Question

1）Solve the differential equation for v:

m$\frac{dv}{dt}$+$\frac{eE}{vf}$v=eE

2）Polarizability of electronic displacement polarization:

Under low applied electric field, the induced dipole moment μ is proportional to the (local) electric field E, with the coefficient defined as polarizability α, that is: μ=αE.

Now let’s consider the case called as electronic displacement polarization. In this case we assume the atom consists of a nucleus which is surrounded by the spherical shaped electron cloud with a radius as r. When an electric field is applied, the center of charge of the electronic cloud shifts a certain distance from the nucleus of the atom, and thus induces dipole moment in the atom, but the electronic cloud still keeps the spherical shape with the same radius. Please prove the polarizability of this so-called electronic displacement polarization can be written as:

αe=4πr3ε0