CBSE Class 11-science Answered

Dimensions of each quantity should be expressed in terms of dimesions of fundamental quantities like

length(L) , mass (M), Time (T), Ampere (A)

units of charge e  :- C = A s

units of permittivity of free space, ε_o = F/m  = C/(V m )

1 volt = 1 J/ C

units of ε_o = C² / ( J m ) = ( A² s² ) / ( kg m s^-2 m² ) = A² s⁴ kg^-1 m^-3

units of planck constant , h = J s = kg m² s^-1

units of speed of light c = m/s

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If  the combination of  e^α ε_o^β h^γ c^δ  is dimensionless , then we get

[ A T ]^α [ A² T⁴ M^-1 L^-3 ]^β [ M L² T^-1 ]^γ [ L T^-1 ]^δ  = A^o T^o M^o L^o 

By equating exponents of same dimension on both sides

α + 2 β = 0

α + 4 β - γ - δ = 0

-β + γ = 0

-3β + 2γ + δ = 0

By solving above equations , we get

if α = 2 , then β = γ = δ = -1