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Magnetism and Vectors

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Consider an otherwise free particle with charge q moving in a uniform magnetic field whose field vectors are B. For the sake of simplicity, let's orient our reference frame so that the magnetic field points entirely in the z direction.

a) The component definition of the cross product specifies that

(see attached)

b) Show that if we integrate these equations from time t = 0 to some arbitrary time t, we get

(see attached)

where x_0, y_0 and z_0 are the components of the particle's position at t = 0; v_ox, v_oy, v_pz are components of its velocity at t = 0; and v is a constant whose value you should determine. Note that equation E7.22c shows that the particle's velocity component along the field direction is constant.

c) Our freedom to choose the origin of our reference frame in the xy plane implies that we can find a coordinate system where y_0 - bv_0x, and x_0 + bv+0y are zero. Draw a sketch showing how we might do this in an example case (perhaps one where either v_0x or v_0y is zero).

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Solution Summary

This solution includes the formula for the acceleration of the vector a, and integrates with with respect to t to get the velocity. The origin of the new coordinate system is found. This solution is provided in an attached Word document.

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