Chapter 12: Q73P (page 574)

Generalize the laws of relativistic electrodynamics (Eqs. 12.127 and 12.128) to include magnetic charge. [Refer to Sect. 7.3.4.]

Short Answer

Expert verified

The generalized law of relativistic electrodynamics to include magnetic charge is

$K = \frac{1}{\sqrt{1 - \frac{u^{2}}{c^{2}}}} {[q_{e} [E + (μ \times B)]]}_{x} + q_{m} [(B - \frac{1}{c^{2}} (u \times E))]$

Step by step solution

Expression for Maxwell’s equation:

Write the three Maxwell’s equations.

$\partial_{v} F^{m v} = m_{0} J_{e}^{m} \partial_{v} G^{m v} = \frac{m_{0}}{c} J_{e}^{m} K^{m} = (q_{e} F^{mv} + \frac{q_{m}}{c} G^{mv}) h_{v}$

Write all the above three equations with a magnetic charge.

localid="1657699942558" $\nabla . E = \frac{ρ_{e}}{ε_{0}} \nabla \times B = μ_{0} J_{e} + μ_{0} ε_{0} \frac{\partial E}{\partial t} \nabla . B = (\frac{μ_{0}}{c}) (c ρ_{m}) = μ_{0} ρ_{m} - \frac{1}{c} (\frac{\partial b}{\partial t} + \nabla \times E) = \frac{μ_{0}}{c} J_{m}$

Determine the laws of relativistic electrodynamics to include magnetic charge:

Write the equation for the Minkowski force on a charge q.

$K^{m} = q h_{v} F^{m v}$

Let $μ = 1$ :

$K^{1} q η_{v} F^{1 v} = q (- η^{0} F^{10} + η^{1} F^{11} + η^{2} F^{12} + η^{3} F^{13}) K^{1} = q [\frac{- c}{\sqrt{1 - \frac{u^{2}}{c^{2}}}} (- B_{x}) + \frac{u_{y}}{\sqrt{1 - \frac{u^{2}}{c^{2}}}} (- \frac{E_{x}}{c}) + \frac{u_{z}}{\sqrt{1 - \frac{u^{2}}{c^{2}}}} (- \frac{E_{x}}{c})] K = \frac{1}{\sqrt{1 - \frac{u^{2}}{c^{2}}}} {[q_{e} [E + (μ \times B)]]}_{x} + q_{m} [(B - \frac{1}{c^{2}} (u \times E))]$

Therefore, the generalized law of relativistic electrodynamics to include magnetic charge is $K = \frac{1}{\sqrt{1 - \frac{u^{2}}{c^{2}}}} {[q_{e} [E + (μ \times B)]]}_{x} + q_{m} [(B - \frac{1}{c^{2}} (u \times E))]$ .