Verified Solution • 2025–26 Session moderate

QJEEAdv22Ph25f4: JEE Advanced 2022 Physics - Class 11 JEEAdv22Ph25f4 Solution: In a particular system of units, a physical quantity can be…

In a particular system of units, a physical quantity can be expressed in terms of the electric charge ee , electron mass mem_{e} , Planck's constant hh , and Coulomb's constant k=14πϵ0k=\frac{1}{4 \pi \epsilon_{0}} , where ϵ0\epsilon_{0} is the permittivity of vacuum. In terms of these physical constants, the dimension of the magnetic field is [B]=[e]α[me]β[h]γ[k]δ[B]=[e]^{\alpha}\left[m_{e}\right]^{\beta}[h]^{\gamma}[k]^{\delta} . The value of α+β+γ+δ\alpha+\beta+\gamma+\delta is _______.

Step-by-Step Verified Solution

amp;B=eα(me)βhγkδamp;[ B]=[eα][me]β[h]γ[kδ]amp;[M1 T−2 A−1]=[AT]α[m]β[ML2 T−1]γ[ML3 A−2 T−4]δamp;MlT−2 A−1=mβ+γ+δL2r+3δTα−γ−4δAα−2δamp;Compare :β+γ+δ=1;2γ+3δ=0,α−γ−4δ=−2,α−2δ=−1amp;On solving α=3,β=2,γ=−3,δ=2amp;α+β+γ+δ=4\begin{aligned}& \mathrm{B}=\mathrm{e}^\alpha\left(\mathrm{m}_{\mathrm{e}}\right)^\beta \mathrm{h}^\gamma \mathrm{k}^\delta \\\\& {[\mathrm{~B}]=\left[\mathrm{e}^\alpha\right]\left[\mathrm{m}_{\mathrm{e}}\right]^\beta[\mathrm{h}]^\gamma\left[\mathrm{k}^\delta\right]} \\\\& {\left[\mathrm{M}^1 \mathrm{~T}^{-2} \mathrm{~A}^{-1}\right]=[\mathrm{AT}]^\alpha[\mathrm{m}]^\beta\left[\mathrm{ML}^2 \mathrm{~T}^{-1}\right]^\gamma\left[\mathrm{ML}^3 \mathrm{~A}^{-2} \mathrm{~T}^{-4}\right]^\delta} \\\\& \mathrm{M}^{\mathrm{l}} \mathrm{T}^{-2} \mathrm{~A}^{-1}=\mathrm{m}^{\beta+\gamma+\delta} \mathrm{L}^{2 \mathrm{r}+3 \delta} \mathrm{T}^{\alpha-\gamma-4 \delta} \mathrm{A}^{\alpha-2 \delta} \\\\& \text {Compare }: \beta+\gamma+\delta=1 ; 2 \gamma+3 \delta=0, \alpha-\gamma-4 \delta=-2, \alpha-2 \delta=-1 \\\\& \text {On solving } \alpha=3, \beta=2, \gamma=-3, \delta=2 \\\\& \alpha+\beta+\gamma+\delta=4\end{aligned}

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