A large number of turns are wound in a single layer on the surface of a plastic sphere. What is the magnetic field at the center of the sphere when current flows through the it.

Consider a large number of turns N of closely spaced (insulated) fine wire are wound in a single layer on the surface of a plastic sphere of radius a. All the planes of the turns are perpendicular to the axis of the sphere (assume the z-axis) and the windings entirely cover the surface. If the current winding is I, what is the magnetic field at the center of the sphere?

A bar slides on two long horizontal parallel conducting rails in a vertical magnetic field. A resistor R connects the rails. Find the equations of motion of the bar.

1. A bar of mass m, slides without friction on two long horizontal parallel conducting rails. The bar lies across the rails (perpendicular to them) and remains that way while sliding. The rails are separated by a distance d. There is a uniform, vertical magnetic field. A resistor R connects the rails. At time t=0 the bar is given velocity v0 to the right. (i) Find the equations of motion of the bar at arbitrary time and solve for the motion as a function of time. (ii) How far does it go? Where does initial kinetic energy of the bar end up?

Magnetic field

1. A long cylindrical beam (with circular cross section) of charged particles has velocity v in the axial direction. The beam has a uniform charge density f (assume fixed). Find B inside and outside the beam as a function of distance from the axis, in terms of given quantities.

Vector potential

(See attached file for full problem description) --- 1. Consider a uniformly charged sphere with total charge Q, mass M of uniform mass density, and radius R. The sphere rotates about a diameter (assume the z-axis) with angular velocity . Find the vector potential anywhere on the axis of rotation.

please see attachment below

Answer only question number (3). Please give detailed solution