Question

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

• A. \( qI / \mu_0 \)
• B. \( q \mu_0 I \)
• C. \( q^2 \mu_0 I \)
• D. \( q \mu_0 / I \)

Hint:

To match the dimensions of momentum, use the fact that momentum has the dimensions \( \text{M} \cdot \text{L} \cdot \text{T}^{-1} \) and check for the correct expression.

Answer 1

The Correct Option is B

We are given: \[ Q = AT \] \[ I = A \] \[ \mu_0 = \text{ML}^3 \text{T}^{-2} \text{A}^{-2} \] Now, we need to find the dimensions of the product \( P = Q \mu_0 I \). The dimension of \( P \) is calculated as follows: \[ P = Q \mu_0 I = [AT] [\text{ML}^3 \text{T}^{-2} \text{A}^{-2}] [A] \] This simplifies to: \[ P = [\text{M}^1 \text{L}^1 \text{T}^{-2} \text{A}^1] \] Now, we check the dimensions of momentum: \[ \text{Momentum} = \text{M} \cdot \text{L} \cdot \text{T}^{-1} \] We find that the dimensions of \( P \) are the same as that of momentum. Therefore, the correct answer is: \[ q \mu_0 I \]