1) A magnetic potential vector A is given in the cylindrical coordinates system (up,uo,k) by the relation A=5 psin (9)k. Prove that its associated magnetic field is given in cartesian coordinates system (i, j,k) by the relation B=5i.

Suppose there were an asteroid approaching the Earth-Moon system starting from rest at infinity.   Under what conditions would it experience forces directing it  toward a collision with the Moon rather than toward a collision with Earth?     If it were on a path within a narrow cone where  the field lines converge directly toward the Moon it would go there.  Otherwise it would end at Earth.     All the field lines converge at the Earth rather than the Moon.     The field  at a distance would take take it with equal probability toward the Earth or toward the Moon.     The field would take it mostly through the empty space between Earth and Moon with no collision

A particle of mass m, charge q and position x moves in a constant, uniform magnetic field B which points in a horizontal direction. The particle is also under the influence of gravity, g , acting vertically downwards. Write down the equation of motion and show that it is invariant under translations x -> x + x0. Obtain x = \alpha x \times n + gt + a where \alpha = qB/m, n is a unit vector in the direction of B and a is a constant vector. Show that, with a suitable choice of origin, a can be written in the form a = an. By choosing suitable axes, show that the particle undergoes a helical motion with a constant horizontal drift. Suppose that you now wish to eliminate the drift by imposing a uniform electric field E. Determine the direction and magnitude of E. Training

A magnetic dipole moment of m = (7.50, -8.38, 4.59) Am² is inside a uniform magnetic field of B = (3.34, -6.77, 6.79) T. What is the potential energy of the dipole? (a) What is the angle between the magnetic dipole moment vector and the magnetic field vector? (b) What is the magnitude of the torque acting on the dipole?