question_answer

The magnetic induction at the centre O in the figure shown is  

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 are O due to sections 1, 2, 3 and 4 are considered as \[{{B}_{1}},\,\,{{B}_{2}},\,\,{{B}_{3}}\,\,and\,\,{{B}_{4}}\] respectively. \[{{\operatorname{B}}_{1}}={{B}_{3}}=0\] \[{{B}_{2}}=\frac{{{\mu }_{0}}}{4\pi }.\frac{\pi i}{{{R}_{1}}}\otimes \] \[{{B}_{4}}=\,\,\frac{{{m}_{0}}}{4p}.\frac{pi}{{{R}_{1}}}\odot \,\,\,As\,\,\,\left| {{B}_{2}} \right|>\left| {{B}_{4}} \right|\] \[{{B}_{net}}={{B}_{2}}-{{B}_{4}}\,\,\,\Rightarrow \,\,{{B}_{net}}=\frac{{{\mu }_{0}}i}{4}\left( \frac{1}{{{R}_{1}}}-\frac{1}{{{R}_{2}}} \right)\otimes \]