Question

Dimensions of electrical resistances are:
A: \[[M{{L}^{2}}{{T}^{-3}}{{A}^{-1}}]\]
B: \[[M{{L}^{2}}{{T}^{-3}}{{A}^{-2}}]\]
C: \[[M{{L}^{3}}{{T}^{-3}}{{A}^{-2}}]\]
D: \[[M{{L}^{3}}{{T}^{3}}{{A}^{2}}]\]

Answer 1

Hint: Dimensional analysis is used to represent the dimensions of any quantity in terms of 7 fundamental quantities. We have 7 basic quantities in physics. These are length, time, mass, temperature, amount of substance and luminous intensity. Their SI units are called fundamental units and they cannot be derived by using other units of measurement.

Complete step by step answer:
In order to find the dimensions of resistance, we shall write first a relationship involving the same.
We know electrical energy or heat is given by the relation: \[H={{i}^{2}}Rt\], where H is the heat, i is the current and t is the time. The dimensions of heat is \[[M{{L}^{2}}{{T}^{-2}}]\], for current it is \[[A]\]and for time it is \[[T]\]
  $H={{i}^{2}}Rt \\ $
 $\Rightarrow R=\dfrac{H}{{{i}^{2}}t} \\ $
 $\Rightarrow R=\dfrac{[M{{L}^{2}}{{T}^{-2}}]}{[{{A}^{2}}][T]} \\ $
 $\therefore R=[M{{L}^{2}}{{T}^{-3}}{{A}^{2}}] \\ $

So, the correct answer is “Option B”.

Additional Information:
Foot is used as the standard unit of measurement of length in the United States of America. Foot is given preference to the meter there in commercial and engineering measurements. The foot was used by the Greeks, Romans, Chinese & French.

Note:
Resistance is not a fundamental quantity but electric current is. When we are doing problems of dimensional analysis then we have to keep in mind that we have to write every given quantity in terms of those 7 fundamental quantities. When we say that the unit is a fundamental quantity then we imply we cannot derive that. For example, Watt is not a fundamental unit because it cannot be derived and can be written as Joules/s.