Magnetic Fields?

Question

You are standing in the middle of the recreation room of a nuclear submarine that is lurking in the Arctic Ocean at the magnetic south pole of the earth, where the magnetic field lines point straight down toward the ocean floor, in the negative z direction. You can fire positively charged particles from the origin at various speeds in any one of the four directions along the positive or negative x and y axes. There is a vertical pole in the position shown. In which direction(s) (positive x, negative x, positive y, negative y) can you fire positively charged particles and have them reach the pole? Assume that the room is large enough that you don't have to worry about hitting the walls.

See diagram.

http://i338.photobucket.com/albums/n418/gregismighty/p6e.jpg

Answer 1

So the particle goes in a circle.  Use your handy right hand to get the direction--fingers go like velocity, palm goes like B-field (down), thumb points in the direction of the magnetic centripetal force.  Looks to me like from the origin, you could go down or right. and curve up to the pole.  Shooting up or left, you curve away.

Aquaboy: you realize that the cross product is cyclic, right?

i x j = k

k x i = j

j  x k = i

So there are different ways you could say it.  The way I learned it was fingers x palm = thumb.  That picture is different, but it doesn't matter--they're equivalent.  You could also say thumb x fingers = palm or palm x thumb = fingers.  If you actually do the cross product using the method you showed, our answers agree--the particle goes counter-clockwise.

Because the force is perpendicular to the velocity, it will not change the speed--it is a centripetal force that drives the particle in a circular path.  So your assertion that a particle fired from the origin goes through all four quadrants is false.  When fired in a cardinal direction, it only goes through two of the quadrants.  Draw some circles through the origin tangent to the axes and you'll see what I mean.  You have to shoot down or right and with just the right velocity to get the right curvature to hit the pole.

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Aquaboy--I was imprecise in my language.  High school students in their first week of introductory physics should have been all over me for that one.  I should have said the speed doesn't change.  Of course the velocity changes when the particle changes direction.  Fixed it. :)

Answer 2

If you don't have to worry about the particle hitting any walls and you can fire the particle at any speed; then theoretically, you can fire the positively-charged particle in any direction from the origin, and it should be able to hit the pole.

If you neglect the force of gravity exerted on the particle, then the only other force acting on the particle is the force due to the magnetic field. This means that if, for example, you fire the particle in the +y direction, (i.e. the velocity is in the +y direction), then according to the Lorentz Force Law, the net force exerted on the particle would be in the -x direction. This causes the velocity of the particle to gradually shift from the +y direction to the -x direction. But due to Lorentz Force, the particle with a velocity in the -x direction, now experiences a net force in the -y direction, so the velocity of the particle gradually shifts from the -x direction to the -y direction.

All this means that since the magnetic field is in the -z direction, (i.e. into the screen), the particle will travel counter-clockwise in a curvilinear motion about the origin. If that's the case, and since the particle won't hit any walls, it doesn't matter which direction you fire the particle. In other words, the particle can hit the pole regardless of the direction in which you fire it, as long as it's fired with the correct initial speed, (because the particle is traveling counter-clockwise about the origin).

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The other response is incorrect.

For the Right-Hand-Rule, the fingers point in the direction of the magnetic field, while the thumb points in the direction of the velocity of the particle. The resulting force (NOT centripetal force) exerted onto the particle comes out of the palm. Here's a diagram of the Right-Hand-Rule:

http://www.britannica.com/EBchecked/topi...

Besides, like I said before, since the particle doesn't hit any walls, if the particle is fired in the positive y direction, it will simply travel a curved path around the origin, (through all 4 quadrants), before eventually hitting the pole.

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Ok, I'll agree that because we're working with the assumption that the magnetic field is uniform throughout that the particle doesn't just follow general curvilinear motion, but in particular, a circular path. Meaning that the only way for the particle to hit the pole is if it is fired in the negative y direction or positive x direction, as Bekki suggested.

But I still disagree that the velocity of the particle is unaffected by the force exerted onto the particle due to the magnetic field. It is this force that is exerted onto the particle that changes the direction of the particle. If the force didn't exist, (i.e. if the magnetic field didn't exist), the particle would simply travel in a straight line away from the origin from whence it is fired. In other words, the force does change the velocity of the particle, which is precisely what causes the particle to travel in a circular path.

Answer 3

Recreation room on a submarine?  LOL.  That's funny!