Question

A toroidal inductor with an inductance of $90\,mH$ encloses a volume of $0.0200\,{m^3}$ if the average energy density in the toroid is $70.0\,J{m^{ - 3}}$ , what is current through inductor?

Answer 1

Hint: In order to solve this question we need to understand Faraday’s law of electromagnetic induction which states that whenever there is change in magnetic flux through loop an emf induced in it and hence current flows in loops according to Lenz’s law. So whenever a loop is connected with an emf and emf is eventually starts to decrease then back emf is induced in loop and it is directly proportional to rate of change of current where proportionality constant is known as inductance and this phenomena is known as self-inductance.

Complete step by step answer:
Given Inductance, $L = 90 \times {10^{ - 3}}H$
Energy density is given by, $u = \dfrac{1}{2}L{i^2} \to (i)$
Where $U = \dfrac{u}{V}$
Given, $U = 70\,J{m^{ - 3}}$
Given, $V = 0.02\,{m^3}$
Putting values we get, $u = UV$
$u = (70 \times 0.02)\,J$
$\Rightarrow u = 1.4\,J$
Putting value of $u$ in equation $(i)$
We get, ${i^2} = \dfrac{{2u}}{L}$
${i^2} = \dfrac{{(2 \times 1.4)}}{{90 \times {{10}^{ - 3}}}}$
$\Rightarrow {i^2} = 31.1{A^2}$
$\Rightarrow i = \sqrt {31.1} A$
$\therefore i = 5.576\,A$

So the current through the inductor is, $i = 5.576\,A$.

Note: It should be remembered that toroid is several layer binding of wire so when current is decreasing in any loop then it induces emf in another loop according to the rule of induction. Lenz law states that the direction of current induced is such that it opposes the cause of its generation. While flux through loops can be changed by three ways, first is by changing magnetic field, second is by changing area under loop and third is by changing the orientation of loop in magnetic field. Lenz law is in accordance with inertia in classical dynamics of the body where the body opposes the change in its state.