Question

Assume there is one electron charge $-e$ inside a spherical region of uniform charge density $\rho$ (a Thomson hydrogen atom). If the sphere of the radius is given to be $1.0\times {10}^{-10}m$ , find the frequency of the radiation emitted by the atom is ${10}^{N}Hz$. The value of $N$ is​​​

Answer 1

This question is based on the Thomson's model where the atom(solid sphere) can be considered as series of concentric spheres.
By taking a corrollory from electrostatics,we can say that for an electron present somewhere inside a imaginary shell having radius $r$, force exerted by the shells having more radius will be zero. Only force experience by this electron is by the positive charge inside the shell.
$F=\frac{ke\frac{\rho 4\pi x^3}{3}}{x^2}$
$F=\frac{e\frac{\rho 4\pi x^3}{3}}{4\pi {ϵ}_0{x}^2}$
$F=\frac{\rho ex}{3{ϵ}_0}$
$F=-kx$ where $k$ is constant.

$m_{e}a=-kx$
$a=\frac{-kx}{m_e}$
${\omega }^{2}x=-kx$
${\omega }^2=\frac{k}{m_e}$, where $k=\frac{\rho e}{3{ϵ}_0}$
Now, frequency can be written as.
$\nu =\frac{\omega }{2\pi }$
$\nu =\frac{1}{2\pi }\times \sqrt{\frac{\rho e}{3{ϵ}_0{m}_e}}$

Putting the order of values we get $\nu$ in the range of ${10}^{15}Hz$.
So $N=15$