# Time-dependent electric field

## Main Question or Discussion Point

Hello! This is my doubt:

I have a particle with charge q and mass m with a relativistic velocity v·u_x in a region of electric field E = E_0 cos (wt)·u_z. I want to calculate the time evolution of the velocity.

I would use the 2nd law of Newton, so that dp / dt = F

where p = gamma·m·v·u_x and F = q (E + v·u_x ^ B).

The thing is that as the electric field is variable, it is assumed that there will be an induced magnetic field acting on the particle (or not?). The first thing is that I don't know if I have to consider a field B acting on the particle and the second is that, in that case, when I try to calculate B, I use Maxwell's equations but I get results that do not fit.

So is there only electric force or no electric and magnetic force?

Thank you.

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mfb
Mentor
There should be a spot without magnetic fields - in a symmetric setup, this is the center. Everywhere else, it can get tricky.

Dotini
Gold Member
In his book, The Lightning Discharge, Uman considers models of varying levels of sophistication for computing time dependent electric and magnetic fields from Maxwell's equations. Buy it, it's cheap.

Here is a link to a variety of publications and patents dealing in transient electrical events, Helmholtz equations, etc. http://www.ee.psu.edu/Directory/FacultyInfo/Pasko/PaskoPublications.aspx

Respectfully,
Steve

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lailola - your problem is not specified terribly well, but a few general observations can be made. in the rest frame of said particle, only electric forces qE operate. In any other Lorentzian frame, the relativistically exact Lorentz force law expression F = q(E+vxB) applies, where E, B, and v are all evaluated in that frame. Sticking with that one frame, integration over time using F = q(E+vxB) = dp/dt = d(γmv)/dt then gives the velocity evolution, where γ = 1/√(1-(v/c)2). This assumes no implied back-reaction coupling to whatever source produces the applied field, and further assumes radiative back reaction is negligible. You have correctly identified the relativistic energy part of all this re coupling to velocity - the remainder seems straightforward to me (well ok it gets to be a messy integro-differential equation likely needing numerical evaluation - but then what's in the detail )

But..in the 'laboratory' rest frame, where the particle is moving with velocity v, is there an external magnetic field B (due to time-dependent E) that I have to consider? I don't know how could it be calculated..

But..in the 'laboratory' rest frame, where the particle is moving with velocity v, is there an external magnetic field B (due to time-dependent E) that I have to consider? I don't know how could it be calculated..

Ah ok - it's the external source relation between time varying E and B that's bothering you. Details will depend on the source configuration - i.e. is it idealized to the field between the plates of a capacitor with time-varying voltage, or that owing to a solenoid with time-varying current etc. In general, one has the Maxwell eqn's as guide. Bottom line is that except at a particular point in space and time, time variation of E implies the existence of a time varying B also, and vice versa. That follows from the relations curl E = -dB/dt, and curl B = c-2dE/dt (in vacuo). These curl relations imply there must be both fields present if time variation of either one exists.

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mfb
Mentor
In the lab frame, you can have a magnetic field.
And to evaluate this in the electron rest frame, you have to convert your fields - where the conversion (given by the velocity of the electron) changes in time. Not sure whether this helps or not...

Ah ok - it's the external source relation between time varying E and B that's bothering you. Details will depend on the source configuration - i.e. is it idealized to the field between the plates of a capacitor with time-varying voltage, or that owing to a solenoid with time-varying current etc. In general, one has the Maxwell eqn's as guide. Bottom line is that except at a particular point in space and time, time variation of E implies the existence of a time varying B also, and vice versa. That follows from the relations curl E = -dB/dt, and curl B = c-2dE/dt (in vacuo). These curl relations imply there must be both fields present if time variation of either one exists.
The thing is that i haven't got a conductor as capacitor or a solenoid, I have a point charge suffering a time-dependent external E. My problem is that I'm not sure if I have to consider that the point charge is also suffering an induced external B, or if there's no B.

The thing is that i haven't got a conductor as capacitor or a solenoid, I have a point charge suffering a time-dependent external E. My problem is that I'm not sure if I have to consider that the point charge is also suffering an induced external B, or if there's no B.
Well sorry but in that case your problem is ill-defined. The time and space relations between externally acting E and B do depend on the details of whatever system of charges and currents generate the E and B. If as it seems your home-work assignment specifies only a time-varying E, then my guess is it implies such time-variation is slow enough to assume a negligible B field - so just work from an E only. And blame your teacher for being sloppy in asking the question if that's 'the wrong answer'! :rofl:

Well sorry but in that case your problem is ill-defined. The time and space relations between externally acting E and B do depend on the details of whatever system of charges and currents generate the E and B. If as it seems your home-work assignment specifies only a time-varying E, then my guess is it implies such time-variation is slow enough to assume a negligible B field - so just work from an E only. And blame your teacher for being sloppy in asking the question if that's 'the wrong answer'! :rofl:
Ok, I think you must be right. Only a last question... Why don't I get a correct result when I use the two Maxwell equations that you have writen? I have the expression of E, so..the only value that isn' known is B.

Ok, I think you must be right. Only a last question... Why don't I get a correct result when I use the two Maxwell equations that you have writen? I have the expression of E, so..the only value that isn' known is B.
Those curl eqn's must be applied to a specific system and then solved using boundary values - e.g. tangent component of E vanishes at surface of a perfect conductor, likewise for normal component of B. it can get quite messy in detail! You probably have some good textbooks that give worked examples.