total heat flow from $A$ to $B$ is $1.2$ W. When $\theta ={90}^o$, this rate will be

• A. 0.6 W
• B. 0.9 W
• C. 1.6 W
• D. 1.8 W

Answer 1

The correct option is C 1.6 W

Let $R$ = total thermal resistance of the ring, $\Delta T$ = difference in temperature between $A$ and $B$. For $\theta ={180}^o$, two sections of resistance $R/2$ each are in parallel. Equivalent resistance = $R/4$.
Rate of total heat flow $l_1$ = 1.2 = $\frac{\Delta T}{R/4}$ or 0.3 = $\frac{\Delta T}{R}$
For $\theta ={90}^o$, two sections of resistance R/2 and 3R/4 are in parallel.
Equivalent Resistance = $\frac{\left(R/4\right)\left(3R/4\right)}{R/4+3R/4}=\frac{3R}{16}$
Rate of total heat flow $l_2=\frac{\Delta T}{3R/16}W=1.6W$