# Charge above a conductor; effects due to Lorentz force law for moving charges

Currently working through a practice preliminary examination problem. I have your standard charge situated a distance d from a infinite conductor(lets say in the $\hat{z}$ direction and neglecting gravity). The question asks when $v = 0$ at $t = 0$, determine the equation of motion of the electron.

We know the charge induces a oppositely charged particle a distance 2d from the original charge. The electric field due to this charge at the point of the original charge is

$$\vec{E} = - \frac{1}{4 \pi \epsilon_0} \frac{q}{(2d)^2} \hat{z}$$ giving a force of $\vec{F} = q \vec{E}$

Therefore $\ddot{z}(t) = -\frac{1}{16 \pi \epsilon_0} \frac{q^2}{m d^2}$ and straightforward integration gives $z(t) = - \frac{1}{32 \pi \epsilon_0} \frac{q^2 t^2}{m d^2} + d$

Now here's where I'm getting confused. The question then asks a "You know a moving electron produces a magnetic field, does this magnetic field vary with time?"

Intuitively I would say yes because as the speed increases we will have two $\vec{B}$ fields created in the new moving frame from the acceleration of the charges(one original charge and one image charge)

The force felt on the original charge will now have to be the Lorentz force law, but I'm not sure how to include the velocity terms in here.

Thanks.

EDIT I - (7/23/13): I think if I was to see the precise equations of motion for the entire range of the motion as the charge eventually comes to the plate. I would understand what I'm missing.

EDIT II - (7/26/13):

Since we are looking at an image charge that is also in motion(accelerating and moving at some instantaneous velocity) A poster here has given an idea to use the Leinard-Wiechert expression for the Lorentz force and then calculate the EOM from here. The expression of the LW Lorentz force is quite complicated. See for example 10.67 in Griffiths EM in SI units. Where is the acceleration of the test charge in this expression?

This question and differential equation seem entirely too complicated for a qualification examination but it has sparked some interest. Is it even possible to write down the EOM for this situation?

• Are you asking for a formal proof or just a confirmation of your intuitive guess? – Frederic Brünner Jul 24 '13 at 22:28
• I'd like to understand precisely what the equations of motion are for this situation. – John M Jul 24 '13 at 22:29
• Interesting. The method of images is traditionally used for electrostatics, but I don't see why it wouldn't also apply here. By the way, I think you should look again at your equation of motion: the force is not constant but increases as the charge approaches the conductor. I confess I don't see the catch in what looks like a trick question; obviously the magnetic field changes with time (see the Lienard-Wiechert fields). – Art Brown Jul 26 '13 at 3:42
• See my edit. Thanks. Using the LW fields makes sense but I'm unclear as to how to go about doing it. Seems extremely complicated. – John M Jul 26 '13 at 18:51