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

Given:\[Q = AT\]\[I = A\]\[\mu_0 = \text{ML}^3 \text{T}^{-2} \text{A}^{-2}\]Calculate the dimensions of \( P = Q \mu_0 I \).The dimensions of \( P \) are derived as:\[P = Q \mu_0 I = [AT] [\text{ML}^3 \text{T}^{-2} \text{A}^{-2}] [A]\]Simplifying yields:\[P = [\text{M}^1 \text{L}^1 \text{T}^{-2} \text{A}^1]\]Comparing with the dimensions of momentum:\[\text{Momentum} = \text{M} \cdot \text{L} \cdot \text{T}^{-1}\]The dimensions of \( P \) match those of momentum. The correct expression is:\[q \mu_0 I\]