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 $k=\frac{1}{4 \pi \epsilon_{0}}$, where $\epsilon_{0}$ is the permittivity of vacuum. In terms of these physical constants, the dimension of the magnetic field is $[B]=[e]^{\alpha}\left[m_{e}\right]^{\beta}[h]^{\gamma}[k]^{\delta}$. The value of $\alpha+\beta+\gamma+\delta$ is \_\_\_\_.

JEE Advanced · 2022 · 28 Aug 2022 · Paper 2 · Units & Measurement

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 ____.

Answer: 4

Dimensional analysis:

[B] = [M^{1} T^{- 2} A^{- 1}]

Express in terms of given quantities:

So:

[M^{1} T^{- 2} A^{- 1}] = [A T]^{α} [M]^{β} [M L^{2} T^{- 1}]^{γ} [M L^{3} A^{- 2} T^{- 4}]^{δ}

Comparing powers:

Solving:

α = 3, β = 2, γ = - 3, δ = 2

Therefore:

α + β + γ + δ = 3 + 2 - 3 + 2 = 4

JEE Advanced · 2022 · 28 Aug 2022 · Paper 2 · Integer answer