- 4,222

- 1

**1. Homework Statement**

There is a propagating planar wave of the Coulomb potential, [tex] \phi = sin(kx - \omega t) [/tex]. What other fields result when it is assume the magnetic potential, [tex]\textbf{A}[/tex] is everywhere constant?

[tex]\phi[/tex], Coulomb potential

[tex]\textbf{B}[/tex], magnetic field strength

[tex]\textbf{E}[/tex], electric field strength

[tex]\textbf{A}[/tex], magnetic potential

[tex]\textbf{J}[/tex], current density

[tex]c[/tex], speed of light in a vacuum

[itex]\rho[/itex], charge density

**2. Homework Equations**

[tex] \nabla\times\textbf{B} - (1/c) \partial \textbf{E} / \partial t = \textbf{J} [/tex]

[tex] \nabla \cdot \textbf{E} = \rho [/tex]

[tex] \nabla\times\textbf{E} + (1/c) \partial \textbf{B} / \partial t = \textbf{0} [/tex]

[tex] \nabla \cdot \textbf{B} = 0 [/tex] ,

where

[tex] \textbf{E} = - \nabla \phi - (1/c) \partial \textbf{A} / \partial t [/tex]

[tex] \textbf{B} = \nabla \times \textbf{A} [/tex]

and

[tex] \omega/k = c [/tex] .

**3. The Attempt at a Solution**

Starting with

[tex] \phi = sin(kx - \omega t) [/tex] , and [tex] \textbf{A} = \textbf{0} [/tex] ,

I get

[tex]\textbf{E} = -kx\ cos(kx - \omega t) [/tex]

[tex]\textbf{B} = \textbf{0} [/tex]

[tex]\textbf{J} = (-k \omega /c) sin(kx- \omega t) [/tex]

[tex]\rho = k^2 sin(kx- \omega t) [/tex] .

But in each case the velocity of propagation is [tex]c=\omega/k[/tex]. This includes nonzero charge density [tex]\rho[/tex] traveling at c, so I came up with a nonphysical solution. Where did I go wrong?