Magnetic field of a wire, a loop and an inductor

This week's homework assignment is pretty difficult,  so I thought it might help to summarize a few essential things here. (These things are also discussed in the book (Ch's 26 and 27).)  [By the way, we have new constant: the Greek letter mu with a subscript zero. For this discussion, let's just call that mu; also let's use pi for the Greek pi (3.14159). That should make equations a little easier to read and write here. i think mu is 4 pi 10^{-7}.  what are its units?]

First, regarding magnetic fields:
The magnetic field of a straight wire is: B=mu I/(2 pi r).
The magnetic field in the center of a wire loop is:  B=mu I/(2 r) (larger by a factor of pi).
(r  is the distance from the wire; B is the magnitude of the magnetic field associated with the current flowing in the wire.)  These results come from the Biot-Savart equation and/or Ampere's relation.  Mostly we will just use these results and not get bogged down in the derivations.

There is a problem on the homework in which I believe you are asked for the magnitude of the magnetic field due to the current in a wire with a single loop in it. I think the answer involves combining the two results above, thus the total field is the contribution from the straight wire, B=mu I/(2 pi r), plus the contribution from the loop, B=mu I/(2 r).  (please post a comment if that doesn't work.)

A solenoid involves many loops of wire very close together with current flowing through all of them. The magnetic field everywhere inside a the solenoid is given by:

B=mu N/Z I = mu n I,

where I is the current, N is the number of loops in the solenoid, and Z is the length of the solenoid. ( Note that the units of B are the same as above because Z has units of length. Also, in truth, this i an approximation, in case you were wondering. It is not exactly correct, but it is a pretty good approximation in many cases of interest.)

Chapter 26 might be summarized by saying: current flowing in wires produces magnetic fields,  and we are particularly interested in the field produced inside a wire loop or a solenoid. Chapter 27 can be summarized as: changing magnetic fields generate electric fields.

In Chapter 27  the most important thing to learn and remember is that when a magnetic field changes, as a function of time, that creates an electric field.  That is what Faraday's law is all about.   Faraday's law is expressed in terms of a quantity called magnetic flux, or just flux, which is magnetic field strength multiplied by an area. For a solenoid, the flux is:
Phi = pi r^2  B
       = pi r^2 mu (N/Z) I

The reason that flux is important to us is that when it changes it produces an emf (a voltage-like quantity) given by:

emf = -d(Phi)/dt

i.e., an emf equal to the time derivative of the flux.

This relationship is the basis for the new circuit device known as an Inductor.  It means that when the current in a solenoid changes, that change will produce a voltage.   An inductor is a solenoid. The name inductor focuses on its function. The name solenoid focuses on its physical appearance. They both refer to the same thing.

The magnetic field inside the solenoid is B= mu N I/Z and the associated flux is pi r^2 times that. The emf produced by a changing current is then all those factors time dI(t)/dt, i.e.,

emf=-(pi r^2) mu N/Z dI/dt.

Actually, that is the emf per loop or turn. The total emf is N times that. From that, one obtains the inductance of a solenoid.

With this knowledge you should be able to do many of the homework problems. If not, please feel free to ask any questions here.

**Note added on Sunday:    Do you think that it might make sense to extend the due date  for some of the homework problems?  The reason I ask is that I believe that the homework problems can be divided into two categories: those that can be done using just the information provided above, and, those that can't. I would appreciate some feedback on this. Would some people go through the problems and analyze and explain which ones should be extended and why?   Your efforts will be much appreciated. (PS. It is especially helpful if you include a summary of  what the problem is asking as well as it's number.)