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Mathematics for Physics

161 Electromagnetic Field Transformations

The rules for how electric and magnetic fields transform between inertial frames, so a pure electric field in one frame can appear mixed with a magnetic field in another.

Relativistic mixing of \(\mathbf{E}\) and \(\mathbf{B}\). An electric field in one inertial frame is a mixture of electric and magnetic fields in a relatively moving frame. Field mixing is that exchange of \(\mathbf{E}\) and \(\mathbf{B}\) under a boost. This principle is used to treat electricity and magnetism as one electromagnetic field.

Longitudinal invariance and transverse mixing. Components of \(\mathbf{E}\) and \(\mathbf{B}\) parallel to the relative motion are unchanged. Transverse components mix and scale with the Lorentz factor. A longitudinal component is a field part along the boost. A transverse component is a field part perpendicular to the boost. This principle is used to compute moving-frame fields from rest-frame fields.

The field transformations for a boost along \(x\) are

\[ E'_{x} = E_{x},\qquad B'_{x} = B_{x} \]

\[ E'_{y} = \gamma\left(E_{y} - v B_{z}\right),\qquad E'_{z} = \gamma\left(E_{z} + v B_{y}\right) \]

\[ B'_{y} = \gamma\left(B_{y} + \dfrac{v}{c^{2}} E_{z}\right),\qquad B'_{z} = \gamma\left(B_{z} - \dfrac{v}{c^{2}} E_{y}\right) \]

where

  • \(\mathbf{E}\) and \(\mathbf{B}\) are the fields in the first inertial frame.
  • \(\mathbf{E}'\) and \(\mathbf{B}'\) are the fields in the second inertial frame.
  • \(v\) is the relative speed of the primed frame along \(x\).
  • \(c\) is the speed of light.
  • \(\gamma\) is the Lorentz factor.

The unified field tensor. The six independent field components form one antisymmetric field tensor. The electromagnetic field tensor is a \(4\times 4\) array whose entries are those components. This principle is used to write the transformations as a Lorentz transformation of \(F^{\mu\nu}\).

Frame-invariance of the total force. All inertial observers agree on the deflection of a moving charge, while they split that force into different mixtures of \(\mathbf{E}\) and \(\mathbf{B}\). The Lorentz force is the net electromagnetic force on a moving charge. This principle is used to keep the physical trajectory the same while the field mix changes.

The Lorentz force is

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

where

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

161.1 References

  1. Griffiths, D. J. Introduction to Electrodynamics. — source for the heading explanation.
  2. Griffiths, D. J. Introduction to Electrodynamics. Cambridge University Press, 2024. — \(E'_{\parallel}=E_{\parallel}\); transverse mixing; \(\mathbf{F}=q(\mathbf{E}+\mathbf{v}\times\mathbf{B})\); \(F^{\mu\nu}\).
  3. Knight, R. D. Physics for Scientists and Engineers: A Strategic Approach with Modern Physics. Pearson, 2023. — electric and magnetic fields mix between frames; Lorentz force.
  4. Shankar, R. Fundamentals of Physics II. Yale University Press, 2020. — direction-dependent field transformations.
  5. Carroll, S. M. Spacetime and Geometry. Cambridge University Press. — unified electromagnetic field; \(F_{\mu\nu}\).
  6. Emam, M. H. Covariant Physics. Oxford University Press, 2021. — electromagnetic field tensor.