Question

The dimensions of self-induction are:
A. $[ML{{T}^{-2}}]$
B. $[M{{L}^{2}}{{T}^{-1}}{{A}^{-2}}]$
C. $[M{{L}^{2}}{{T}^{-2}}{{A}^{-2}}]$
D. $[M{{L}^{2}}{{T}^{-2}}{{A}^{2}}]$

Answer 1

Hint: Dimension of self-induction is the same as that of inductance. Write inductance in terms of magnetic flux and current. Next, express the magnetic flux in terms of field and area. Since the dimensions of simple terms like field, area and current are known, using dimensional analysis derive the formula for induction.

Formula used: Formula of flux relating it with current and inductance:
\[\phi =LI\]
Formula for flux relating it with magnetic field and area:
\[\phi =B.A\]
Basic dimensional formulae for some terms:
$\begin{align}
  & [B]=[{{M}^{1}}{{L}^{0}}{{T}^{-2}}{{A}^{-1}}] \\
 & [I]=[{{M}^{0}}{{L}^{0}}{{T}^{0}}{{A}^{1}}] \\
 & [A]=[{{M}^{0}}{{L}^{2}}{{T}^{0}}{{A}^{0}}] \\
\end{align}$

Complete step-by-step answer:
Every quantity can be expressed in the terms of the following seven dimensions
Dimension - Symbol
Length - L
Mass - M
Time - T
Electric charge - Q
Luminous intensity - C
Temperature - $\theta $
Angle - None

The dimensions of self-inductance must be the same as that of an inductance. We will stick with impedance as finding the dimensions for it is easier. Inductance is defined as the property of an electric conductor that causes an electromotive force to be generated by a change in the current flowing.
It is related to the flux as the following equation:
\[\phi =LI\]
Where Flux is represented as $\Phi $, flux is defined as the number of field lines passing through a surface. Inductance is represented by $L$. And $I$ is the current in the wire.
From the basic definition of flux, the following formula is used,
\[\phi =B.A\]
Where, $B$ is the magnetic field.
Dimensions of magnetic field, area and current are known to us.
$\begin{align}
  & [B]=[{{M}^{1}}{{L}^{0}}{{T}^{-2}}{{A}^{-1}}] \\
 & [I]=[{{M}^{0}}{{L}^{0}}{{T}^{0}}{{A}^{1}}] \\
 & [A]=[{{M}^{0}}{{L}^{2}}{{T}^{0}}{{A}^{0}}] \\
\end{align}$
Where the formula for inductance becomes:
\[\begin{align}
  & L=\dfrac{BA}{I} \\
 & [L]=\dfrac{[B][A]}{[I]} \\
 & \Rightarrow \dfrac{[{{M}^{1}}{{L}^{0}}{{T}^{-2}}{{A}^{-1}}][{{M}^{0}}{{L}^{2}}{{T}^{0}}{{A}^{0}}]}{[{{M}^{0}}{{L}^{0}}{{T}^{0}}{{A}^{1}}]} \\
 & \Rightarrow [M{{L}^{2}}{{T}^{-2}}{{A}^{-2}}] \\
\end{align}\]
Hence, option C. is the correct answer.

Note: Firstly, the options given to us have an ‘A’ dimension. This dimension is used for current and represents ‘Amperes’. It is not a basic unit. Current is defined as the flow of charge per unit time, therefore in terms of basic dimensions, $[A]=[Q{{T}^{-1}}]$. Keep this in mind as it may lead to some mistakes. Secondly, since we are asked about inductance, the flux and the fields are magnetic flux and fields. Do not apply dimensions of electrical flux and field in this question as it will give you the incorrect answer.