Question

The $SI$ unit of inductance, the henry can be written as
(A) ${\text{weber}}/{\text{ampere}}$
(B) ${\text{volt}}\,{\text{second}}/{\text{ampere}}$
(C) ${\text{joule}}/{\text{amper}}{{\text{e}}^2}$
(D) all of the above

Answer 1

The inductance is the change in the electric current that results in the electromotive force of the inductor. The Henry is the standard system of unit for the inductance. It is the derived unit from many fundamental units in many ways and this quantity is named after the scientist Henry.

Useful formula:
(1) The formula of the self-inductance is given by

$L = \dfrac{\phi }{i}$

Where $L$ is self-inductance, $\phi$ is the magnetic flux and $i$ is the current.

(2) The formula of the electromotive force is given as

$e = - L\left( {\dfrac{{di}}{{dt}}} \right)$

Where $e$ is the emf and $\dfrac{d}{{dt}}$ is the change with respect to time.

(3) The formula of the energy is given as

$U = \dfrac{1}{2}L{I^2}$

Where $U$ is the energy and $I$ is the current.

Complete step by step solution:
Use the formula of the self-inductance,

$L = \dfrac{\phi }{i}$

The unit of the magnetic flux is weber and the current is ampere. Hence the unit of the inductance is obtained as ${\text{weber}}/{\text{ampere}}$ . ------------(1)
Using the formula of the emf,

$e = - L\left( {\dfrac{{di}}{{dt}}} \right)$
Rearranging the above equation,
$L = - e\left( {\dfrac{{dt}}{{di}}} \right)$
The unit of the emf is volt, time is second and the current is ampere. Hence the unit of the self-inductance is obtained as ${\text{volt}}\,{\text{second}}/{\text{ampere}}$ . -------------(2)
Using the formula of the energy,

$U = \dfrac{1}{2}L{I^2}$

By rearranging the above equation, we get

$L = \dfrac{{2U}}{{{I^2}}}$

The unit of the energy is Joule and the unit of the current is ampere. Hence the unit of the self-inductance is obtained as ${\text{joule}}/{\text{amper}}{{\text{e}}^2}$ . ---------------(3)
From the equation (1), (2) and (3), the self-inductance of the inductance is obtained as all the three options.

Thus the option (C) is correct.

Note: The inductance is the ability of the conductor to oppose the change in electric current. Hence this change produces the magnetic field around the conductor. The substituted units are from the standard system of units, since the Henry is the $SI$ unit.