Question

Two rods of same length and material transfer a given amount of heat in $12$ seconds , when they are joined in parallel. But when they are joined in series , then they will transfer same heat in same condition in

• A. $24s$
• B. $3s$
• C. $1.5s$
• D. $48s$

Answer 1

The correct option is D $48s$

Both rods of same length, material and axis sectional area so their thermal conductivities $\left(k\right)$ would be $be same

also, thermal resistance

$P_m=\frac{L}{KA}\Rightarrow \frac{R_1}{R_2}=\frac{K_2}{K_1}$ [$K\to$ conductivity in series / parallel]

also thermal transfer

$\frac{\Delta Q}{\Delta t}=\frac{\Delta T}{R}$ [$\Delta Q=$ Heat transformed $\Delta T: Temperature difference]

That mass for same $\Delta Q$ & $\Delta T$

$\Delta t\alpha Rt\left(2\right)$ [$\Delta t=$ time taken]

From $\left(1\right)$ & $\left(2\right)$ gives

$\frac{\Delta t_1}{\Delta t_2}=\frac{R_1}{R_2}=\frac{K_2}{K_1}$ given $\Delta t_1=125$

In parallel connection $K_1=2K$ But in series $K_2=K/2$ so:-

$\frac{\Delta t_1}{\Delta t_2}=\frac{K/2}{2K}=\frac{1}{4}\Rightarrow \Delta t_2=4\Delta t_1$

$=4\times 12=48$