Question

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

Answer 1

Hint: Use the expression for Ohm’ law. This expression gives the relation between the potential difference across the ends of the conductor, current flowing through the conductor and resistance of the conductor. Substitute the dimensions of all the remaining physical quantities and determine the dimensional formula for the resistance.

Formula used:
The expression for Ohm’s law for the potential difference is as follows:
\[V = IR\] …… (1)
Here, \[V\] is the potential difference across the ends of the conductor, \[I\] is the electric current flowing through the conductor and \[R\] is the resistance of the conductor.

Complete step by step solution:
We can determine the dimensional formula for resistance of a conductor using equation (1).
Rearrange equation (1) for the resistance of the conductor.
\[R = \dfrac{V}{I}\] …… (2)

If we substitute the dimensions of potential difference and electric current in the above equation, we can determine the dimensional formula for resistance of the conductor.
The dimensional formula potential difference is \[\left[ {M{L^2}{T^{ - 3}}{A^{ - 1}}} \right]\].
The dimensional formula for electric current is \[\left[ A \right]\].

Substitute \[\left[ {M{L^2}{T^{ - 3}}{A^{ - 1}}} \right]\] for \[V\] and \[\left[ A \right]\] for \[I\] in equation (2).
\[R = \dfrac{{\left[ {M{L^2}{T^{ - 3}}{A^{ - 1}}} \right]}}{{\left[ A \right]}}\]
\[ \Rightarrow R = \left[ {M{L^2}{T^{ - 3}}{A^{ - 1}}} \right]{\left[ A \right]^{ - 1}}\]
\[ \Rightarrow R = \left[ {M{L^2}{T^{ - 3}}{A^{ - 2}}} \right]\]

Therefore, the dimensional formula for resistance is \[\left[ {M{L^2}{T^{ - 3}}{A^{ - 2}}} \right]\] which is not given in any of the options.

So, the correct answer is “Option D”.

Note:
One can also derive the dimensional formula for resistance of the conductor using the equation for resistance of wire in terms of specific resistance of the material of the wire, length of the wire and cross-sectional area of the wire. Substitute the dimensions of all these physical quantities and derive the dimensional formula for resistance.