Question

What is the dimensions of the ratio \[\dfrac{{{e^2}}}{{4\pi { \in _0}G{M_p}^2}}\] where, $G = $ Gravitational constant and ${M_p} = $ Mass of a proton.
A. $[ML{T^{ - 3}}{A^{ - 1}}]$
B. $[{M^0}{L^0}{T^0}{A^0}]$
C. $[M{L^3}{T^{ - 3}}{A^{ - 1}}]$
D. $[{M^{ - 1}}{L^{ - 3}}{T^4}{A^2}]$

Answer 1

Hint: In physics, there are seven fundamental base units which are used to express every other physical quantity and these units are Mass Length Time Temperature Mole Electric current Amount of light. They all have their own respective representation symbol. These seven fundamental physical units can derive any other complex physical quantity units and dimensions.

Complete step by step answer:
In given ration which we have, \[\dfrac{{{e^2}}}{{4\pi { \in _0}G{M_p}^2}}\] now, $e$ Is the charge on electrons whose dimension can be written as $[AT]$. $\dfrac{1}{{4\pi { \in _0}}}$ Is the permittivity of free space constant term used in force between two charges defined by coulomb’s law whose dimension can be written as $[N{L^2}{A^{ - 2}}{T^{ - 2}}]$.

$G$ Is the gravitational constant which appears in the force formula governed by the gravitational force of attraction between two bodies and this dimension can be written as $[N{L^2}{M^{ - 2}}]$. ${M_p}$ Is the mass of a proton and its dimension is simply of mass which can be written simply as $[{M^2}]$.

Now, in order to find dimensions of ratio \[\dfrac{{{e^2}}}{{4\pi { \in _0}G{M_p}^2}}\] we will add all parameters dimension simultaneously, we get $[{A^2}{T^2}]$ $[N{L^2}{A^{ - 2}}{T^{ - 2}}]$ divided by the dimensions of $[N{L^2}{M^{ - 2}}]$ $[{M^2}]$. Remember $N$ is a unit force newton which can simply be written as $[ML{T^{ - 2}}]$ hence,
$[M{L^3}{T^{ - 2}}]$ divided by the dimension $[M{L^3}{T^{ - 2}}]$.
Or we can write,
$\dfrac{{[M{L^3}{T^{ - 2}}{A^0}]}}{{[M{L^3}{T^{ - 2}}]}} \\ $
$[{M^0}{L^0}{T^0}{A^0}]$

Hence, the correct option is B.

Note: Remember the basic formulas used to determine dimensions of various quantities like ${F_e} = \dfrac{1}{{4\pi { \in _0}}}\dfrac{{{q_1}{q_2}}}{{{r^2}}}$ which a force between two charges and and is called the electrostatic force between two charges ${F_g} = G\dfrac{{{m_1}{m_2}}}{{{r^2}}}$ which is a force of gravitation between two bodies which was discovered by Newton and $I = \dfrac{q}{t}$ which is current equals to charge over time.