The Electromagnetic Field

Abstract

F ij =     0 −E x −E y −E z E x 0 B z −B y E y −B z 0 B x E z B y −B x 0     (9) G ij =     0 −B x −B y −B z B x 0 −E z E y B y E z 0 −E x B z −E y E x 0     (10) We can now calculate some invariants by finding F ij F ij = −2E 2 + 2B 2 (11) G ij G ij = −2B 2 + 2E 2 (12) F ij G ij = −2E · B − 2E · B = −4E · B (13) Comparing these results with those got earlier by directly calculating the Lorentz transformation of these quantities, we see that the tensor products give the same invariants (after restoring the factors of c). As simple of example of calculating the elements in the tensors, suppose we have an infinite straight wire along the z axis with linear charge density λ moving at speed v. From Gauss's law the electric field a distance x from the wire is E = λ 2ππ 0 x ˆ r (14) and from Ampère's law the magnetic field is B = µ 0 λv 2πx ˆ φ φ φ (15) so at point (x, 0, 0) we have E x = λ 2ππ 0 x (16) E y = E z = 0 (17) B y = µ 0 λv 2πx (18) B x = B z = 0 (19)