Given the following coil in a magnetic field we can make an AC (alternating current) generator. An example of a DC (direct current) source would be a battery where the current flows in only 1 direction.

Consider the current generated by the system:

• Position 1: No current flows--force is perpendicular to the wire
• Position 2: Current flows in the CW-sense (as viewed from above). From Faraday's Law, emf = - N x (delta Phi / delta t) = - N x (delta [B x A cosine theta] / delta t). Here theta is the angle made by the normal of the loop and the direction of the magnetic field. Using a little calculus, we can show that

  emf = N x ( 2 pi frequency) x B x A x sine theta

  Here, frequency is the "rate" at which the loop is spun. Since theta is given by 2 x pi x frequency x time. At theta = 90 degrees at this point the emf is 2 pi frequency x B x A.

• Position 3: No current flows
• Position 4: theta = 270 degrees at this point and the emf is - 2 pi frequency x B x A

The emf and hence I in this generator flips polarity every half cycle of spin. The variation is sinusoidal in that not only does the direction of the current change, but the amplitude also changes.

Question: How can we arrange to make the current have one sign? That is, how can we make a DC generator? AC generator versus DC generator

Recall that for the transmission of power, we noted that it is much better to transmit the power at high voltage because for a given power need,

• P = IV ===> at high V, need a smaller current I
• small I ===> Joule heating losses in the resistive transmission line are smaller, Loss = I^2 R.
• To transmit a given amount of power through power lines with resistance R at different voltages means that we will lose
  P(V=100 V)/P(V=100,000 V)=(100,000/100)^2=1,000,000 !

That is, the transmission of power at high voltages is much more efficient than the transmission of power at low voltages.

So, the need is to take the low voltage current produced by some generator and to step it up to high voltage for transmission and then to step it down for commercial and private use. This exercise is accomplished by using things known as transformers. The transformer is a large inductive device which takes variable current at a certain V and transforms it into a current with different V. [Does this require AC?] How does this work?

Take a loop made out of some ferro-magnetic material (like iron) and wrap two coils around it:

The left coil (N_p turns) is referred to as the primary and the right coil (N_s turns) is referred to as the secondary coil.

• Run an AC current through the left coil. Because the current is AC, the coil sees a changing I ===> the coil is threaded by a changing magnetic flux.
  Phi_m ~ mu x (N/l) x I

• Because the iron has a high mu ===> Phi_m is large. Further, because the iron loop strongly enhances the field inside of the material, the induced flux is confined mainly to the loop material and the same amount of flux threads through the secondary loop as threads the primary loop!

• The secondary loop thus sees the same changing Phi_m as does the primary loop, so that

• ===> Primary: emf_p = - N_p x [changing flux]
• ===> Secondary: emf_s = -N_s x [changing flux]
• ======> (emf_p/emf_s) = (N_p/N_s)

Chapter 23

Discussion Question 1