Why is electricity not transmitted wirelessly such that we don't need to span cables on the earth's surface? As in: electricity is transmitted wirelessly from the power plant to the household.
Electricity is the flow of electrical charge - generally electrically charged particles called electrons in a wire. It can't flow through air, except in the form of electrically charged particles of air - as in a spark or lightning stroke.
Magnetic fields can travel in air, so you can send electricity by using it to make a magnetic field and then using the magnetic field at the other end to make electricity. This is how a transformer works - but it only works efficiently if the two sets of wire making the magnetic field are very close.
You can use it for sending small amounts of electricity a short distance where a wire (or connector) would be difficult, such as charging an electric toothbrush - but it's not efficent for large amounts or a long distance.
Power is transmitted wirelessly in many applications, just not where transmission efficiency is high importance. And that is the reason it is not done more: efficiency. At low frequencies, galvanic (that means metal) conduction of electricity is many, many, many orders of magnitude more efficient than, say, air.
At higher frequencies, one can use electricity to create E-M waves which travel well in air (or even space); but the problem is that the medium in which they travel is typically 3D (rather than the functionaly 1D path in an electric circuit), so the stuff spreads out a lot and the receiving station only gets a very small fraction of it (your car or home radio, receiving a signal from a braodcast station, is an example).
Techniques have been and are continually being developed to force power to travel in a 1-D manner, instead of 3D, in air without wave guides. Some successes include phased array RADARs, lasers, microwave antennas for telecommunications, Yagi antennas, and parabolic dish antennas. However, no commercially viable technique has surfaced to move large amounts of electrical power over long distances without long stretches of metal.
It can be done via electromagnic waves. The higher the tansmission frequency the better. Tesla wanted to do it. But is is very inefficient, much of the power will be dissapated where it won't do you any good (such as heating the ground), or be poached by non-paying people who have learned how to harvest it. There is some work on antennas that can be used to obtain energy from the environment (mainly radio waves), but these are for very low power remote devices, they are not suitable for bulk power applications.
Actually there have been made some wireless chargers where the energy is transferred by using electromagnetic field. Anyway the efficiency of such chargers is much lower than wired chargers due to following reasons:
The antenna (remote charger) doesn't catch the whole electromagnetic field and therefore only a certain ammount of available energy is received
The power decreases with distance.
The first issue could be solved with directional antenna (in this case in my opinion it is much easier to plug the cable directly into the power source than fixing the receiver - remote charger - into the optimal receiving point)
Regarding the decreasing field amplitudes the electromagnetic field consists of far field and near field. the near field dominates near the source and the far field extends from about two wavelengths distance from the antenna to infinity. The near field however decreases faster (1/d^3) whereas the far field decreases with 1/d^2. Due to these facts wireless communications exploit far field and the frequencies are high.
To conclude the answer I would say although it is possible to transfer the energy wirelessly it is not affordable.
Though I don't, by any means, have a collegiate degree in Physics or Electromagnetism, I'd like to attempt to give an answer.
First, we need to understand what electricity is. Fundamentally, electricity is the flow of particles with any electric charge. There are two common matter particles with a charge: protons and electrons. Since the proton's mass is far greater than that of the electron, and the fact that protons are held together in larger nuclei via gluons (force carriers of the strong force), electricity can be thought of as the flow of electrons. They are far easier to move as they are held in atoms less strongly and have less than 1/1000th the mass of a proton despite having an identical charge (opposite charge). Simple electrodynamics dictates that electrons flow from a more negatively charged system to a less negatively charged system, much the same with heat.
That being said, when you are talking about wireless technology, I would assume that you are speaking of, for example, a Wireless Router or Access Point which uses electromagnetism (photons at various wavelengths) to carry information, not electrons. In physics, of course high energy photons can be converted to equal quantities of electrons and positrons, but a particle accelarator in a laptop would seem... inefficient.
Electromagnetism would be the best bet for transferring energy (more specifically, electrons) over wireless technology, but as of now, a long distance electromagnetic charging station is neither practical nor reasonable.
I'm sorry that I'm not able to provide formulas or advanced topics, but I hope I've been able to provide some assistance.
I have one example of useful power being transmitted over long distances.
When I was a kid I got an electronics kit and one of the circuits you could build was an unpowered radio receiver - the energy of the broadcast signal was enough to drive the earbud. It is also called a crystal set and you can read more on wikipedia.
Nicola Tesla was actually perusing this wireless energy transfer by means of his telsa coils. He actually designed large EM coils that could be placed near the generating station and smaller coils at subscriber end.
Well, people at that time, especially under the influence of Edison (they were arch enemies), made a mockery of him and the attempt was soon discarded after his death.(One may call this a conspiracy theory)...
But then it is possible....
EDIT1: One viable solution is Tesla Coil.