This is meant to be a quick and not comprehensive post about the reasons (or non-reasons) for prohibiting electricity on Shabbos. Note, this only applies to non-incandescent appliances.
Various poskim offer the following reasons why electricity should be prohibited
1. Molid (Beit Yitzchak 2:31)- Turning on an appliance is analogous to creating something new which is prohibited on Shabbat.
2. Boneh (Chazon Ish Orach Chaim 50:9) - Completion of a circuit is prohibited because it is a form of building.
3. Makeh B'Patish (Chazon Ish Orach Chaim 50:9)- Turning on an appliance completes it.
4. Sparks (Chazon Ish Orach Chaim 50:9) - Completion of a circuit creates sparks and therefore is prohibited because it creates a flame.
5. Increased fuel consumption (Chashmal Leor Halacha 2:6) - The use of electrical appliances leads to an increase in fuel consumption at the power station, which is prohibited.
6. Heating of metal (Chazon Ish, Orach Chaim 50:9) - Heating of a metal transistor or wire, even when no visible light is emitted, is prohibited because of cooking or burning.
These were rejected by RSZA for the following reasons:
1. Only a limited number of actions were prohibited by Chazal because of molid, and therefore we may not extrapolate from these limited examples that creating anything else new (like electrical current) is rabbinically prohibited.
2. Closing a circuit is analogous to closing a door (which is permitted) because it is meant to be opened and closed.
3. Since the appliance is made to be turned on and off it cannot be makeh b'patish
4. This is not factually true anymore
5. This is at most grama and in fact is many times not true.
6. This is not factually true anymore
Based on the above RSZA (Minchat Shlomo 74, 84), writes the following:
In my opinion there is no prohibition [to use electricity] on Shabbat or Yom Tov... There is no prohibition of ma'keh bepatish or molid... (However, I am afraid that the masses will err and turn on incandescent lights on Shabbat, and thus I do not permit electricity absent great need...) ... This matter requires further analysis.
However, the key point in my opinion is that there is no prohibition to use electricity on Shabbat unless the electricity causes a prohibited act like cooking or starting a flame.
He states unequivocally that since the minhag is to prohibit the use of electricity, and this minhag received near unanimous approval from the poskim absent great need we should accept this tradition.
My point in my previous post was that soon we will reach a point where it will be very hard to refrain from using electricity in some form given the ubiquity of electronics and sensors everywhere and this minhag may need to be revisited.