Magnetic fields! Help!?

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) 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.

The pole is in the upper left quadrant. The magnetic field is pointed down away from the screen.

Answer 1

let position coordinate of pole be [-x, y] in said quadrant

let unit vecorts along [+/- x, +/- y, +/- z] be [+/- i, +/- j, +/- k]

before chosing firing direction, let us commit condition of (v) only having four possible direction each being at right angles to (B the meg field) fixes the mag force to be at right angles to both (B) and (v)

reaching pole (-x, y) quadrant, let in general

vector F = Fx (i) + Fy (+j)

charge = +q

B = B (-k)

velocity = v (p^)

let p^ be direction in [x, y plane] which charge was fired> equation will decide it

F = q [v cross B]

Fx (i) + Fy (+j) = q { [v (p^)] cross [B(- k)]}

Fx (i) + Fy (+j) = - qvB { p^ cross k}

if p^ = (j) >> fired in +y direction > [j cross k] = + i

Fx (i) + Fy (+j) = qvB {- i}

comparing coefficientof unit vectors gives

Fx = - qvB, Fy=0

F = qvB (- i)

both F and v (mutually perpendicular) vectors will confine the charge to be in [- x, y] quadrant and

the general coordinate of pole [- F, v] or [- qvB, v]

so you must fire the particle in +y direction, and magnetic force (guided by the right hand rule) will take it to pole> just like a person trying to cross a river in north direction lands in [N_W] direction when opposed by a water current (F) in west direction

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for other directions of (p^) you may replace

p^ by [-j , then +i, then – i] but will find that resultant of 2 vectors goes in other 3 quadrants