આપેલ પરિપથ માટે $O $ બિંદુ પાસે ચુંબકીયક્ષેત્ર આપેલ છે તો નીચે પૈકી કયું સાચું થાય?

$(i)$ $(ii)$ $(iii)$

(A) $\frac{{{\mu _0}i}}{r}$ $\otimes$ (A) $\frac{{{\mu _0}i}}{4}\left( {\frac{1}{{{r_1}}} - \frac{1}{{{r_2}}}} \right)$ $\otimes$ (A) $\frac{{{\mu _0}i}}{4}\left( {\frac{1}{{{r_1}}} - \frac{1}{{{r_2}}}} \right)$ $\otimes$

(B) $\frac{{{\mu _0}i}}{{2r}}$ $\odot$ (B) $\frac{{{\mu _0}i}}{4}\left( {\frac{1}{{{r_1}}} + \frac{1}{{{r_2}}}} \right)$ $\otimes$ (B) $\frac{{{\mu _0}i}}{4}\left( {\frac{1}{{{r_1}}} + \frac{1}{{{r_2}}}} \right)$ $\otimes$

(C) $\frac{{{\mu _0}i}}{{4r}}$ $\otimes$ (C) $\frac{{{\mu _0}i}}{4}\left( {\frac{1}{{{r_1}}} - \frac{1}{{{r_2}}}} \right)$ $\odot$ (C)$\frac{{{\mu _0}i}}{4}\left( {\frac{1}{{{r_1}}} - \frac{1}{{{r_2}}}} \right)$ $\odot$

(D) $\frac{{{\mu _0}i}}{{4r}}$ $\odot$ (D) $0$ (D) $0$

Question

આપેલ પરિપથ માટે $O $ બિંદુ પાસે ચુંબકીયક્ષેત્ર આપેલ છે તો નીચે પૈકી કયું સાચું થાય?

$(i)$	$(ii)$	$(iii)$
(A) $\frac{{{\mu _0}i}}{r}$ $\otimes$	(A) $\frac{{{\mu _0}i}}{4}\left( {\frac{1}{{{r_1}}} - \frac{1}{{{r_2}}}} \right)$ $\otimes$	(A) $\frac{{{\mu _0}i}}{4}\left( {\frac{1}{{{r_1}}} - \frac{1}{{{r_2}}}} \right)$ $\otimes$
(B) $\frac{{{\mu _0}i}}{{2r}}$ $\odot$	(B) $\frac{{{\mu _0}i}}{4}\left( {\frac{1}{{{r_1}}} + \frac{1}{{{r_2}}}} \right)$ $\otimes$	(B) $\frac{{{\mu _0}i}}{4}\left( {\frac{1}{{{r_1}}} + \frac{1}{{{r_2}}}} \right)$ $\otimes$
(C) $\frac{{{\mu _0}i}}{{4r}}$ $\otimes$	(C) $\frac{{{\mu _0}i}}{4}\left( {\frac{1}{{{r_1}}} - \frac{1}{{{r_2}}}} \right)$ $\odot$	(C)$\frac{{{\mu _0}i}}{4}\left( {\frac{1}{{{r_1}}} - \frac{1}{{{r_2}}}} \right)$ $\odot$
(D) $\frac{{{\mu _0}i}}{{4r}}$ $\odot$	(D) $0$	(D) $0$

Diffcult

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Solution

b
$(i)$ $(c)$ $B_1 = B_3 = 0$

${B_2} = \frac{{{\mu _0}}}{{4\pi }}.\frac{{\pi i}}{r}$ $\otimes$

${B_{net}} = {B_1} + {B_2} + {B_3} = \frac{{{\mu _0}}}{{4\pi }}.\frac{{\pi \,i}}{r}$ $\otimes$ $⇒$ ${B_{net}} = \frac{{{\mu _0}i}}{{4r}} \otimes $

$(ii)$ $(b)$ $B_1 = B_3 = 0$

${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}}} \otimes $

So, ${B_{net}} = {B_2} + {B_4} = \frac{{{\mu _0}}}{{4\pi }}.\pi \,i\left( {\frac{1}{{{r_1}}} + \frac{1}{{{r_2}}}} \right) \otimes $

$(iii)$ $(a)$ $B_1 = B_3 = 0$

${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}}} \otimes $ $|{B_2}|\,\, > \,\,|{B_4}|$

${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 $

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