In a particular system of units, a physical quantity can be expressed in terms of the electric charge e, electron mass $m_e,$ plank’s constant h, and Coulombs constant where $k=\frac{1}{4\pi {\in }_0},$ is the permittivity of vacuum. In terms of these physical constants, the dimension of the magnetic field is $\left[B\right]={\left[e\right]}^{\alpha }{\left[m_e\right]}^{\beta }{\left[h\right]}^{\gamma }{\left[k\right]}^{\delta }.$ The value of $\alpha +\beta +\gamma +\delta$ is________

Substituting dimensions in given dimensional expression of magnetic field, we have
$$\begin{gathered} \left[B\right]={\left[e\right]}^{\alpha }{\left[m_e\right]}^{\beta }{\left[h\right]}^{\gamma }{\left[k\right]}^{\delta } \\ \Rightarrow \left[M^1{T}^{-2}{A}^{-1}\right]={\left[AT\right]}^{\alpha }{\left[m\right]}^{\beta }{\left[M{L}^2{T}^{-1}\right]}^{\gamma }{\left[M{L}^3{A}^{-2}{T}^{-4}\right]}^{\delta } \\ \Rightarrow M^1{T}^{-2}{A}^{-1}=m^{\beta +\gamma +\delta }{L}^{2r+3\delta }{T}^{\alpha -\gamma -4\delta }{A}^{\alpha -25} \end{gathered}$$
Comparing dimensions in LHS and RHS gives
$$\begin{gathered} \beta +\gamma +\delta =1 \\ 2\gamma +3\delta =0 \\ \alpha -\gamma -4\delta =-2 \\ \alpha -2\delta =-1 \end{gathered}$$
Solving above equations, we get
$\alpha =3, \beta =2, \gamma =-3, \delta =2 \Rightarrow \alpha +\beta +\gamma +\delta =4$