Question

Explain What formula you have used to determine the charge to mass ratio of the electron and what is

the origin of it?

Answer 1

Suppose the electron moves in electric $\mathbf{E}$ and magnetic $\mathbf{B}$ fields. The charge to mass ratio of the electron can be found from two formulas: second Newton's law:

$$\mathbf{F} = m\mathbf{a}$$

and Lorentz force law:

$$\mathbf{F} = e(\mathbf{E} + \mathbf{v}\times \mathbf{B})$$

where $\mathbf{v}$ is the velocity of electron. Since forces are equal, obtain:

$$\left( \dfrac{e}{m} \right)(\mathbf{E} + \mathbf{v}\times \mathbf{B}) = \mathbf{a}$$

This is the most general expression to determine the charge to mass ratio of the electron.