Aurora borealis or australis, i.e. northern or southern lights, are caused by ac

Question

Aurora borealis or australis, i.e. northern or southern lights, are caused by accelerated motion of charged particles in earth's magnetic field. These phenomena are often observed in high latitudes, because the magnetic field forces charged particles to move in a certain manner. Our simplified model for this motion consists of a unit charge with unit mass in a uniform magnetic field (vector) B= (unit vector) i (This means that the magnetic field at any point is (unit vector) i) From basic physics, we know that the acceleration experienced by this charge is (vector) a= (vector) v X (vector) B, where (vector) v=v1i+v2j+v3k is the velocity of the charge.

(a) find (vector) a in terms of v1,v2, and v3.

a =   (1)
We recall that a is the derivative of v with respect to time t

a = dv1/dt i + dv2/dt j + dv3/dt k  (2)

(b) Set up a system of ODEs by matching components of a in (1) with those in (2)

dv1/dt = (3)

dv2/dt = (4)

dv3/dt = (5)

(c) Differentiate (4) with respect to t and rewrite the right hand side of the resulting equation using (5).

Explanation / Answer

(vector) a= (vector) v X (vector) B

(vector) a= (v1i+v2j+v3k) X (i)

(vector) a= -v2k+v3j (iXi=0; jXi=-k; kXi=j)

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