Figure shows a circular loop with radius -r- The resistance of are ABC is 5- and that of ADC is 10- Magnetic field at the centre of the loop is-

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Question

Figure shows a circular loop with radius $^{'} r^{'}$ . The resistance of are $A B C$ is $5 Ω$ and that of $A D C$ is $10 Ω$ . Magnetic field at the centre of the loop is:-

\frac{μ_{0} I}{3 r} ⨂

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\frac{μ_{0} I}{15 r} ⨂

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\frac{μ_{0} I}{12 r} ⨀

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\frac{μ_{0} I}{6 r} ⨀

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Solution

The correct options are
A $\frac{μ_{0} I}{3 r} ⨂$
C $\frac{μ_{0} I}{12 r} ⨀$

Resistance $A B C = 5 Ω$ , Resistance $A D C = 10 Ω$
Current $A B C = \frac{2}{3} I$ , Current $A D C = \frac{1}{3} I$
Magnetic field due to $A B C$
$B_{1} = \frac{μ_{0} I_{A B C}}{4 R}$ out of plane.
$= \frac{μ_{0} I}{6 R}$
Similarly, $B_{2} = \frac{μ_{0} I_{A D C}}{4 R}$
$= \frac{μ_{0} I}{12 R}$
$⟹ B_{n e t} (B_{1} - B_{2}) = \frac{μ_{0} I}{12 R}$

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