question_answer

In the circuit diagram, heat produces in R, 2R and 1.5R are in the ratio of

A)  \[4:2:3\]        

B)  \[8:4:27\]

C)  \[2:4:3\]       

D)  \[27:8:4\]

Correct Answer: B

Solution :

:          Current in resistance R is \[{{I}_{1}}=\frac{12R}{R+2R}=\frac{2}{3}I\] Heat produced in resistance R in time t is \[{{H}_{1}}=I_{1}^{2}Rt={{\left( \frac{2}{3}I \right)}^{2}}Rt\] Current in resistance 2R is \[{{I}_{2}}=\frac{IR}{2R+R}=\frac{1}{3}I\] Heat produced in resistance 2R in same time t is \[{{H}_{2}}=I_{3}^{2}2RT=\left( \frac{1}{3}I \right)2RT\] Current in resistance 1.5R is \[{{I}_{3}}=I\] Heat produced in resistance 1.5R in same time t is \[{{H}_{3}}=I_{3}^{2}(1.5R)t={{I}^{2}}(1.5R)t\] \[\therefore \]\[{{H}_{1}}:{{H}_{2}}:{{H}_{3}}={{\left( \frac{2}{3}I \right)}^{2}}Rt:{{\left( \frac{1}{3}t \right)}^{2}}2Rt:{{I}^{2}}(1.5R)\] \[=\frac{4}{9}:\frac{2}{9}:\frac{3}{2}=8:4:27\]