Question

If e is the electronic charge, c is the speed of light in free space and h is Planck's constant, the quantity $\frac{1}{4\pi\epsilon_0}\frac{e^2}{hc}$ has dimensions of:

• A. $[M L T⁰]$
• B. $[M L T⁻¹]$
• C. $[M⁰ L⁰ T⁰]$
• D. $[L C⁻¹]$

Hint:

The expression $\alpha = \frac{e^2}{4\pi\epsilon_0 \hbar c}$ (where $\hbar = h/2\pi$) is known as the fine-structure constant. It is a fundamental physical constant that is dimensionless. Recognizing this can provide a quick check for the answer.

Answer 1

The Correct Option is C

The given quantity is $\frac{1}{4\pi\epsilon_0}\frac{e^2}{hc}$.
We can determine the dimensions by analyzing the components of the expression.
First, consider the electrostatic force $F$ between two charges $e$ separated by a distance $r$: $F = \frac{1}{4\pi\epsilon_0}\frac{e^2}{r^2}$.
From this, the dimensions of the numerator part can be found: $[\frac{e^2}{4\pi\epsilon_0}] = [F][r^2]$.
The dimensions of force $[F]$ are $[MLT^{-2}]$ and dimensions of distance squared $[r^2]$ are $[L^2]$.
So, $[\frac{e^2}{4\pi\epsilon_0}] = [MLT^{-2}][L^2] = [ML^3T^{-2}]$.
Next, consider the energy of a photon, $E = \frac{hc}{\lambda}$.
From this, the dimensions of the denominator part can be found: $[hc] = [E][\lambda]$.
The dimensions of energy $[E]$ are $[ML^2T^{-2}]$ and dimensions of wavelength $[\lambda]$ are $[L]$.
So, $[hc] = [ML^2T^{-2}][L] = [ML^3T^{-2}]$.
Finally, the dimensions of the entire quantity are the ratio of the dimensions of the two parts:
$\left[\frac{1}{4\pi\epsilon_0}\frac{e^2}{hc}\right] = \frac{[ML^3T^{-2}]}{[ML^3T^{-2}]} = [M^0L^0T^0]$.
Therefore, the quantity is dimensionless.