The attached figure shows the electric field inside a cylinder of radius R=3.0mm. The field strength is increasing with time as E = 1.0x10^8 t^2 V/m (where 't' is in seconds). The electric field outside the cylinder is always zero, and the field inside the cylinder was zero for t < 0.

a: Find an expression for the electric flux through the entire cylinder, as a function of time.

b: Find an expression for the magnetic field strength as a function of time, at a distance r<R from the center.

c: Find an expression for the magnetic field strength as a function of time, at a distance r>R from the center.

d: Draw a picture showing the magnetic field lines inside AND outside the cylinder.

 

A hollow metal cylinder has an inner radius , outer radius , length , and conductivity . The current is radially outward from the inner surface to the outer surface.

a. Find an expression for the electric field strength inside the metal as a function of the radius r from the cylinder’s axis.

b. Evaluate the electric field strength at the inner and outer surfaces of an iron cylinder if , , , and .

Faraday's law describes how electric fields and electromotive forces are generated from changing magnetic fields. This problem is a prototypical example in which an increasing magnetic flux generates a finite line integral of the electric field around a closed loop that surrounds the changing magnetic flux through a surface bounded by that loop. A cylindrical iron rod with cross-sectional area is oriented with its symmetry axis coincident with the z axis of a cylindrical coordinate system as shown. It has a uniform magnetic field inside that varies according to  .

1. Find  , the electromotive force (EMF) around a loop that is at distance R from the z axis, where R is restricted to the region outside the iron rod as shown. Take the direction shown in the figure as positive

2. Due to the cylindrical symmetry of this problem, the induced electric field   can depend only on the distance R from the z axis, where R is restricted to the region outside the iron rod. Find this field