In a particular system of units, a physical quantity can be expressed in terms of the electric charge , electron mass , Planck's constant , and Coulomb's constant where is the permittivity of vacuum. In terms of these physical constants, the dimension o

Question

In a particular system of units, a physical quantity can be expressed in terms of the electric charge $e$ , electron mass $m_{e}$ , Planck's constant $h$ , and Coulomb's constant $k = \frac{1}{4 π ε_{0}},$ where $ε_{0}$ is the permittivity of vacuum. In terms of these physical constants, the dimension of the magnetic field is $[B] = {[e]}^{α} {[m_{e}]}^{β} {[h]}^{γ} {[k]}^{δ} .$ The value of $α + β + γ + δ$ is _____ . [2022]

Smart Achievers · Accepted Answer

(4)

$[B] = {[e]}^{α} {[m_{e}]}^{β} {[h]}^{γ} {[k]}^{δ}$

$\Rightarrow M T^{- 2} A^{- 1} = A^{α} T^{α} M^{β} M^{γ} L^{2 γ} T^{- γ} M^{δ} L^{3 δ} A^{- 2 δ} T^{- 4 δ}$

$\Rightarrow β + γ + δ = 1, - 2 = α - γ - 4 δ, - 1 = α - 2 δ$

$2 γ + 3 δ = 0$

$\Rightarrow α = 3, β = 2, γ = - 3, δ = 2$

$\Rightarrow α + β + γ + δ = 4$

Solution

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Solution

Ans. (4) [B]=[e]α[me]β[h]γ[k]δ ⇒MT-2A-1=AαTαMβMγL2γT-γMδL3δA-2δT-4δ ⇒β+γ+δ=1, -2=α-γ-4δ, -1=α-2δ 2γ+3δ=0 ⇒α=3, β=2, γ=-3, δ=2 ⇒α+β+γ+δ=4

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