Electron Beams and Their Interactions

When two parallel beams of electrons move in the same direction, they experience two primary forces: an electrostatic force and a magnetic force.

Electrostatic Force (Repulsion)

• Like Charges Repel: Electrons are fundamental particles that carry a negative electric charge. Since both beams consist of electrons, which are all negatively charged, there will be a repulsive electrostatic force between them. This force tries to push the beams apart because like charges repel each other.
• This electrostatic repulsion is often referred to as the "space charge effect" in electron beam physics, causing a beam to spread out.

Magnetic Force (Attraction)

• Current Creation: Even though electrons are individual particles, a continuous flow of electrons in a beam constitutes an electric current. Therefore, two parallel beams of electrons moving in the same direction can be thought of as two parallel electric currents flowing in the same direction.
• Magnetic Field Generation: According to Ampere's law, any moving electric charge (or current) generates a magnetic field in the space around it. So, each electron beam creates its own magnetic field.
• Force on Moving Charges: When the electrons in one beam move through the magnetic field generated by the other beam, they experience a magnetic force. This force is described by the Lorentz force law: $$ \vec{F} = q(\vec{v} \times \vec{B}) $$ where:
  • $\vec{F}$ is the magnetic force.
  • $q$ is the charge of the electron (negative).
  • $\vec{v}$ is the velocity of the electron.
  • $\vec{B}$ is the magnetic field produced by the other beam.
• Direction of Force (Attraction):
  • Consider the first electron beam (Beam 1) producing a magnetic field. Using the right-hand rule for currents, if Beam 1 moves to the right, the magnetic field lines around it would curl in a specific direction.
  • Now, consider the electrons in the second beam (Beam 2), which are also moving to the right and are located in the magnetic field produced by Beam 1.
  • When the Lorentz force equation is applied, accounting for the negative charge of the electron, it is found that the magnetic force on the electrons in Beam 2 is directed towards Beam 1. Similarly, the force on Beam 1 due to Beam 2's magnetic field is directed towards Beam 2.
  • This mutual force pulls the two beams towards each other, resulting in an attraction. This is a fundamental principle: two parallel currents flowing in the same direction attract each other.

Overall Interaction

While there is an electrostatic repulsion between the negatively charged electrons, there is also a magnetic attraction due to the currents they form. In typical scenarios discussed in introductory physics, for two parallel currents moving in the same direction, the magnetic force causes them to attract.

Therefore, two parallel beams of electrons moving in the same direction will attract each other due to the magnetic interaction between the currents they represent.