Classical Electrodynamics · Problem 11.14

Radiating Systems, Multipole Fields and Radiation — John David Jackson

Physics — E&M · electromagnetic-field-invariants

Problem

(a) Express the Lorentz scalars

F^{\alpha\beta}F_{\alpha\beta}

\mathcal{F}^{\alpha\beta}F_{\alpha\beta}

, and

\mathcal{F}^{\alpha\beta}\mathcal{F}_{\alpha\beta}

in terms of

\mathbf{E}

and

\mathbf{B}

. Are there any other invariants quadratic in the field strengths

\mathbf{E}

and

\mathbf{B}

? (b) Is it possible to have an electromagnetic field that appears as a purely electric field in one inertial frame and as a purely magnetic field in some other inertial frame? What are the criteria imposed on

\mathbf{E}

and

\mathbf{B}

such that there is an inertial frame in which there is no electric field? (c) For macroscopic media,

\mathbf{E}

\mathbf{B}

form the field tensor

F^{\alpha\beta}

and

\mathbf{D}

\mathbf{H}

the tensor

G^{\alpha\beta}

. What further invariants can be formed? What are their explicit expressions in terms of the 3-vector fields?

Related theory

Electric Field Of A Dipole And Behaviour In Uniform Field

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