question_answer

When a certain current is passed in the circuit as shown in figure, \[10\,\,kcal\] of heat is              produced in \[5\,\,\Omega \] resistance. How much heat      is produced in \[4\,\,\Omega \] resistance?                                      [BHU M-2003]

A)  \[4\,\text{kcal}\]                                            

B)  \[2\,\text{kcal}\]

C)  \[5\,\text{kcal}\]                                            

D)  \[3\,\text{kcal}\]

Correct Answer: B

Solution :

                 Key Idea: When resistors are combined in parallel potential difference between their ends is same. The heat produced in a wire carrying \[i\] and resistance \[R\] for time \[t\] is\[H={{i}^{2}}\,Rt\]. The \[4\,\Omega \] and \[6\,\Omega \] resistances are connected in series, hence equivalent resistance is                                 \[R'=4\,\Omega +6\,\Omega =10\,\Omega \] Now the\[10\,\Omega \] and \[5\,\Omega \] resistors are connected parallel. Let current through \[10\,\Omega \] be\[i\]and through \[5\,\Omega \] be\[{{i}_{2}}\], also potential across ends is same, hence \[V={{i}_{1}}\,{{R}_{1}}={{i}_{2}}\,{{R}_{2}}\]                 \[\Rightarrow \]                               \[{{i}_{1}}\times 10={{i}_{2}}\times 5\]                 \[\Rightarrow \]                               \[{{i}_{2}}=2{{i}_{1}}\]                 \[{{H}_{1}}-i_{1}^{2}{{R}_{1}}\,t\] Is heat produced across \[4\,\Omega \]                 \[{{H}_{2}}=i_{2}^{2}\,{{R}_{2}}\,t\]Is heat produced across\[5\,\Omega \]                                 \[\frac{{{H}_{1}}}{{{H}_{2}}}=\frac{i_{1}^{2}\,4}{{{\left( 2{{i}_{2}} \right)}^{2}}\times 5}=\frac{1}{5}\] Given, \[{{H}_{2}}=10\,kcal\]                                                 \[{{H}_{1}}=\frac{{{H}_{2}}}{5}=\frac{10}{5}=2\,kcal\]