Why does the USA use 110V and UK use 230-240V? What are the advantages? Explain me with calculation. Why do they use different frequencies like 50Hz, 60Hz? What is the reason?
You should not be surprised that they use different voltages and frequencies, you should be surprised that there are only two big voltage/frequency standards.
When electricity was first introduced each producer provided a different voltage and frequency (or even DC instead of AC). Gradually producers merged, governments set standards, and market pressure demanded that appliances could be used everywhere. This lead to the current situation, where the pressure for a world standard is counterbalanced by the invested interests.
For the same amount of energy 110V requires more current, hence thicker wires. 230V requires better isolation. In some rare situations 220V might be more dangerous to touch.
I don't think 50 or 60 Hz makes any significant difference. (An iron core for a transformer might be a little smaller at 60 Hz. But iron cores are soooo last century...)
It's hard to be definitive. But before AC power distribution was widely adopted, there was a bitter battle in the United States between Edison and Westinghouse about DC versus AC power distribution.
Edison's DC system used +110V, 0V and -110V . There was a campaign by Edison to portray AC as dangerous, even going so far as to introduce an electric chair powered by AC electricity as an execution device, thereby demonstrating the "danger of AC". Once AC was widely accepted as being superior to DC for power distribution, 110V became the standard for AC distribution presumably because it used the "safer" Voltage level of the DC system.
After metal filament lamps became feasible, 220V became common in Europe because of the lower distribution costs.
As for 50Hz versus 60Hz.. well that's just the Metric system.
The same reason we still pave roads that go around buildings that were knocked down half a century ago.
Historically someone, or some group, picked a number in each country, others followed suit, and it became "a standard". Now we are stuck with them.
Each has it's advantages and disadvantages. You can argue them forever.
The 110V issue is simply that once Tesla and Westinghouse proved long distance AC transmission feasible, the #1 issue driving the proliferation of electrification was lighting in houses, replacing the gas and oil lighting that was a major fire hazard of the day. Edison's lamps were 100V, but a lamp does not care if it gets AC or DC. So our AC distribution system, AT THE RESIDENTIAL LEVEL, was designed to take advantage of the existing installed base and inventory availability of Edison's lamps. Then as personal appliances began proliferating, they were designed to take advantage of the 110VAC lighting circuits already used in houses and the concept cemented itself into our culture to where there was no going back.
The 50/60Hz issue is different and is not "metric" at all (what's metric about the number 50?). Despite Westinghouse/Tesla having championed it, AC only really took off here once Edison gave in to the inevitability. Edison, despite having investing in AEG as Europe began electrifying, was reluctant to allow a system in which Europeans could enter our market by selling electrical products here. So after also experimenting with different frequencies (40Hz was the first major industrial installation, at the Folsom Power House in California), Edison and Steinmetz settled on 60Hz, partly because of the flicker issue, then also because it would make European equipment incompatible. He wanted it all to himself... which is the same motivation behind his initial push to discredit AC distribution in the first place. He wanted DC because he owned the US patent rights to his DC dynamo (even though he actually bought his first one, for proof of concept, from Werner von Siemens. Yes, THAT Siemens... Siemens had not patented it in the US). So if DC had won, we would have had Edison DC dynamos every 5 miles or so. Only the rich would be able to afford it, and they would all be paying Edison for the privilege. Tesla's egalitarianism ruined his vision.
This was extensively covered in The Simpsons:
You know, Europe's no place for a six-year-old. He can handle 110 volt, but 220 would kill him.
Jokes aside, once you pick a value and produce a substantial amount of compatible devices, the price of switching to a different value becomes prohibitively high.
We learnt that it was about resources. Europe had abundant iron and was short of copper, hence 50Hz. Whereas (mistakenly) America and particularly Pennsylvania, had a surplus of copper, hence 60Hz. 110v, 60Hz has 4 times distribution loss compared with 220V (power loss = I squared R) That is why American wire is so much thicker and the overhead (not underground) distribution puts on such a spectacular display when a pole mounted transformer goes up. Also, lights flicker at twice the AC frequency, one flash on the positive half cycle and one flash on the negative half cycle, not at the supply frequency
In the UK wiring was available nationwide in the late 1950's. The rest of europe followed shortly after the US. Because it took the UK a bit of time to catch up, they had time to learn an important thing about the previous experience with household electricity - wiring houses was expensive! They had to use a lot of wire and by doubling the voltage they reduced the current to half thus reducing the wire gauge needed.
The AC freq. is a bit better known story... Up until 1890 there was no standard for the mains freq. (obviously) AEG who had a monopoly on electricity production in Europe set the standard at 40Hz, however a bit later on they had noticed lamps flicker at that freq. so they increased it to 50Hz which was fine. Westinghouse learned about this becoming a standard but they figured the lights still flicker a bit so they increased the freq. to 60Hz. In the following years they started wiring the entire US and motors and other devices were designed for that freq. so it was not easy to change it any longer.