question_answer

For ensuring dissipation of same energy in all three resistors \[({{R}_{1}},\,{{R}_{2}},\,{{R}_{3}})\] connected as shown in figure, their values must be related as ]                                                            [AIIMS 2005]

A)            \[{{R}_{1}}={{R}_{2}}={{R}_{3}}\]                                    

B)            \[{{R}_{2}}={{R}_{3}}\]and \[{{R}_{1}}=4{{R}_{2}}\]

C)            \[{{R}_{2}}={{R}_{3}}\]and \[{{R}_{1}}=\frac{1}{4}{{R}_{2}}\]

D)            \[{{R}_{1}}={{R}_{2}}+{{R}_{3}}\]

Correct Answer: C

Solution :

                   As the voltage in \[{{R}_{2}}\]and \[{{R}_{3}}\]is same therefore, according to,                    \[H=\frac{{{V}^{2}}}{R}.t,\] \[{{R}_{2}}={{R}_{3}}\]                               Also the energy in all resistance is same.                    \         \[{{i}^{2}}{{R}_{1}}t=i_{1}^{2}{{R}_{2}}t\]                    Using \[{{i}_{1}}=\frac{{{R}_{3}}}{{{R}_{2}}+{{R}_{3}}}i=\frac{{{R}_{3}}}{{{R}_{3}}+{{R}_{3}}}i=\frac{1}{2}i\] Thus \[{{i}^{2}}{{R}_{1}}t=\frac{{{i}^{2}}}{4}{{R}_{2}}t\] or, \[{{R}_{1}}=\frac{{{R}_{2}}}{4}\]