In this problem, we will consider the force on a moving charged particle in a ma

Question

In this problem, we will consider the force on a moving charged particle in a magnetic field.

Consider an electron (charge -q = -1.6 times 10^-19 C, mass m = 9.1 times 10^-31 kg) moving in the earth's magnetic field at a velocity v = V_o (i + k), where v_0 = 8 times 10^6 m/s. Let's consider the earth's magnetic field, which we will take to be constant: B = B_0j + B_0k, where B_0 = 2 times 10^-5 T. What is the force F on the electron? In the previous question, it turns out the electron will follow a helical trajectory. What is the radius of the helix? What is the "pitch" of the helix (that is, how far along the long axis of the helix does the electron have to move, before it has made one entire revolution around the helix)? In reality, the earth's magnetic field is not constant, but depends on where you are. We can approximate the radial and tangential components of the earth's field as roughly: Br = -2B_0(R_e/r)^3 cos (theta) and B_theta = -B_0(R_e/r)^3 sin (theta), where R_e is the radius of the earth, theta is the angle away from the magnetic north pole, r is the distance you are away from the center of the earth, and B_0 3 times 10^--5 T. Based on this information, provide a brief (e.g. 2-3 sentence) explanation why the aurora borealis is observed near the magnetic north pole (and its counterpart, the aurora australis, near the south pole). Consider what the force on an electron would be if it was coming from outer space, directly head-on to the magnetic north pole (theta = 0), versus what the force would be if it was coming in along the equatorial plane (theta = 90 degree).

Explanation / Answer

a)

B = magnetic field = (2 x 10-5) j^ + (2 x 10-5) k^

V = velocity = (8 x 106) i^ + (8 x 106) k^

q = charge = -1.6 x 10-19 C

magnetic force is given as

F = q (V x B )

F = ( -1.6 x 10-19 ) [( (8 x 106) i^ + (8 x 106) k^) x ((2 x 10-5) j^ + (2 x 10-5) k^)]

F = ( -1.6 x 10-19 ) [160 k^ - 160 j^ - 160 i^ ]

F = (2.56 x 10-17) i^ + (2.56 x 10-17) j^ - (2.56 x 10-17) k^

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