question_answer

The magnetic induction at the centre O in the figure shown is                     [IIT 1988; KCET 2002]

A)            \[\frac{{{\mu }_{0}}i}{4}\left( \frac{1}{{{R}_{1}}}-\frac{1}{{{R}_{2}}} \right)\]            

B)            \[\frac{{{\mu }_{0}}i}{4}\left( \frac{1}{{{R}_{1}}}+\frac{1}{{{R}_{2}}} \right)\]

C)            \[\frac{{{\mu }_{0}}i}{4}({{R}_{1}}-{{R}_{2}})\]          

D)            \[\frac{{{\mu }_{0}}i}{4}({{R}_{1}}+{{R}_{2}})\]

Correct Answer: A

Solution :

                   In the following figure, magnetic fields at O due to sections 1, 2, 3 and 4 are considered as \[{{B}_{1}},\,{{B}_{2}},\,{{B}_{3}}\] and \[{{B}_{4}}\] respectively. \[{{B}_{1}}={{B}_{3}}=0\]                    \[{{B}_{2}}=\frac{{{\mu }_{0}}}{4\pi }.\frac{\pi \,i}{{{R}_{1}}}\otimes \]                    \[{{B}_{4}}=\frac{{{\mu }_{0}}}{4\pi }.\frac{\pi \,i}{{{R}_{2}}}\]¤       As \[|{{B}_{2}}|\,\,>\,\,|{{B}_{4}}|\]            So \[{{B}_{net}}={{B}_{2}}-{{B}_{4}}\Rightarrow {{B}_{net}}=\frac{{{\mu }_{0}}i}{4}\left( \frac{1}{{{R}_{1}}}-\frac{1}{{{R}_{2}}} \right)\otimes \]