Basic Wireless Communication for Microcontrollers
Chapter 1 - Electricity and Magnetism
The Postulates
We have discussed the basics of electricity and magnetism in words. The relationships which we have already mentioned can be condensed and re-expressed mathematically. While the more concise mathematical equations require more thought to understand, they ultimately offer a more powerful and complete understanding.#1) Conservation of charge - Charge can neither be created nor destroyed. Equal and opposite charges can cancel each other, but the charge itself does not cease to exist, as evidenced by the E field which exists between the opposing charges.
#2) Coulomb's Law - Coulomb's Law gives the force between charged bodies: F=(1/(4*pi*epsilon_o))* (Q1 * Q2)/r^2), stated in the MKS (Meter Kilogram Second) system of units. Q1 and Q2 are in Coulombs, F is in Newtons, and r is in meters. Epsilon_o is a constant called the permitivity of free space. It is equal to approximately 8.85e-12 Farads/meter. The force, F, is directed along the line between the two charged bodies.
#3) Lorentz's Law - Lorentz's Law yields the force on a moving charge due to a magnetic field: F=qV*B*sin(theta). In the MKS system, q is in Coulombs, V in meters/second, B in Amperes/meter, and theta is the angle between V and B. F is directed according to the right-hand rule (take your right hand, point your fingers along the direction of V, curl them along the direction of B, then your thumb will point along F).
#4) Gauss's Law - Gauss's Law tells you how to determine the electric field from a charge distribution. Phi=Q/epsilon_o in MKS units. Phi, called the electric flux, is the integral of the outward-pointing component of the E field over the area of any closed immaginary surface containing the charge Q. If you immagine a surface such that the E field is constant over it (i.e., like a spherical shell around a point charge at its center), then you can get the magnitude of E by just dividing phi by the area of the surface. Since the force on a charged particle by an electric field is just F=qE with F and E parallel, you can also use Coulomb's law to determine the E field of a point charge, from which you get E=(1/(4*pi*epsilon_o))*Q/r^2. There are many cases where it is not possible to easily define such a surface over which E is constant. In those cases, the problem has to be approached differently and a modified form of Gauss's law must be used (such as breaking the charge distribution up into pieces and using the E field equation, given above, treating each piece as a point charge).
#5) Phi_B=0 - This tells us that the integral of the outward-pointing component of the magnetic field over the area of any closed surface will be zero. This is equivalent to saying that there is no such thing as magnetic charges, and that B field vectors do not emerge or converge on any point, unlike electric field lines, which do eminate from charged objects. This postulate is effectively Gauss's law for magnetism.
#6) curl B = Uo*J+Uo*Epsilon_o*dE/dt - Postulate 6 shows how magnetic fields are created: due to both current and changing electric fields. Uo is a constant, like Epsilon_o, called the permeability of free space. It's value is about 1.2566e-6 Henries/meter. Curl B is a special type of derivative of a vector with respect to distance, and it tells you how much the field is increasing or decreasing along the directions which are perpendicular to the direction the vectors are pointing. Because Phi_B=0, B fields do not increase or decrease along the direction they point, only in directions perpendicular to that. J is the current density, in amperes/meter^2.
The postulate tells us that when we have either current flowing, or a changing E field, or both, a B field will be present and we also know how much it will be changing over distance. Postulate 6 and the next one, Faraday's law, are differential equations. They are very useful for thinking about E&M, but in order to get actual solutions to them is very difficult in most cases, as is discussed in the section "Solving General Problems."
#7) Faraday's Law: curl E = -dB/dt. Faraday's Law is the basis of electromagnetic induction. It says that a changing B field produces an E field which changes over distance, in the same way as the B field discussed in postulate 6. As discussed under "AC E and B fields", E fields such as this do not exhibit path invariance. If you place a wire loop in such an E field, there will be a net voltage around the loop, because the E field all around the loop pushes charge in the same direction. It is like traveling along a closed loop of road which is always downhill, never going uphill again (impossible for roads but possible for E fields). So, changing B fields produce voltage around closed loops (the basis for transformers and inductors) and, equivalently, they produce E fields in free space (the basis for EM waves, as we shall see).
Incidentally, if a loop of wire were placed in this E field, and the wire had no resistance, infinite current would flow (a finite voltage across no resistance means infinite current). This current would prouduce its own B field, which would tend to cancel the original. What would actually happen is that the current in the loop would change exactly as needed to cancel the changes in applied B field, and there would be no actual changes in the B field.
For any real loop of wire, with resistance, the cancelation would be imperfect and some B field change would occur. This variation of Faraday's law is called Lenz's Law, which says that perfectly conducting loops of wire prevent changes in the B field passing through them by creating an AC current in the loop.