Question

Find the dimensional formula for $\dfrac{hc}{G}$.

Answer 1

Hint: We will find out the dimensional formulas of different quantities separately. Then we will put them in the given expression. After adding and subtracting different powers of M, L and T the answer will be easily found.

Formula used:
$E=h\nu$
$F=G\dfrac{m_1.m_2}{R^2}$

Complete step by step solution:
First of all, we have to know the basic dimensional notations. Like, the dimensional formula for mass is M. The dimensional formula for length and time are L and T respectively. Now, let’s find out the dimensional formulas of h, c and G separately.
If $\nu$ be the frequency of light, then its energy is given by, $E=h\nu$
Here, h is called Planck’s constant. Now, dimensional formula for energy is,
$[E]=[\text {force}]\times[\text{distance}]=[\text{mass}]\times[\text {acceleration}]\times[\text{distance}]=M.LT^{-2}.L=M.L^2.T^{-2}$
(While denoting dimensional formula, we write the quantity in [].)
The dimensional formula for frequency is , $[\nu]=T^{-1}$
So, dimensional formula for h is,
$[h]=\dfrac{[E]}{[\nu]}=\dfrac{ML^2T^{-2}}{T^{-1}}=ML^2T^{-1}$
Now, c= speed of light in vacuum
So, its dimensional formula is,
$[c]=LT^{-1}$
Now, we know gravitational force as,
$F=G\dfrac{m_1m_2}{R^2}$
$G=F.\dfrac{R^2}{m_1m_2}$
R is the distance and m's are the masses. So, dimensional formula for G is given by,
$[G]=[F].\dfrac{L^2}{M^2}=MLT^{-2}.L^2.M^{-2}=M^{-1}.L^3.T^{-2}$
So, finally the dimensional formula for the given expression is,
$\left[\dfrac{hc}{G}\right]=\dfrac{ML^2T^{-1}.LT^{-1}}{M^{-1}.L^3.T^{-2}} =M^2$
So, the required dimensional formula is $M^2$.

Additional information:
The value of Planck’s constant is $h=6.625\times10^{-34} J.s$
The value of universal gravitational constant is, $G=6.674\times10^{-11} N.m^2/kg^2$
Again, there are some more fundamental dimensional formulas like,
$[\text{Current}]=I ,[\text{Temperature}]=\theta, [\text{Amount of matter}]=N , [\text{Luminous intensity}]=J$.
If the dimensional formula of a physical quantity be equal to unity, it is called a dimensionless quantity. An example of dimensionless quantity is Angle.

Note: Remember the following things,
1. Be very careful while adding or subtracting the powers of M, L or T.
2. All velocities, may it be of light or of sound, has the same dimensional formula that is of the velocity, $LT^{-1}$
3. The dimensional formulas for h and G could be obtained by any other known formula. But, always choose the easiest one.