Question

The unit of \(\dfrac{1}{\mu_0 \varepsilon_0}\) is:

• A. \(\text{m s}^{-1}\)
• B. \(\text{m}^{-1}\,\text{s}\)
• C. \(\text{m}^{2}\,\text{s}^{-2}\)
• D. \(\text{m}^{-2}\,\text{s}^{2}\)

Hint:

Recall \(c = 1/\sqrt{\mu_0\varepsilon_0}\), so \(1/(\mu_0\varepsilon_0)\) equals \(c^2\). Find the unit of speed squared.

Answer 1

The Correct Option is C

Step 1: Recall Maxwell's result linking electromagnetism and light: the wave speed of an electromagnetic wave in vacuum is \(v = 1/\sqrt{\mu_0\varepsilon_0}\), and this equals the measured speed of light \(c\).
Step 2: Rearrange for the required quantity. Squaring gives \(\dfrac{1}{\mu_0\varepsilon_0} = v^2 = c^2\).
Step 3: Dimensional check. Speed has dimension \([L T^{-1}]\), so \(c^2\) has dimension \([L^2 T^{-2}]\).
Step 4: Convert dimension to SI units: length in metre (m), time in second (s), giving \(\text{m}^2\,\text{s}^{-2}\). This confirms option (iii).
\[\boxed{[L^2T^{-2}] = \text{m}^2\,\text{s}^{-2}}\]