The problem (From Electromagnetism by Pollack and Stump, exercise 11.34)(adsbygoogle = window.adsbygoogle || []).push({});

A stationary charge of charge e and mass m encounters a electromagnetic wave with vector potential

[tex]\vec{A}=\vec{j}f(x-ct)[/tex]

where

[tex]\vec{j}[/tex]

is the unit vector in the y-direction. The scalar potential is zero.

What are the components of the velocity, as a function of time,

[tex]v_{x}, v_{y},v_{z}[/tex] ?

What I´m asking for is some insight into why the y-velocity is not a more complicated function. The y-velocity is:

[tex]v_{y}=\frac{e}{m}f(x-ct)[/tex]

More specifically, when the charge is stationary, it will begin to move in the y-direction because of the changing vector potential but the motion in the y-direction will result in the magnetic field causing a motion in the x-direction which in turn should (?) effect the y-velocity, but apparently it does not. Why?

I wrote the equations of motion but they don´t seem to give any insight into the y-motion. I determined the equations of motion by starting with the following equations:

[tex]\vec{E}=-\frac{\partial A}{\partial t}[/tex]

(In the equation above, A is a vector)

[tex]\vec{B}=\nabla X \vec{A} [/tex]

[tex]\vec{F}=e(\vec{E}+\vec{v}X\vec{B})[/tex]

The function f(x-ct) isn´t a problem if we use the chain rule

w = x -ct

[tex]\frac{\partial f}{\partial t}=\frac{df}{dw}\frac{\partial w}{\partial t}=-c\frac{df}{dw} [/tex]

likewise,

[tex]\frac{\partial f}{\partial x}=\frac{df}{dw}\frac{\partial w}{\partial x}=\frac{df}{dw} [/tex]

Note, I´ve been a little sloppy about notation regarding partials and full derivatives.

The resulting equations of motion are:

[tex]\frac{dv_{x}}{dt}=\frac{e}{m}v_{y}\frac{df}{dw}[/tex]

[tex]\frac{dv_{y}}{dt}=\frac{e}{m}(c\frac{df}{dw}-v_{x}\frac{df}{dw})[/tex]

[tex]\frac{dv_{z}}{dt}=0[/tex]

Thus the z-component of the velocity is zero since it started at rest.

The equations do give the right answer for the x-velocity if I assume the value of the y-velocity. But they aren´t helpful in giving me the y-velocity. Note that the two equations involving vy and vx have an additional unknown df/dw. I´ve tried everything, even eliminating df/dw and getting an equation that appears to have just x-dependence on one side and y-dependence on the other which one could solve by setting each to a constant but that doesn´t seem to work either.

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# Homework Help: Insight needed into electromagnetism problem

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