Question

The lengths and radii of two rods made of same material are in the ratios $1:2$ and $2:3$ respectively. If the temperature difference between the ends for the two rods be the same, then in the steady state, the amount of heat flowing per second through them will be in the ratio :

• A. 1 : 3
• B. 4 : 3
• C. 8 : 9
• D. 3 : 2

Answer 1

Answer: (C)

8 : 9

The rate of heat flow in a rod of length $l$ , is given by ,

$Q/t=KA\Delta \theta /l$ ,

where $A=$ cross- sectional area of rod ,

$K=$ thermal conductivity of material of rod ,

$\Delta \theta =$ temperature difference between two ends of rod ,

now given that ratio of lengths of two rods $=1:2$ ,

let their lengths are $l_1=L$ , and $l_2=2L$ respectively ,

and given that ratio of radii of two rods $=2:3$ ,

let their radii are $r_1=2r$ , and $r_2=3r$ respectively ,

hence , their cross-sectional areas are $A_1=2\pi (2r{)}^2$ and $A_2=2\pi (3r{)}^2$ ,

therefore , rate of heat flow in a rod of length $L$ is ,

$X=Q/t=K2\pi (2r{)}^2\Delta \theta /L$ , ...........eq1

and rate of heat flow in a rod of length $2L$ is ,

$Y=Q^{\prime }/t=K2\pi (3r{)}^2\Delta \theta /2L$ , ...........eq2 ,

dividing eq1 by eq2 ,

$X/Y=\left(K2\pi \right(2r{)}^2\Delta \theta /L\left)/\right(K2\pi \left(3r{)}^2\Delta \theta /2L\right)=8:9$ ,