Which of the following quantity has the same dimensions as \( \sqrt{\frac{\mu_0}{\epsilon_0}} \)?
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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.
The given quantity is \( \sqrt{\frac{\mu_0}{\epsilon_0}} \), where:
- \( \mu_0 \) is the permeability of free space, and its dimensions are:
\[
[\mu_0] = \frac{\text{kg} \cdot \text{m}^3}{\text{A}^2 \cdot \text{s}^2}
\]
- \( \epsilon_0 \) is the permittivity of free space, and its dimensions are:
\[
[\epsilon_0] = \frac{\text{A}^2 \cdot \text{s}^4}{\text{kg} \cdot \text{m}^3}
\]
Now, let's calculate the dimensions of \( \sqrt{\frac{\mu_0}{\epsilon_0}} \):
\[
\sqrt{\frac{\mu_0}{\epsilon_0}} = \sqrt{\frac{\frac{\text{kg} \cdot \text{m}^3}{\text{A}^2 \cdot \text{s}^2}}{\frac{\text{A}^2 \cdot \text{s}^4}{\text{kg} \cdot \text{m}^3}}}
\]
Simplifying:
\[
\sqrt{\frac{\mu_0}{\epsilon_0}} = \sqrt{\frac{\text{kg}^2 \cdot \text{m}^6}{\text{A}^4 \cdot \text{s}^6 \cdot \text{kg}^2 \cdot \text{m}^6}} = \frac{\text{m}}{\text{A} \cdot \text{s}}
\]
The dimensional formula \( \frac{\text{m}}{\text{A} \cdot \text{s}} \) is the same as the dimension of inductance.
Thus, the correct answer is inductance.
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