Question

Find the dimension formula for inductance and also the dimension formula for resistance.

Answer 1

In order to find the dimensional formula for inductance and resistance, first of all we will learn a few basic things about dimensional formulas. We will then write the equations for inductance and resistance and then convert them into fundamental quantities to find their dimensional formula.

Formula used:
(1) Formula for self-induced voltage: $V=L\dfrac{dI}{dt}$
Where $V$ is the voltage, $L$ is the inductance and $\dfrac{dI}{dt}$ is the rate of change in current.
(2) Ohm’s law: $V=IR$
Where $V$ is the voltage, $I$ is the current and $R$ is the resistance.

Complete step by step solution:
(A) In order to find out the dimensional formula for inductance, first of all we will write down the formula for induced voltage:
$V=L\dfrac{dI}{dt}$
On simplifying and making inductance as subject
$\begin{aligned} & \Rightarrow L=\dfrac{V}{\dfrac{dI}{dt}} \\\ & \\\ \end{aligned}$
$\Rightarrow L=\dfrac{W}{q\times \dfrac{dI}{dt}}$
\left\\{ \begin{aligned} & Voltage=\dfrac{Work}{Ch\arg e} \\\ & \Rightarrow V=\dfrac{W}{q} \\\ \end{aligned} \right\\}
Now, writing the above equation in terms of their fundamental quantities
$\Rightarrow L=\dfrac{\left[ M{{L}^{2}}{{T}^{-2}} \right]}{\left[ AT \right]\times \left[ \dfrac{A}{T} \right]}$
Where, \left\\{ \begin{aligned} & W=\left[ M{{L}^{2}}{{T}^{-2}} \right] \\\ & q=\left[ AT \right] \\\ & \dfrac{dI}{dt}=\left[ \dfrac{A}{T} \right] \\\ \end{aligned} \right\\}
$\Rightarrow L=\left[ M{{L}^{2}}{{T}^{-2}}{{A}^{-2}} \right]$
Therefore the dimensional formula for inductance is $\left[ M{{L}^{2}}{{T}^{-2}}{{A}^{-2}} \right]$ where dimensions $M$(Mass), $L$(Length), $T$(Time) and $A$(Ampere) are used.
(B) Similarly to find out the dimensional formula for resistance, we write the Ohm’s law
$V=IR$
On simplifying and making the resistance as subject
$\Rightarrow R=\dfrac{V}{I}$
$\Rightarrow R=\dfrac{W}{qI}$
\left\\{ \begin{aligned} & Voltage=\dfrac{Work}{Ch\arg e} \\\ & \Rightarrow V=\dfrac{W}{q} \\\ \end{aligned} \right\\}
Now, writing the above equation in terms of fundamental quantities
$\Rightarrow R=\dfrac{\left[ M{{L}^{2}}{{T}^{-2}} \right]}{\left[ AT \right]\times \left[ A \right]}$
Where, \left\\{ \begin{aligned} & W=\left[ M{{L}^{2}}{{T}^{-2}} \right] \\\ & q=\left[ AT \right] \\\ & I=A \\\ \end{aligned} \right\\}
$\Rightarrow R=\left[ M{{L}^{2}}{{T}^{-3}}{{A}^{-2}} \right]$
Therefore, the dimensional formulae for resistance is $\left[ M{{L}^{2}}{{T}^{-3}}{{A}^{-2}} \right]$ where dimensions $M$(Mass), $L$(Length), $T$(Time) and $A$(Ampere) are used.

Additional information:
The Dimensional Formula of the Physical Quantity is a collection of expressions or formulae that tell us how and which of the fundamental quantities are present in a physical quantity. Dimensional formulae may also be used to convert units between systems. It is a fundamental feature of units and measures and has many real-world applications.
If there is a physical quantity X that is dependent on the base dimensions $M$(Mass), $L$(Length), and $T$(Time) with powers$a$, $b$, and$c$, then its dimensional formula is:
$\left[ {{M}^{a}}{{L}^{b}}{{T}^{c}} \right]$
A square bracket $\left[ {} \right]$ is often used to close a dimensional formula.
Dimensional Equations are the equations that result when we equal a physical quantity with its dimensional formulae. The dimensional equation aids in the expression of physical quantities in terms of fundamental or base quantities.
If there is a physical quantity X that is dependent on the base dimensions $M$(Mass), $L$(Length), and $T$(Time) with powers$a$, $b$, and$c$, then its dimensional equation is:
$X=\left[ {{M}^{a}}{{L}^{b}}{{T}^{c}} \right]$

Note:
Fundamental quantities are quantities that are not dependent on other quantities. Fundamental units are the units that are used to calculate certain fundamental quantities. The derived quantities are those that are derived from the fundamental quantities. Derived units are the units that are used to calculate these derived quantities.