Question

Define the unit of self inductance

Answer 1

Hint: Change in flux produces induced emf. If we are equating this induced emf and change in current through the coil, then the proportionality constant will be the self inductance.

Complete step by step answer:
As we know the flux is proportional to current.
\[\phi =Li\], where \[\phi \] is the flux, \[L\] is the self inductance and \[i\] is the current.
Change in flux in a coil produces an induced emf. This is known as inductance effect. The induced emf is proportional to the rate of change of current through coil and this proportionality constant is called the self inductance.

\[e=-L\dfrac{di}{dt}\], where L is the self inductance, e is the induced emf and \[\dfrac{di}{dt}\] is the rate of change of current through the coil. The negative sign indicates that generated emf is opposing the cause producing it. Henry (H) is the SI unit of inductance.
Both self inductance and mutual inductance use the Henry to represent the inductance.

\[H=kg{{m}^{2}}{{s}^{-2}}{{A}^{-2}}\]

One Henry can write like this also, one-kilogram meter squared per second square per ampere squared.

When a current change at the rate of 1 ampere per second and induced emf is one volt, the self inductance of the coil will be one Henry.

Additional information: We can find the self inductance of a solenoid.

Consider a solenoid of length l with n number of turns. If a current i flows through this solenoid, a magnetic field will generate inside.

Magnetic field, \[B={{\mu }_{0}}\dfrac{Ni}{l}\]

Each turn have area A, then the total magnetic flux through the solenoid is given by; \[{{\phi }_{\mathbf{B}}}={{\mu }_{0}}\dfrac{Ni}{l}AN\]

i.e. \[{{\phi }_{\mathbf{B}}}={{\mu }_{0}}\dfrac{{{N}^{2}}i}{l}A\]

Varying current will generate induced emf.

So, induced emf, \[e=-\dfrac{d}{dt}\left[ {{\mu }_{0}}\dfrac{{{N}^{2}}i}{l}A \right]\]

Negative sign indicates the generated emf is opposing the cause producing it.

Constant terms can be taken outside, \[e=-{{\mu }_{0}}{{N}^{2}}\dfrac{A}{l}\dfrac{di}{dt}\]
These constant terms are collectively treated as proportionality constant and known as self inductance.

\[e=-L\dfrac{di}{dt}\]

Therefore, \[L=\dfrac{{{\mu }_{0}}{{N}^{2}}A}{l}\]

\[L={{\mu }_{0}}{{n}^{2}}Al\], where n is the number of turns (N) per unit length (l)

Note: Self inductance and mutual inductance are using the same unit Henry. One Henry can be written like this also; one-kilogram meter squared per second square per ampere squared.