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 $k=\frac{1}{4\pi {ϵ}_0}$, where ${ϵ}_0$ is the permittivity of vacuum. In terms of these physical constants, the dimension of the magnetic field is $[B]=[e{]}^{\alpha }{\left[m_e\right]}^{\beta }[h{]}^{\gamma }[k{]}^{\delta }$. The value of $\alpha +\beta +\gamma +\delta$ is _____

Answer 1

Answer: 4

$B=e^{\alpha }{\left(m_e\right)}^{\beta }{h}^{\gamma }{k}^{\delta }$
$[B]=\left[e^{\alpha }\right]{\left[m_e\right]}^{\beta }[h{]}^{\gamma }\left[k^{\delta }\right]$
$\left[M^1{T}^{-2}{A}^{-1}\right]=[AT{]}^{\alpha }{\left[m^{\beta }\right]}^{[}{\left[M{L}^2{T}^{-1}\right]}^{\gamma }{\left[M{L}^3{A}^{-2}{T}^{-4}\right]}^{\delta }$
$M^1{T}^{-2}{A}^{-1}=m^{\beta +\gamma +\delta }{L}^{2r+3\delta }{T}^{\alpha -\gamma -4\delta }{A}^{\alpha -2\delta }$
Compare : $\beta +\gamma +\delta =1;2\gamma +3\delta =0,\alpha -\gamma -4\delta =-2,\alpha -2\delta =-1$
On solving $\alpha =3,\beta =2,\gamma =-3,\delta =2$
$\alpha +\beta +\gamma +\delta =4$