Particle in constant electic and magnetic field

[Decan]

This isn't really a homework question, just something to satisfy my curiosity but if it belongs in the homework section, I really apologize and if the mods could move it to that section, it would be much appreciated.

Anywho, in class we talked about a particle accelerated by an electric field generated by capacitors. It then leaves through a hole and enters a constant magnetic field. That got me thinking, what if there is a constant magnetic field AND an electric field acting on the object at the same time?

http://img442.imageshack.us/img442/406/physicsix5.png

Above is my interpretation of what would happen. Let's say a proton is released, it would travel toward the negative plate because it is accelerated in that direction because of the electric force. The motion of the proton in a magnetic field would generate a magnetic force and cause the proton to move in a circle. The net force on the object = magnetic force + electric force. My question is, how would the partice move? I know the manetic force and the electric force are components of the net force but can anyone help me understand the direction of movement? Since the particle does not have constant acceleration (so no constant velocity), is it possible to figure out the velocity of the particle? Here's what I think...

F net = qvB + qE = ma; so v = (ma-qE)/qB but since the acceleration is changing constantly, would this be instantaneous velocity? If so, is the trajectory created by this velocity also changing constantly (since Fmag = mv^2/R)?

Finally, what if gravity was involved? Now the net force has 3 components...but I can't imagine/figure out how the partice would move. The professor is out of town and will be for the next few days. I think that understanding this would really help me on the exam so any help would be greatly appreciated. Thanks!

 

[f3nr15]

From memory, I looked at a few study guide diagrams, if both an electric field and magnetic field were present in a cathode ray tube an electron would pass through undeflected.

 

[Decan]

I wonder why that is...

 

[cesiumfrog]

f3, that'd be Thomson's experiment rather than the general case.

Decan, the simplest (brute force) approach to this problem (using your F net = qvB + qE = m$\frac{\triangle v}{\triangle t}$) is to consider one ${\triangle t}$ at a time, draw the particle's velocity initially, calculate the initial change in velocity, use that to deduce the approximate position and velocity a moment later, then re-calculate the force and change in velocity at this later moment, and step by step you'll draw out the shape of the motion. (Within a few steps you'll understand why physicists use computers so frequently.)

 

[marcusl]

I don't believe fnr15's memory is serving correctly here. You are on the right track but you need more sophisticated tools (namely vector calculus and differential equations) to describe the motion mathematically. For instance, the Lorentz force is written as
$$\vec{F}=q\vec{E}+q\vec{v}\times\vec{B}$$

The situation you describe is actually used in the magnetron, the famous pulsed microwave source that powered radar during WW2 and that now powers your microwave oven. Take a look at Fig. 4 in
http://www.radartutorial.eu/08.transmitters/tx08.en.html"
which shows the motion of electrons between the negative inner conductor (cathode) and positive shell (anode), where the potential difference is usually on the order of 10kV. A magnetic field (oriented perpendicular to the paper) makes the electrons spiral.

Here's a picture of a magnetron like that used in GE microwave ovens
http://www.gallawa.com/microtech/magnetron.html"

In addition to the simple fields you envisioned, a magnetron adds local RF fields in the cavities. You can read down the page to see how it works.

 

[Decan]

So, would the net force then constantly change because the particle does not have constant velocity? If so, would the motion of the particle be in a spiral?

 

[marcusl]

1. Yes
2. Almost. There will be a net motion along the field between your capacitor plates, so the path can have loops.

 

[Decan]

I drew out what I think might happen...which one is closer to what really happens?

http://img137.imageshack.us/img137/9149/physicsil6.png

 

[marcusl]

I haven't worked this out exactly, but I think you can get either depending on the relative strengths of E and B.