question_answer

A wire loop PQRSP is constructed by joining two semi-circular coils of radius\[{{\operatorname{r}}_{\,1}}\]and\[{{\operatorname{r}}_{\,2}}\] respectively, as shown in the figure. If the current flowing in the loop is I, then the magnetic induction at the point o is:

A) \[\frac{{{\mu }_{\operatorname{o}}}\operatorname{I}}{4}\left[ \frac{1}{{{r}_{1}}}-\frac{1}{{{r}_{2}}} \right]\]              

B) \[\frac{{{\mu }_{\operatorname{o}}}\operatorname{I}}{4}\left[ \frac{1}{{{r}_{1}}}+\frac{1}{{{r}_{2}}} \right]\]

C) \[\frac{{{\mu }_{\operatorname{o}}}\operatorname{I}}{2}\left[ \frac{1}{{{r}_{1}}}-\frac{1}{{{r}_{2}}} \right]\]              

D) \[\frac{{{\mu }_{\operatorname{o}}}\operatorname{I}}{2}\left[ \frac{1}{{{r}_{1}}}+\frac{1}{{{r}_{2}}} \right]\]

Correct Answer: A

Solution :

Magnetic field due to coil of radii \[{{\operatorname{r}}_{1}}\] \[{{\vec{B}}_{1}}=\frac{1}{2}\times \frac{{{\mu }_{0}}\operatorname{I}}{2{{\operatorname{r}}_{1}}}\odot \,\,\,\operatorname{out}\,\,of\,\,plane\] Magnetic field due to coil radii \[{{\operatorname{r}}_{2}}\] \[{{\vec{B}}_{2}}=\frac{1}{2}\times \frac{{{\mu }_{0}}\operatorname{I}}{2{{\operatorname{r}}_{2}}}\odot \,\,\,\operatorname{in}\,\,of\,\,plane\] \[\operatorname{B}\propto \frac{1}{\operatorname{r}}\,\,\operatorname{hence}\,\,{{\vec{B}}_{\operatorname{net}}}=\frac{{{\mu }_{0}}\operatorname{I}}{4{{\operatorname{r}}_{1}}}-\frac{{{\mu }_{0}}\operatorname{I}}{4{{\operatorname{r}}_{1}}}\Rightarrow \frac{{{\mu }_{0}}\operatorname{I}}{4}\left( \frac{1}{{{r}_{1}}}-\frac{1}{{{r}_{2}}} \right)\]