What effect do an electric field and magnetic field have on the electron beam?

An electron beam is made up of fast-moving electrons, which are charged particles. Both electric fields and magnetic fields can exert significant effects on the motion of these electrons. Let’s look at how each field influences the electron beam.

Effect of Electric Field on Electron Beam

An electric field exerts a force on charged particles, and since electrons are negatively charged, they experience a force in the direction opposite to the electric field. The force is described by Coulomb’s law and is given by:

F=−eE

Where:

• F is the force acting on the electron.
• $e$ is the charge of the electron (negative).
• E is the electric field.

Effects on Electron Beam:

• Deflection: When an electron beam passes through an electric field, the electrons experience a force that causes them to accelerate or decelerate in the direction of the electric field. For example, if the electric field is applied perpendicular to the motion of the beam, it will deflect the electron beam in a direction perpendicular to the beam’s original path.

• Speed Change: If the electric field is parallel or anti-parallel to the velocity of the electron, the electrons will either speed up or slow down, depending on the direction of the field. The change in speed happens due to the work done on the electron by the electric field.

• Energy Change: The work done by the electric field on the electron can also change the kinetic energy of the electron. If the electron is moving in the same direction as the field, its kinetic energy increases, and if it moves against the electric field, its kinetic energy decreases.

Effect of Magnetic Field on Electron Beam

A magnetic field interacts with a moving charged particle (such as an electron) through the Lorentz force. The force on an electron moving with velocity V in a magnetic field B is given by:

F=−e(v×B)

Where:

• F is the force on the electron.
• $e$ is the charge of the electron.
• v is the velocity of the electron.
• B is the magnetic field.
• The cross product indicates that the force is perpendicular to both the velocity of the electron and the direction of the magnetic field.

Effects on Electron Beam:

• Deflection: The magnetic field causes the electron beam to change direction. Since the force is always perpendicular to the velocity of the electron, it doesn’t do work (so the kinetic energy remains constant), but it changes the direction of the velocity. This results in a circular or spiral motion of the electrons if the magnetic field is uniform and perpendicular to the direction of the electron beam.

  • In a uniform magnetic field, the trajectory of the electron beam will form a circular path with a radius determined by the velocity of the electrons, the strength of the magnetic field, and the mass of the electron. The radius of curvature r is given by the equation:

  r=mv/eB​

  Where:

  • $m$ is the mass of the electron.
  • $v$ is the speed of the electron.
  • $e$ is the charge of the electron.
  • $B$ is the magnetic field strength.

• Helical Motion: If the magnetic field is not perfectly perpendicular to the velocity of the electron, the electron will follow a helical path, combining circular motion in the direction of the magnetic field and linear motion along the direction of the electron’s initial velocity.

• No Energy Change: Unlike the electric field, the magnetic field does not change the speed of the electron (because the magnetic force is always perpendicular to the velocity). Therefore, the kinetic energy of the electron remains constant.

Combined Effect of Electric and Magnetic Fields on an Electron Beam

When both electric and magnetic fields are applied simultaneously, the motion of the electron beam is determined by the combined Lorentz force. The total force on the electron is:

F=−e(E+v×B)

This means that the electron beam can experience both deflection due to the electric field and a change in direction (but not speed) due to the magnetic field.

• Crossed Fields: If the electric and magnetic fields are perpendicular to each other, and the velocity of the electrons is also perpendicular to both fields, the beam may experience a complex trajectory, often called a crossed field effect. This is used in devices like cathode ray tubes (CRTs) or mass spectrometers to manipulate the electron path precisely.

• Velocity Selector: In some applications, electric and magnetic fields can be used together to create a velocity selector, which allows only electrons with a specific velocity to pass through, as the forces from both fields balance each other out.