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Introduction to Electrodynamics

David J. Griffiths

Physics

4th edition Edition 路 613 pages

ISBN: 9780321856562

Chapter 12: 53 (page 568)

Obtain the continuity equation (Eq. 12.126) directly from Maxwell鈥檚 equations (Eq. 12.127).

Short Answer

Expert verified

The continuity equation is obtained as . $\frac{未 J^{渭}}{未 X^{渭}} = 0$

Step by step solution

01

Expression for Maxwell’s equation:

Using equation 12.127, write the expression for Maxwell鈥檚 equation.

$\frac{未 F^{渭谓}}{未 X^{谓}} = 渭_{0} J^{渭}$ 鈥︹€� (1)

It is known that:

$\frac{未}{未 X^{v}} = 未_{v}$

Substitute $未_{v}$ for $\frac{未}{未 X^{v}}$ in equation (1).

$未_{v} F^{渭谓} = 渭_{0} J^{渭}$ 鈥︹€� (2)

02

Determine the continuity equation from Maxwell’s equation:

Differentiate the equation (2).

$未_{v} 未_{渭} F^{渭谓} = 渭_{0} 未_{渭} J^{渭}$

From the above equation, observe the symmetric and anti-symmetric combination.

$未_{渭} 未_{v} = 未_{谓} 未_{渭} (s y m m e t r i c) F^{渭谓} = - F^{渭谓} (A n t i - s y m m e t r i c)$

Since it is known that:

$未_{渭} 未_{v} F^{渭谓} = 0$

As the above indices are summed from 0 to 3, the term and v can be pronounced as the same. Hence,

$未_{渭} 未_{v} F^{渭谓} = 未_{v} 未_{渭} F^{渭谓} = 未_{渭} 未_{v} (- F^{渭谓}) = - 未_{渭} 未_{v} F^{渭谓}$

Now, as the above quantity is equal to minus itself, it must be zero. Hence,

$\frac{未 J^{渭}}{未 X^{渭}} = 0$

Therefore, the continuity equation is obtained as . $\frac{未 J^{渭}}{未 X^{渭}} = 0$

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Most popular questions from this chapter

Use the Larmor formula (Eq. 11.70) and special relativity to derive the Lienard formula (Eq. 11. 73).

$\begin{array}{l} P = \frac{渭_{0} q^{2} a^{2}}{6 蟺 c} 鈥夆赌夆赌� (11.70) \\ P = \frac{渭_{0} q^{2} 纬^{6}}{6 蟺 c} (a^{2} - {| \frac{蠀脳 a}{c} |}^{2}) 鈥夆赌夆赌� (11.73) \end{array}$

An ideal magnetic dipole moment m is located at the origin of an inertial system $\overset{炉}{S}$ that moves with speed v in the x direction with respect to inertial system S. In $\overset{炉}{S}$ the vector potential is

$\overset{炉}{A} = \frac{渭_{0}}{4 蟺} \frac{\overset{炉}{m} 脳 \overset{炉}{\hat{r}}}{{\overset{炉}{r}}^{2}}$

(Eq. 5.85), and the scalar potential $\overset{炉}{V}$ is zero.

(a) Find the scalar potential V in S.

(b) In the nonrelativistic limit, show that the scalar potential in S is that of an ideal electric dipole of magnitude

$p = \frac{v 脳 m}{c^{2}}$

located at $\overset{炉}{O}$ .

Obtain the continuity equation (Eq. 12.126) directly from Maxwell鈥檚 equations (Eq. 12.127).

Prove that the symmetry (or antisymmetry) of a tensor is preserved by Lorentz transformation (that is: if $t^{渭 v}$ is symmetric, show that ${\overset{炉}{t}}^{渭 v}$ is also symmetric, and likewise for antisymmetric).

In classical mechanics, Newton鈥檚 law can be written in the more familiar form $F = ma$ . The relativistic equation, $F = \frac{dp}{dt}$ , cannot be so simply expressed. Show, rather, that

$F = \frac{m}{\sqrt{1 - u^{2} / c^{2}}} [a + \frac{u (u 鈥� a)}{c^{2} 鈥� u^{2}}]$

where $a = \frac{du}{dt}$ is the ordinary acceleration.

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