- #1
Phrak
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- 6
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
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] .
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?