Question

Consider the following two statements A and B. Identify the correct answer.
A) The quantity $\frac{e^2}{{ϵ}_{0}ch}$ is dimensionless.
B) $\frac{1}{\sqrt{{\mu }_0{ϵ}_0}}$ has the dimensions of velocity and is numerically equal of velocity of light.

• A. B is true but A is false
• B. A is true but B is false
• C. Both A and B are false
• D. Both A and B are true

Answer 1

Answer: (D)

Dimensions of permittivity ${ϵ}_0$: $A^2{T}^4{M}^{-1}{L}^3$

Dimensions of $e^2$ : $A^2{T}^2$

Dimensions of $c$ : $L{T}^{-1}$

Dimensions of $h$ : $M{L}^2{T}^{-1}$

Therefore, dimensions of the given expression:

$\frac{A^2{T}^2}{\left(A^2{T}^4{M}^{-1}{L}^3\right)\times \left(L{T}^{-1}\right)\times \left(M{L}^2{T}^{-1}\right)}=1$ which is dimensionless.

Hence, Statement A is true.

In statement B, the expression given is for the speed of light.Hence, it has dimensions of velocity.