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Mathematics for Physics

47 Electromagnetic Interaction

A force between charged particles that is used to combine the electric-field force with the velocity-dependent magnetic force.

The Lorentz force. The electromagnetic force on a point charge is the Lorentz force. This principle is used to compute the mechanical force felt by a charge in given fields.

The Lorentz force is

\[ \mathbf{F} = q\bigl(\mathbf{E} + \mathbf{v}\times\mathbf{B}\bigr) \]

where

  • \(\mathbf{F}\) is the force.
  • \(q\) is the charge.
  • \(\mathbf{E}\) is the electric field.
  • \(\mathbf{v}\) is the velocity.
  • \(\mathbf{B}\) is the magnetic field.

The electric part. The electric part is independent of velocity. This principle is used to recover the electrostatic force on a charge at rest.

The electric force is

\[ \mathbf{F}_{\mathrm{el}} = q\mathbf{E} \]

where

  • \(\mathbf{F}_{\mathrm{el}}\) is the electric force.
  • \(q\) is the charge.
  • \(\mathbf{E}\) is the electric field.

The magnetic part of the electromagnetic interaction. The magnetic part is perpendicular to the velocity and does no work. This principle is used to change the direction of a moving charge without changing its kinetic energy.

The magnetic force is

\[ \mathbf{F}_{\mathrm{mag}} = q\bigl(\mathbf{v}\times\mathbf{B}\bigr) \]

where

  • \(\mathbf{F}_{\mathrm{mag}}\) is the magnetic force.
  • \(q\) is the charge.
  • \(\mathbf{v}\) is the velocity.
  • \(\mathbf{B}\) is the magnetic field.