parametric equation of the circle calculus homework help

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I’ve tried to answer the problem but I’m pretty sure that everything I did was wrong.

Here’s the problem.

This fun problem relates to physics. In case you’re not a physics major, we urge you to follow the immortal advice of Douglas Adams: Don’t Panic! You don’t need to be Marie Curie to solve this one. Let M be a vector in three-space.

A constant magnetic field of value M exists at all points of three-space.

(a) Suppose M = <2, 1, 5>. What is the value of the magnetic field experi- enced at the point (3, 2, −3)? [Hint, this is not a trick question, it’s a warm-up]

A particle of unit mass and unit electric charge is in the field and it experiences velocity along the vector v. The field acts on the particle via the Lorentz force F = v × M. Since this force is perpendicular to v and M, by the laws of physics it does no work, and thus it can change the direction of the particle but not its speed. If v is perpendicular to M then the particle will circle in a plane perpendicular to M. The radius of the circle is equal to |v|/|M|.1

(b) Suppose the particle is at the point (3, 2, −3) and v = <0, 1, 0>. What is the particle’s charge and in which direction is it going to move? [Hint, this is also meant to be a softball question.]

(c) Suppose that the particle is observed to be travelling along the circle (cos t, sin t, 0) in the x,y-plane. What is the velocity v of the particle at time t? Use Mathematica to plot in a single graph the circle and the velocity vector at some point on the circle. What magnetic field M could be acting on the particle?

(d) What sort of trajectory do you expect if the particle acquires a velocity component parallel to M? For example, what sort of trajectory do you expect if M = <0, 0, 1> and v = <1, 1, 1>? A qualitative answer is accept- able here.

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