In this section a few examples of motion of charged particles in electric and magnetic fields are given. The set ups in these examples chosen from experiments that led to important discoveries. The experiments are

- Thomson's parabola method
- Measurement of \(e/m\) using crossed electric and magnetic fields
- Cyclotron motion
- Hall effect

The charged particles traveling perpendicular to electric field, experience a constant force and move along a parabola In this experiment J. J. Thomson discovered that cathode rays consist of charged particles, the electrons, by demonstrating that the properties of cathode rays did not depend on the gas in the cathode ray tube. He measured \(e/m\) of the electrons by analyzing the deflection of cathode rays in electric field. The computational details are presented here.

### Thomsonâ€™s Method for e/m

Thomson passed electrons through a region having mutually perpendicular electric and magnetic field, and both perpendicular to the velocity of the electrons. The fields were adjusted so as to produce no deflection. This enabled him to measure the \(e/m\) of electrons.

### Cyclotron Motion

A charged particle entering a region of uniform magnetic field \(B\) perpendicular to the velocity \(v\) of the charged particle moves in a circle. This motion is called cyclotron motion. The radius and frequency of the circular motion are\(R=\left(\frac{mv}{qB}\right)\qquad \qquad \omega =\left(\frac{qB}{2\pi m}\right)\).

### Hall Effect

The Hall effect described here provided an early method to study the effect of magnetic field on a current. An expression for Hall voltage and Hall resistance is obtained in this section.