Question

Which of the following quantity has the same dimensions as \( \sqrt{\frac{\mu_0}{\epsilon_0}} \)?

• A. Voltage
• B. Resistance
• C. Inductance
• D. Capacitance

Hint:

The quantity \( \sqrt{\frac{\mu_0}{\epsilon_0}} \) has the same dimensions as inductance, as it relates to the speed of light in vacuum and appears in the formula for inductance.

Answer 1

The Correct Option is C

The quantity provided is \( \sqrt{\frac{\mu_0}{\epsilon_0}} \). The dimensions are as follows:- \( \mu_0 \) (permeability of free space): \[ [\mu_0] = \frac{\text{kg} \cdot \text{m}^3}{\text{A}^2 \cdot \text{s}^2} \] - \( \epsilon_0 \) (permittivity of free space): \[ [\epsilon_0] = \frac{\text{A}^2 \cdot \text{s}^4}{\text{kg} \cdot \text{m}^3} \]Calculating the dimensions of \( \sqrt{\frac{\mu_0}{\epsilon_0}} \):\[\sqrt{\frac{\mu_0}{\epsilon_0}} = \sqrt{\frac{\frac{\text{kg} \cdot \text{m}^3}{\text{A}^2 \cdot \text{s}^2}}{\frac{\text{A}^2 \cdot \text{s}^4}{\text{kg} \cdot \text{m}^3}}}\]Simplification yields:\[\sqrt{\frac{\mu_0}{\epsilon_0}} = \sqrt{\frac{\text{kg}^2 \cdot \text{m}^6}{\text{A}^4 \cdot \text{s}^6}} = \frac{\text{m}}{\text{A} \cdot \text{s}}\]The dimensional formula \( \frac{\text{m}}{\text{A} \cdot \text{s}} \) corresponds to inductance.Therefore, the correct answer is inductance.