Question

What is the dimensional formula of resistance?
$\text{A}\text{. }\left[ {{M}^{-1}}L{{T}^{4}}{{A}^{-2}} \right]$
$\text{B}\text{. }\left[ M{{L}^{2}}{{T}^{-5}}{{A}^{-3}} \right]$
$\text{C}\text{. }\left[ {{M}^{-1}}{{L}^{-2}}{{T}^{4}}{{A}^{2}} \right]$
$\text{D}\text{. none of these}$

Answer 1

Hint: Use the formula given by Ohm’s law, i.e. V=iR to find the dimensional formula of resistance. The dimensional formula of current (i) is [A]. Potential difference is the work done per unit charge. Calculate the dimensional formula of work done and charge.

Complete step by step answer:
Resistance is the ability of a conductor of given length (l) and cross sectional area (A), to resist or oppose the flow of charges through it when a potential difference is created across the conductor.
Resistance of a conductor is directly proportional to its length and inversely proportional to its cross sectional area.
Resistance of a conductor also depends on the resistivity of the material of the conductor. Resistance is directly proportional to the resistivity of the material of the conductor. Therefore, the resistivity will be the resistance of that conductor.
Consider a conductor of length l and cross sectional area A. Let the resistivity of the material be $\rho$. Then the resistance R of the conductor is given as $R=\dfrac{\rho l}{A}$.
We know that when a conductor of resistance R is connected to a potential difference of V, a current i flows in the conductor. According to Ohm’s law, V = iR.
This gives us $R=\dfrac{V}{i}$.
Therefore, dimensional formula of resistance will be $\left[ R \right]=\left[ \dfrac{V}{i} \right]=\dfrac{\left[ V \right]}{\left[ i \right]}$ …… (i).
Potential difference between two points is the work done by the electric force to move a unit charge from one point to another. This means that potential difference or voltage is $\dfrac{\text{work done}}{\text{charge}}$.
Therefore, $\left[ V \right]=\left[ \dfrac{\text{work done}}{\text{charge}} \right]=\dfrac{\left[ \text{work done} \right]}{\left[ \text{charge} \right]}$ ….. (ii).
Since work done is equal to forces times displacement, (i.e. W = Fd) the dimensional formula of work done is $\left[ W \right]=\left[ Fd \right]=\left[ F \right]\left[ d \right]$.
We know that the dimensional formula of force is $\left[ F \right]=\left[ ML{{T}^{-2}} \right]$
The dimensional formula of displacement is [L].
Hence, $\left[ W \right]=\left[ F \right]\left[ d \right]=\left[ ML{{T}^{-2}} \right]\left[ L \right]=\left[ M{{L}^{2}}{{T}^{-2}} \right]$.
The dimensional formula of charge is [AT].
[A] is the dimensional formula of current.
Substitute the dimensional formulas of work done and charge in equation (ii).
Therefore,
$\left[ V \right]=\dfrac{\left[ \text{work done} \right]}{\left[ \text{charge} \right]}=\dfrac{\left[ M{{L}^{2}}{{T}^{-2}} \right]}{\left[ AT \right]}=\left[ M{{L}^{2}}{{T}^{-3}}{{A}^{-1}} \right]$.
And [i] = [AT].
Substitute the dimensional formulas of potential difference and current in equation (i).
Therefore,
$\left[ R \right]=\dfrac{\left[ V \right]}{\left[ i \right]}=\dfrac{\left[ M{{L}^{2}}{{T}^{-3}}{{A}^{-1}} \right]}{\left[ A \right]}=\left[ M{{L}^{2}}{{T}^{-3}}{{A}^{-2}} \right]$.
Therefore, the dimensional formula of resistance is $\left[ M{{L}^{2}}{{T}^{-3}}{{A}^{-2}} \right]$.
However, none of the first three options match with the answer we found.
Hence, the correct option is D.

Note: Do not confuse between resistance and resistivity of a conductor. Resistivity is material property. It depends only on the material of the conductor. It does not depend on the shape and size of the conductor. Whereas the resistance does depend on the shape and size of the conductor.
Also, note that charge is not a fundamental physical quantity. Current is considered as one of the fundamental quantities. Since current is charge flowing in one unit of time, charge will be current times the time.