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1.Units, Dimensions and Measurement
hard
The dimensions of ${e^2}/4\pi {\varepsilon _0}hc$, where $e,\,{\varepsilon _0},\,h$ and $c$ are electronic charge, electric permittivity, Planck’s constant and velocity of light in vacuum respectively
A$[{M^0}{L^0}{T^0}]$
B$[{M^1}{L^0}{T^0}]$
C$[{M^0}{L^1}{T^0}]$
D$[{M^0}{L^0}{T^1}]$
Solution
(a) $[e] = [AT],$${ \in _0} = [{M^{ – 1}}{L^{ – 3}}{T^4}{A^2}],$ $[h] = [M{L^2}{T^{ – 1}}]$
and $[c] = [L{T^{ – 1}}]$
$\therefore $$\left[ {\frac{{{e^2}}}{{4\pi { \in _0}hc}}} \right] = \left[ {\frac{{{A^2}{T^2}}}{{{M^{ – 1}}{L^{ – 3}}{T^4}{A^2} \times M{L^2}{T^{ – 1}} \times L{T^{ – 1}}}}} \right]$
$ = [{M^0}{L^0}{T^0}]$
and $[c] = [L{T^{ – 1}}]$
$\therefore $$\left[ {\frac{{{e^2}}}{{4\pi { \in _0}hc}}} \right] = \left[ {\frac{{{A^2}{T^2}}}{{{M^{ – 1}}{L^{ – 3}}{T^4}{A^2} \times M{L^2}{T^{ – 1}} \times L{T^{ – 1}}}}} \right]$
$ = [{M^0}{L^0}{T^0}]$
Standard 11
Physics
