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  • Two concentric coplanar circular loops 1 and 2 are shown in the Fig 19. The radii of the loops are r and R. Current I flows in the loop 1. Find the magnetic flux <img alt="\Phi_2" src="https

Two concentric coplanar circular loops 1 and 2 are shown in the Fig 19. The radii of the loops are r and R. Current I flows in the loop
1. Find the magnetic flux \Phi_2 through the loop 2 if r << R

Option: 1

\mathrm{\frac{\mu_0 \pi r^2}{2 R}}


Option: 2

\mathrm{\frac{\mu_0 \pi r^2}{R}}


Option: 3

\mathrm{\frac{2 \mu_0 \pi r^2}{R}}


Option: 4

\mathrm{\frac{\mu_0 \pi r^2}{4 R}}


Answers (1)

best_answer

The direct calculation of the flux \Phi_2 is very complicated because the magnetic field is not uniform over the big loop 2. However, the application of the reciprocity theorem greatly simplifies the solution of the problemLet us pass the same current I through the loop 2. Then the magnetic flux \Phi_1 created by this current through loop 1 can be easily found because magnetic field is more or less uniform in the small loop 1 (because r << R) Thus, magnetic field at the centre of the loops is

\mathrm{B=\frac{\mu_0 I}{2 R}}

and the magnetic flux through the loop 1 is

\mathrm{\Phi_1=\left(\pi r^2\right) B=\frac{\mu_0 \pi r^2 I}{2 R}}

Using reciprocity theorem

\mathrm{\Phi_2=\Phi_1=\frac{\mu_0 \pi r^2 I}{2 R}}

And the coefficient of mutual inductance is

\mathrm{M=\frac{\mu_0 \pi r^2}{2 R}}

 

 

Posted by

Devendra Khairwa

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