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  • Find the magnetic energy stored in the system of two concentric loops of radii r and R (R>>r). Find the mutual inductance of the system. If the loops have currents&nb

Find the magnetic energy stored in the system of two concentric loops of radii r and R (R>>r).

Find the mutual inductance of the system. If the loops have currents of same magnitude io, The magnetic energy stored is: Ignore the effect of self inductance

 

Option: 1

\frac{\pi \mu_0 \mathrm{r}^2 \mathrm{i}_0^2}{\mathrm{R}}


Option: 2

\frac{\pi \mu_0 \mathrm{r}^2 \mathrm{i}_0^2}{2 \mathrm{R}}


Option: 3

\frac{2 \pi \mu_0 \mathrm{r}^2 \mathrm{i}_0^2}{\mathrm{R}}


Option: 4

\frac{3 \pi \mu_0 r^2 i_0^2}{2 R}}


Answers (1)

best_answer

Let a current is set up in the loop 1. It develops a magnetic
field of induction   \mathrm {B_1=\frac{\mu_0 i}{2 R}}    at its center.

The flux linked by the (smaller) loop 2

\begin{aligned} & =\mathrm \phi_{12}=\mathrm{B}_1\left(\pi \mathrm{r}^2\right) \\ & \Rightarrow \quad \phi_{12}=\pi \mathrm{B}_{1 \mathrm{r}^2} \end{aligned}

Putting the value of  \mathrm {B_1=\frac{\mu_0 i}{2 R}}   we obtain mutual inductance

\mathrm {M=\frac{\phi_{12}}{i}=\frac{\pi \mu_0 r^2}{2 R}}

Ignoring the self inductance of the loop, the magnetic energy

Stored =\mathrm{U}=\mathrm{MI}_1 \mathrm{I}_2 \\

\mathrm{U}=\frac{\pi \mu_0 \mathrm{r}^2 \mathrm{i}_0^2}{2 \mathrm{R}} \quad\left(\because \mathrm{i}_1=\mathrm{i}_2=\mathrm{i}_0\right)

Posted by

Deependra Verma

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