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Question: Find the dimensional formula for \(\dfrac{hc}{G}\)....

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

Explanation

Solution

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=hνE=h\nu
F=Gm1.m2R2F=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 ν\nu be the frequency of light, then its energy is given by, E=hνE=h\nu
Here, h is called Planck’s constant. Now, dimensional formula for energy is,
[E]=[force]×[distance]=[mass]×[acceleration]×[distance]=M.LT−2.L=M.L2.T−2[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 , [ν]=T−1[\nu]=T^{-1}
So, dimensional formula for h is,
[h]=[E][ν]=ML2T−2T−1=ML2T−1[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[c]=LT^{-1}
Now, we know gravitational force as,
F=Gm1m2R2F=G\dfrac{m_1m_2}{R^2}
G=F.R2m1m2G=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].L2M2=MLT−2.L2.M−2=M−1.L3.T−2[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,
[hcG]=ML2T−1.LT−1M−1.L3.T−2=M2\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 M2M^2.

Additional information:
The value of Planck’s constant is h=6.625×10−34J.sh=6.625\times10^{-34} J.s
The value of universal gravitational constant is, G=6.674×10−11N.m2/kg2G=6.674\times10^{-11} N.m^2/kg^2
Again, there are some more fundamental dimensional formulas like,
[Current]=I,[Temperature]=θ,[Amount of matter]=N,[Luminous intensity]=J[\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−1LT^{-1}
3. The dimensional formulas for h and G could be obtained by any other known formula. But, always choose the easiest one.