A toy-train cart carrying an object with charge q and total mass M moves freely (no...

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A toy-train cart carrying an object with charge q and total mass M moves freely (no friction) on a cir- cular track with radius a, lying in the x-y plane. A permanent dipole magnet with magnetic moment m is located at z = zo. The distance between the magnet and the track is R = +a. Q1) Find B(R), the radial component of magnetic field on the track by the dipole magnet. (Hint: Recall the relation between the magnetic field and magnetic moment m, or see your lecture note at Lecture #9 or #10). Q2) Find magnetic flux (t = 0) in the track. (Hint: B(R)dS, here the surface integral will be the area of "cap" of sphere around the magnet and it is bordered by the track ring. In other words, integral from 0 = 0 to 0a, where sina = a/+a, and integral from = 0 to 2.) You move the dipole magnet from z = zo to z = 0 by spending t = T. Q3) Find (t = T) (You can use the answer of Q2, with z = %0 = 0) Q4) Faraday's law tells that the time difference of magnetic flux (d) induces the Electric field around the track. Find induced E around the track, by using and radius of track a. Q5) This induced Electric field does "work" to the toy-train cart. Find the velocity v of the toy-train after you move the dipole magnet from z = zo to z = 0. (Hint: v=Fdt = Edt) Toy train cart: Mass M Charge q Dipole magnet Z a -Zo y Figure 1: A schematic image of a train cart on the track. A toy-train cart carrying an object with charge q and total mass M moves freely (no friction) on a cir- cular track with radius a, lying in the x-y plane. A permanent dipole magnet with magnetic moment m is located at z = zo. The distance between the magnet and the track is R = +a. Q1) Find B(R), the radial component of magnetic field on the track by the dipole magnet. (Hint: Recall the relation between the magnetic field and magnetic moment m, or see your lecture note at Lecture #9 or #10). Q2) Find magnetic flux (t = 0) in the track. (Hint: B(R)dS, here the surface integral will be the area of "cap" of sphere around the magnet and it is bordered by the track ring. In other words, integral from 0 = 0 to 0a, where sina = a/+a, and integral from = 0 to 2.) You move the dipole magnet from z = zo to z = 0 by spending t = T. Q3) Find (t = T) (You can use the answer of Q2, with z = %0 = 0) Q4) Faraday's law tells that the time difference of magnetic flux (d) induces the Electric field around the track. Find induced E around the track, by using and radius of track a. Q5) This induced Electric field does "work" to the toy-train cart. Find the velocity v of the toy-train after you move the dipole magnet from z = zo to z = 0. (Hint: v=Fdt = Edt) Toy train cart: Mass M Charge q Dipole magnet Z a -Zo y Figure 1: A schematic image of a train cart on the track.