Match List I with List II $\begin{array}{|l|l|r|l|} \hline \text{ LIST I } & \text{ LIST II } \\ \hline…

Match List I with List II $\begin{array}{|l|l|r|l|} \hline \text{ LIST I } & \text{ LIST II } \\ \hline \text{A.} & \text{Torque} & \text{I.} & {\left[M^1 L^1 T^{-2} A^{-2}\right]} \\ \hline \text{B.} & \text{Magnetic field} & \text{II.} & {\left[L^2 A^1\right]} \\ \hline \text{C.} & \text{Magnetic moment} & \text{III.} & {\left[M^1 T^{-2} A^{-1}\right]} \\ \hline \text{D.} & \text{Permeability of free space} & \text{IV.} & {\left[M^1 L^2 T^{-2}\right]} \\ \hline \end{array}$ Choose the correct answer from the options given below:
  1. A-III, B-I, C-II, D-IV
  2. A-IV, B-II, C-III, D-I
  3. A-IV, B-III, C-II, D-I
  4. A-I, B-III, C-II, D-IV

Solution

$\begin{aligned} & {[\vec{\tau}]=[\overrightarrow{\mathrm{r}} \times \overrightarrow{\mathrm{F}}]=\left[\mathrm{ML}^2 \mathrm{~T}^{-2}\right]} \\ & {[\mathrm{F}]=[\mathrm{qVB}]} \\ & \Rightarrow \mathrm{B}=\left(\frac{\mathrm{F}}{\mathrm{qV}}\right)=\left[\frac{\mathrm{MLT}^{-2}}{\mathrm{ATLT}^{-1}}\right]=\left[\mathrm{MA}^{-1} \mathrm{~T}^{-2}\right] \\ & {[\mathrm{M}]=[\mathrm{I} \times \mathrm{A}]=\left[\mathrm{AL}^2\right]} \\ & \mathrm{B}=\frac{\mu_0}{4 \pi} \frac{\mathrm{Idl} \sin \theta}{\mathrm{r}^2} \\ & \Rightarrow[\mu]=\left[\frac{\mathrm{Br}^2}{\mathrm{Idl}}\right]=\left[\frac{\mathrm{MT}^{-2} \mathrm{~A}^{-1} \times \mathrm{L}^2}{\mathrm{AL}}\right] \\ & =\left[\mathrm{MLT}^{-2} \mathrm{~A}^{-2}\right]\end{aligned}$

Asked in: JEE Main 2024 (06 Apr Shift 1)

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