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Chapter 12: Problem 1836

Match the physical quantities given in column I with their dimensional formulae given in column II - I stands for the dimension of current. \(\begin{array}{ll}\text { Column I } & \text { Column II }\end{array}\) (a) Electromotive force (emf) (p) \(\mathrm{ML}^{2} \mathrm{~T}^{-3} \mathrm{~A}^{-2}\) (b) Resistance (q) \(\mathrm{ML}^{3} \mathrm{~T}^{-3} \mathrm{~A}^{-2}\) (c) Resistivity (r) \(\mathrm{M}^{-1} \mathrm{~L}^{-3} \mathrm{~T}^{3} \mathrm{~A}^{2}\) (d) Conductivity (s) \(\mathrm{ML}^{2} \mathrm{~T}^{-3} \mathrm{~A}^{-1}\) (A) \(a-s, b-p, c-q, d-r\) (B) \(a-p, b-s, c-r, d-p\) (C) \(a-p, b-s, c-r, d-q\) (D) \(a-r, b-p, c-q, d-s\)

Short Answer

Expert verified
The short answer is: (A) \(a-s, b-p, c-q, d-r\)

Step by step solution

01

Find the dimensions of Electromotive force (emf)

The electromotive force (emf) is the work done per unit charge. The dimension of work is [M¹L²T⁻²], and the dimension of charge is [A¹T¹], so the dimension of emf is: \[ \frac{[M^1 L^2 T^{-2}]}{[A^1 T^1]} = [M^1 L^2 T^{-3} A^{-1}]. \] Looking at Column II, emf matches with (s).
02

Find the dimensions of Resistance

Resistance is given by Ohm's Law: V = IR, where V is the voltage (dimensions of emf which is [M¹L²T⁻³A⁻¹]), I is current (dimension [A¹]), and R is the resistance. The dimension of resistance is: \[ \frac{[M^1 L^2 T^{-3} A^{-1}]}{[A^1]} = [M^1 L^2 T^{-3} A^{-2}]. \] Looking at Column II, Resistance matches with (p).
03

Find the dimensions of Resistivity

Resistivity is given by the equation R = ρ(L/A) where R is resistance (dimensions of [M¹L²T⁻³A⁻²]), L is length (dimension [L¹]), and A is the area (dimension [L²]). The dimension of resistivity is: \[ \frac{[M^1 L^2 T^{-3} A^{-2}]}{[L^1]} \cdot [L^2] = [M^1 L^3 T^{-3} A^{-2}]. \] Looking at Column II, Resistivity matches with (q).
04

Find the dimensions of Conductivity

Conductivity is the inverse of resistivity. So the dimensions of conductivity are the inverse of the dimensions of resistivity, which are [M¹L³T⁻³A⁻²]: \[ \frac{1}{[M^1 L^3 T^{-3} A^{-2}]} = [M^{-1} L^{-3} T^3 A^2]. \] Looking at Column II, Conductivity matches with (r).
05

Choose the correct answer

As per our analysis, we have: (a) with (s), (b) with (p), (c) with (q), and (d) with (r). The correct option is: (A) \(a-s, b-p, c-q, d-r\)

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