What is the acceleration of a charge near a solenoid?

[Tusike]

Homework Statement

Hi! Yesterday was the deadline of a monthly competition, and I couldn't solve one problem. Please take a look at it: http://www.komal.hu/verseny/feladat.cgi?a=honap&h=201011&t=fiz&l=en"
It's the last problem, worth 6 points with the image of a solenoid

Homework Equations

E(induced) = (1/2Rpi) * deltaFlux / deltaTime
F = E*q
a = F/m

Since m, q, and deltaFlux/deltaTime is constant, this can be shortly described as a = const/(2rPi)

The Attempt at a Solution

As the current is uniformly increased and then decreased, the magnetic field inside it increases then decreases uniformly, and so for half the time the change in flux is constant, then points in the other direction. So first it will induce a circular electric field around the solenoid in one direction, then another. The acceleration of the particle in a given moment is tangential to the circle it's on, and it's value can be described as a=const/R, where R is it's distance from the middle. However, I couldn't figure out the solution from this.
I also heard something about how the electromagnetic angular momentum doesn't change, and it's supposed to lead to the solution, but I couldn't get anywhere from that.
Any tips on where and how to start are appreciated.
Thanks!

-Tusike

 

[ehild]

At any point at distance r from the centre you have tangential acceleration inversely proportional to r, but zero radial acceleration. What do you think about the velocity of the charged particle?

ehild

 

[Tusike]

Well that's the part I couldn't get anywhere with. Even though there is no radial acceleration, the particle will have a radial velocity, since it moves to another position. So I guess it supposed to move on a spiral-kind of curve, and after a time when it get's far enough should continue in a straight line. I even made a computer simulation, and I was hoping to get a constant alpha value, but it ranged from about 2 to 10 degrees depending on the (constant / 2rPi) value and the starting distance from the center of the coil.

 

[ehild]

You do not need to solve this problem mathematically. Yes, the particle will gain both radial and azimuthal velocity components. What happens to these velocity components if the current is reversed?

ehild

 

[Tusike]

Hmm I guess since the torque M=FR, and F was equal to a constant/R, M is also constant. So for half the time a constant torque was applied to the particle in one direction, then the current was reversed and the same torque was applied to the particle in the opposite direction (for the same time), leading to a total of 0 angular momentum change. And since in the beginning it was 0, the particle should travel directly away from the solenoid, with no tangential velocity? Now why didn't I think of this sooner...

 

[ehild]

Very clever!

ehild