Q6P In Chapter 5, I showed that it i... [FREE SOLUTION]

91影视

Introduction to Electrodynamics

David J. Griffiths

Physics

4th edition Edition 路 613 pages

ISBN: 9780321856562

Chapter 10: Q6P (page 442)

In Chapter 5, I showed that it is always possible to pick a vector potential whose divergence is zero (the Coulomb gauge). Show that it is always possible to choose $鈭� . A = - 渭_{0} 蔚_{0} (鈭� V / 鈭� t)$ , as required for the Lorenz gauge, assuming you know how to solve the inhomogeneous wave equation (Eq. 10.16). Is it always possible to pick $V = 0$ ? How about $A = 0$ ?

Short Answer

Expert verified

It is not always possible to choose $鈭� . A = - 渭_{0} 蔚_{0} \frac{鈭� V}{鈭� t}$ for the Lorentz gauge. For the case of the scalar potential, it is always possible to pick $V = 0$ , and for the vector potential, it is not possible to pick $A = 0$ .

Step by step solution

Expression for the potential:

Write the expression for the potential V and the constant A.

$V = 0, A = \{\begin{cases} \frac{渭_{0} k}{4 c} (c t - {|X|}^{2}) \hat{Z} f o r |X| < c t \\ 0, f o r |X| > c t \end{cases}$

Show that is required for the ∇ . A =μ0ε0 (∂V/∂t) Lorentz gauge:

Take the gradient of A.

$鈭� . A = (\hat{x} \frac{鈭�}{鈭� x} + \hat{y} \frac{鈭�}{鈭� y} + \frac{鈭�}{鈭� z}) 鈥� \frac{渭_{0} k}{4 c} (c t - {|x|}^{2}) \hat{z} 鈭� . A = \frac{鈭�}{鈭� z} (\frac{渭_{0} k}{4 c}) (c t - {|x|}^{2}) 鈭� . A = 0$

Take the derivative of V.

$\frac{鈭� V}{鈭� t} = 0$

Hence, $鈭� . A = 渭_{0} 蔚_{0} \frac{鈭� V}{鈭� t}$

So, it can be observed that both the scalar and potentials are in the coulomb and in the Lorentz gauge.

Suppose $V = 0, A = - \frac{1}{4 蟺蔚_{0} r^{2}} r^{2}$

Take the gradient of A.

localid="1655873040623" role="math" $鈭� . A = - \frac{q t}{4 蟺蔚_{0}} 鈭� . (\frac{\hat{r}}{r^{2}}) 鈭� . A = - \frac{q t}{蔚_{0}} 蟽^{3} (r) = 0$

Take the derivative of V.

$\frac{鈭� V}{鈭� t} = 0$

Hence, $鈭� . A = 渭_{0} 蔚_{0} \frac{鈭� V}{鈭� t}$

So, it can be observed that both the scalar and vector potentials are not in coulomb and in Lorentz gauge.

Similarly, consider $V =, A = A_{0} \sin (k x - 蝇 t) y$

Take the gradient of A.

$鈭� . A = (\hat{x} \frac{鈭�}{鈭� x} + \hat{y} \frac{鈭�}{鈭� y} + \hat{z} \frac{鈭�}{鈭� z}) 鈥� A_{0} \sin (k x - 蝇 t) \hat{y} 鈭� . A = \frac{鈭�}{鈭� y} A_{0} \sin (k x - 蝇 t) \hat{y} 鈭� . A = 0$

Take the derivative of V.

$\frac{鈭� V}{鈭� t} = 0$

Hence, $鈭� . A = 渭_{0} 蔚_{0} \frac{鈭� V}{鈭� t}$

So, it can be observed that both the scalar and vector potentials are not in coulomb and in Lorentz gauge.

Determine the possibility to pick V=0 and A=0:

Yes, it is always possible to pick V=0 but cannot pick A=0 because that would make B=0.

Therefore, it is not always possible to choose $鈭� . A = 渭_{0} 蔚_{0} \frac{鈭� V}{鈭� t}$ for the Lorentz gauge. For the case of the scalar potential, it is always possible to pick $V = 0$ , and for the vector potential, it is not possible to pick $A = 0$ .