Question

The dimension of the quantity $\frac{1}{{\epsilon }_0}\frac{e^2}{hc}$ is $(e=$ charge of electron, $h=$ Planck's constant and $c=$ velocity of light)

• A. $\left[M^{-1}{L}^{-3}{T}^{2}A\right]$
• B. $\left[M^0{L}^0{T}^0{A}^0\right]$
• C. $\left[M{L}^3{T}^{-4}{A}^{-2}\right]$
• D. $\left[M^{-1}{L}^{-3}{T}^4{A}^2\right]$

Answer 1

Answer: (B)

[Energy] $=\left[\frac{hc}{\lambda }\right]$
or $[hc]=[$ Energy $][\lambda ]=[$ Force $][\text{ Distance }{]}^2$
Also, [Force] $=\frac{\left[e^2\right]}{\left[{\epsilon }_0\right][\text{ Distance }{]}^2}$
or $\frac{\left[e^2\right]}{\left[{\epsilon }_0\right]}=[$ Force $][\text{ Distance }{]}^2$
$\therefore [hc]=\frac{\left[e^2\right]}{\left[{\epsilon }_0\right]}$
or $\left[\frac{e^2}{{\epsilon }_0}\cdot \frac{1}{hc}\right]$
$=\left[M^0{L}^0{T}^0{A}^0\right]$