Question

The inductive current in the ring varies with time as:
\[ {{\rm{ I}} = {{\rm{I}}_{\rm{m}}}\sin (\omega {\rm{t}} - \phi )\\
{\rm{ where }}\,\,{{\rm{I}}_{\rm{m}}} = }\omega {\phi _0}\sqrt {{{\bf{R}}^2} + {\omega ^2}\;{{\rm{L}}^2}}\,\, {\rm{ with }}\tan \phi = \omega {\rm{L}}/{\bf{R}}{\rm{ }}{\rm{. }}\]

Answer 1

Hint: The tendency of an electrical conductor to resist a difference in the electric current flowing through it is known as inductance in electromagnetism and electronics. A magnetic field is generated around a conductor by the movement of electric current. An inductor is a type of electronic component that adds inductance to a circuit.

Complete step by step answer:
The tendency of an electrical conductor to resist a difference in the electric current flowing through it is known as inductance in electromagnetism and electronics. A magnetic field is generated around a conductor by the movement of electric current. The field power is proportional to the current magnitude and follows any increases in current.

The shifting magnetic flux causes an emf in the metal ring, causing the ring to produce a large current. The ring is propelled by the Lorentz force, which exists between the magnetic field and the induced current. The current in the ring is induced by the axial magnetic field of the iron heart.

An alternating power source is used to power the induction coil. The metal ring is thrust upwards into the air as an alternating current is applied to the coil. The current induced in the metal ring generates a magnetic field that opposes the induction coil's field.
\[\varepsilon = - \dfrac{{{\rm{d}}\Phi }}{{{\rm{dt}}}} \\
\Rightarrow \varepsilon = \omega {\Phi _0}\sin \omega {\rm{t}} \\
\Rightarrow \varepsilon = {\rm{LI}} + {\rm{RI}}\]
Put
\[{\bf{I}} = {{\bf{I}}_{\rm{m}}}\sin (\omega {\bf{t}} - \phi )\]
Then
\[\omega {\Phi _0}\sin \omega {\rm{t}} = \omega {\Phi _0}\sin (\omega {\rm{t}} - \phi )\cos \phi + \cos (\omega {\rm{t}} - \phi )\sin \phi \]
So,
\[{\bf{R}}{{\bf{I}}_{\rm{m}}} = \omega {\Phi _0}\cos \phi {\rm{\text{ and } L}}{{\rm{I}}_{\rm{m}}}={\bf{R}}{{\bf{I}}_{\rm{m}}} = {{\bf{\Phi }}_0}\sin \phi \]
Or it can be rewritten as
\[{{\bf{I}}_{\rm{m}}} = \dfrac{{\omega {\Phi _0}}}{{\sqrt {{{\bf{R}}^2} + {\omega ^2}{{\bf{L}}^2}} }}\,{\rm{ and }}\,\tan \phi = \dfrac{{\omega {\rm{L}}}}{{\bf{R}}}\]

Note: Any transition of magnetic field through a circuit causes an electromotive force (EMF) (voltage) in the conductors, according to Faraday's law of induction, a phenomenon known as electromagnetic induction. The induced voltage generated by the changing current acts to counteract the current transition. Lenz's law states this, and the voltage is referred to as back EMF.