Question

Planck's constant (h), speed of light in a vacuum (c) and Newton's gravitational constant (G) are three fundamental constants. Which of the following combinations of these has the dimension of length?
A. \[\dfrac{{\sqrt {Gc} }}{{{h^{3/2}}}}\]
B. \[\dfrac{{\sqrt {hG} }}{{{c^{3/2}}}}\]
C. \[\dfrac{{\sqrt {hG} }}{{{c^{5/2}}}}\]
D. \[\sqrt {\dfrac{{hc}}{G}} \]

Answer 1

Hint: SI (standard international) unit is a system of physical units.
Dimensions are written in square brackets.
There are seven SI base units, those are (with dimensions)
Length - meter (m) [$L$]
Mass – kilogram ($kg$) [$M$]
Time – second ($s$) [$T$]
Electric current – ampere ($A$) [${M^{\dfrac{1}{2}}}{L^{1\dfrac{1}{2}}}{T^{ - 2}}$]
Temperature – kelvin ($k$) [$\theta $]
Luminous intensity – candela [$cd$]
Amount of substance – mole [$mol$]
SI unit of $G$ (gravitation constant) is $N.m^2/kg^2$ [${M^{ - 1}}{L^3}{T^{ - 2}}$]
SI unit of $h$ (Planck's constant) is $J.s$ [$M{L^2}{T^{ - 1}}$]
SI unit of $c$ (speed of light) is $ms^{-1}$ [$L{T^{ - 1}}$]
On substituting the SI units instead of their symbol it gets reduced.
SI unit of length is Meter ($m$).

Complete step by step solution:
Step 1:
Substitute the SI units in every formula and check if it reduces to m (SI unit of length)

Step 2:
Substitute SI units in option A
\[\sqrt {\dfrac{{N{m^2}/{{(kg)}^2}(m/s)}}{{{{(Nms)}^{3/2}}}}} = \dfrac{{(kg){m^3}}}{{N{s^5}}} = \dfrac{{(kg){m^3}}}{{((kg)m/{s^2})({s^5})}} = {m^2}/{s^3}\]
It can't be reduced to the SI unit of length, it would not be the answer.

Step 3:
Substitute SI unit in option B
\[\dfrac{{\sqrt {(Nms)(N{m^2}/{{(kg)}^2})} }}{{{{(m/s)}^{3/2}}}} = \dfrac{{N{s^2}}}{{(kg)}} = \dfrac{{((kg)m/{s^2}){s^2}}}{{kg}} = m\]
It is reduced to m which is the SI unit of length so it would be the answer.

Step 4:
Check further by substituting the SI units in option C
 \[\dfrac{{\sqrt {(Nms)(N{m^2}/{{(kg)}^2})} }}{{{{(m/s)}^{5/2}}}} = \dfrac{{N{s^3}}}{{m(kg)}} = \dfrac{{((kg)m/{s^2}){s^3}}}{{m(kg)}} = s\]
It reduces to s which is the SI unit of time so it would not be the answer.

Step 5:
At last check option D by substituting SI units in it
\[\dfrac{{\sqrt {(Nms)(m/s)} }}{{N{m^2}/{{(kg)}^2}}} = kg\]
It is reduced to kg which is an SI unit of mass so it is not a correct option.

$\therefore$ Only option (B) is reduced to SI unit of length so only combination present in option B has the dimension of length.

Note:
Solve unit and dimensions problems by comparing the dimensions of the given formula with the dimension of the required value.
Take SI units of every quantity, avoid taking the derived unit.
Dimensional analysis is very useful in finding errors in the equation.