Switches, relays
A switch is nothing more than a means of connecting and disconnecting two conductors-- A wire nut on two wires could thus be considered to be a crude switch.
Let's stop to consider the wire-nut-as-switch idea further: Not twisting the wire nut onto the conductors tight enough could cause a connection that gets hot with high current and trips the breaker or even causes a fire. In a similar way, switch contacts must also be intimate if they are to pass high currents reliably. So, just as there might be a proper torque for wire nuts (and there actually is a torque specification for lugs in service panels) there is also a proper contact force for a switch.
Almost everybody knows it takes a larger diameter wire to handle higher currents, but what about the area of the switch contacts? When one stops to consider it, larger contacts for higher current makes sense.
So, a high current switch would have larger contacts and heavier contact force. This means closing and opening a high current switch would probably take more work as well-- we're trying to move a larger object that is connected to a stiffer spring. If we wanted the switch to be remotely operated, e.g., a relay, we would have to expect the relay to perform more work in opening|closing the bigger switch.
(Work is directed energy. Power is energy per unit time. You can convince yourself of this by realizing that a Watt is a unit of power, and the power company sells you electricity by the kilowatt-hour.)
So, big relays consume more power than small ones, regardless of the amount of current passing across the contacts.
There are two other things to consider when interrupting the flow of current using a switch. First, the electricity flowing through the switch has momentum. Stopping the electrical momentum by opening the switch will take more work than closing the switch. Second, this electrical momentum will actually cause a spark to be thrown across the opening switch (but not the closing switch). Thus, turning off lights actually presents more of a fire hazard than turning lights on!
When you start your engine, your starter demands a very high current for a relatively short period of time-- like up to 300 amps when the starter is first actuated -- so the starter solenoid has to be able to slam together with lots of force, but it also has to interrupt the current with a lot of force. Force -> work -> power.
Electrical power can be calculated as:
P = I*I*R
So reducing the solenoid resistance by winding it with bigger wire will give it more contact force, but it will also cause the solenoid to consume more power in doing its job. (All solenoids consume power in order to do their jobs.) But, reducing the solenoid resistance can also cause it to produce more thermal energy, and thermal energy over time can cause the solenoid to wear out.
A continuous duty solenoid either has more winding resistance to limit the current through the windings or has some kind of heat dissipating design (heat sinking, so called) to lengthen its life. The trade-off is lower current handling through the contacts, or a bigger more expensive solenoid.
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Before we conclude this post, you might be asking yourself, "Why is it that small wires supplying current to a load, e.g., the "house" battery would get hot, while big wires in the solenoid coil would get hot?"
The answer involves a trick-- In the case of the solenoid, all the charging battery's voltage is delivered to the solenoid coil. In the case of the supply wire and "house" battery, most of the charging battery's voltage is intended delivered to the "house" battery, and only a small amount is intended to be delivered to the supply wire. So the presence of the "house" battery limits the current in the circuit. But the solenoid coil is the load. Making the coil wire smaller limits the current and thus limits the heat dissipated in the wire.