Given a charge $q$, current I and permeability of vacuum $\mu_0$. Which of the following quantity has the…

Given a charge $q$, current I and permeability of vacuum $\mu_0$. Which of the following quantity has the dimension of momentum?
  1. $\mathrm{qI} / \mu_0$
  2. $q \mu_0 I$
  3. $q^2 \mu_0 I$
  4. $q \mu_0 / I$

Solution

$\begin{aligned} & \mathrm{Q}=\mathrm{AT} \\ & \mathrm{I}=\mathrm{A} \\ & \mu_0=\mathrm{MLT}^{-2} \mathrm{~A}^{-2} \\ & \mathrm{P}=\mathrm{Q}^{\mathrm{x}} \mu_0^{\mathrm{y}} \mathrm{I}^{\mathrm{z}}=[\mathrm{AT}]^{\mathrm{x}}\left[\mathrm{MLT}^{-2} \mathrm{~A}^{-2}\right]^{\mathrm{y}}[\mathrm{A}]^{\mathrm{z}} \\ & \mathrm{MLT}^{-1}=\mathrm{M}^{\mathrm{y}} \mathrm{L}^{\mathrm{y}} \mathrm{T}^{\mathrm{x}-2 \mathrm{y}} \mathrm{A}^{-2 \mathrm{y}+\mathrm{z+x}} \\ & \text { Now; } \mathrm{y}=1 \\ & \mathrm{x}-2 \mathrm{y}=-1 \\ & -2 \mathrm{y}+\mathrm{z}=0 \\ & \therefore \mathrm{x}=\mathrm{y}=\mathrm{z}=1\end{aligned}$

Asked in: JEE Main 2025 (02 Apr Shift 2)

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