Q73P Generalize the laws of relativis... [FREE SOLUTION]

91影视

Introduction to Electrodynamics

David J. Griffiths

Physics

4th edition Edition 路 613 pages

ISBN: 9780321856562

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

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

Step by step solution

Expression for Maxwell’s equation:

Write the three Maxwell鈥檚 equations.

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

Write all the above three equations with a magnetic charge.

localid="1657699942558" $鈭� . E = \frac{蚁_{e}}{蔚_{0}} 鈭� 脳 B = 渭_{0} J_{e} + 渭_{0} 蔚_{0} \frac{鈭� E}{鈭� t} 鈭� . B = (\frac{渭_{0}}{c}) (c 蚁_{m}) = 渭_{0} 蚁_{m} - \frac{1}{c} (\frac{鈭� b}{鈭� t} + 鈭� 脳 E) = \frac{渭_{0}}{c} J_{m}$

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 $渭 = 1$ :

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

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