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Physics Question on Youngs double slit experiment

Find the α, β, γ in the Y = CαhβGγ,  C^\alpha h^\beta G^\gamma,\,\,
γ\gamma = Young's Modulus
cc = speed of Light
hh = Plank's Constant
GG = gravitational Constant

A

α=7, β=-1, γ=-2

B

α=1, β=1, γ=2

C

α=1, β=1, γ=-2

D

α=-1, β=0, γ=2

Answer

α=7, β=-1, γ=-2

Explanation

Solution

The correct option is (A)
Y=CαhβGγY=C^\alpha h^\beta G^\gamma
=[LT−1]α[M1L2T−1]β[MLT−2M2]γ=[LT^{-1}]^\alpha[M^1L^2T^{-1}]^{\beta}[\frac{MLT^{-2}}{M^2}]^{\gamma}
=[LT−1]α[M1L2T−1]β [M−1L3T−2]−γ=[LT^{-1}]^\alpha[M^1L^2T^{-1}]^{\beta}\,[M^{-1}L^3T^{-2}]^{-\gamma}
β−γ=1     .....(1)\beta-\gamma=1\,\,\,\,\,.....(1)
α+2β+3γ=−1       ......(2)\alpha+2\beta+3\gamma=-1\,\,\,\,\,\,\,......(2)
α+β+2γ=2       ......(3)\alpha+\beta+2\gamma=2\,\,\,\,\,\,\,......(3)
eq (2)−(3)(2)-(3)
β+γ=−3\beta+\gamma=-3
β−γ=1\beta-\gamma=1
2β=−22\beta=-2
β=−1\beta=-1
γ=−2\gamma=-2
α=2−β−2γ\alpha=2-\beta-2\gamma
=2+1−2(−2)=2+1-2(-2)
=3+4=7=3+4=7