Dimension of 1 0 0 should be equal to: — Solution | JEE Physics

MCQEasyJEE 2023Dimensions & Dimensional Analysis

Dimension of $\frac{1}{\mu_0 \epsilon_0}$ should be equal to:

Answer

Correct answer:D

Standard Method

Given: We use the relationship between $\mu_0$ , $\epsilon_0$ , and the speed of light.

Find: The dimensions of $\frac{1}{\mu_0 \epsilon_0}$ .

From the relation,

c = \frac{1}{\sqrt{\mu_0 \epsilon_0}}

we get,

\frac{1}{\mu_0 \epsilon_0} = c^2

Now, the dimensional formula of speed of light $c$ is

[c] = \frac{L}{T}

Therefore,

\left[\frac{1}{\mu_0 \epsilon_0}\right] = [c^2] = \frac{L^2}{T^2}

Hence, the dimension is $\frac{L^2}{T^2}$ . Therefore, the correct option is D.

Using $c = \frac{L^2}{T^2}$ instead of $[c] = \frac{L}{T}$ is incorrect because $\frac{L^2}{T^2}$ is the dimension of $c^2$ , not $c$ . First write the dimension of speed correctly, then square it.
Forgetting to square the relation after using $c = \frac{1}{\sqrt{\mu_0 \epsilon_0}}$ leads to the wrong dimension $\frac{L}{T}$ . Since $\frac{1}{\mu_0 \epsilon_0} = c^2$ , the final dimension must be squared.

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