$$\Phi_m = \int_{\cal S}\myv B\cdot d\myv a = \int_{\cal S}B_{\perp} \,da.$$
Reading: Chapters 29, ELECTROMAGNETIC INDUCTION
Study guide: Chapter 30
Where should you place a magnet to get a large current in a loop (some loops) of wire?
What is a better question??
How does the motion of a magnet affect the current in a loop of wire?
Magnetic field lines crossing a particular area...
$$\Phi_m = \int_{\cal S}\myv B\cdot d\myv a = \int_{\cal
S}B_{\perp} \,da.$$
Result of our messing around with magnets and coils: $$ I= \frac{\del \Phi_m}{\del t}.$$
But, in order to drive a current, there must be an EMF ${\cal E}$ that produces an electric field $\myv E$ that drives the current.
So,
we'll hypothesize that there is an $\myv E$ field that rings any region
inside of which the magnetic flux is changing, such that...
$${\cal E}=\oint \myv E \cdot d\myv \ell = -\frac{d\Phi_m}{dt}$$This is Faraday's Law of magnetic induction. Units of ${\cal E}$ are volts.
A square wire loop 20 cm on a side, with a 50 $\Omega$ resistor on one side lies in the $x-y$ plane. Initially there is no magnetic field. But then over the course of 1 second, a uniform magnetic field in the $z$- direction is smoothly ramped up from 0.0 to 0.8 T. During this time, what is the current in the resistor?
...through a loop of wire:
[How to steal electric power from transmission lines!!]
...when the flux changes?
Lenz's Law says...
The direction of any magnetic induction effect...
is such as to oppose the cause of that effect.
[direction of current in the square wire loop above...]
As a conducting bar is dragged at constant speed through a region with a magnetic field...
After a while, charges within the bar come to rest once...$$\begineq F_E&=&F_B\\ qE&=&qvB\\ E&=&vB\endeq$$
The total EMF between the two ends is...$$\int_a^b\myv E\cdot d\myv\ell=E\,L=vBL.$$
An alternate point of view: The flux in this circuit
is changing. What is the total ${\cal E}$ from Faraday's law?
$$|{\cal E}|=\frac{d\Phi_M}{dt}=\frac{dBA}{dt}=BL\frac{dx}{dt}=BLv$$
Currents are also induced in conductors by a changing flux, even if not in the form of a loop.
Metal detector:
Loops of wire in the pavement at traffic light intersections... [How can bicycles pretend to be cars??]
Eddy currents in falling coin...