Wires carrying AC current do not get heated up . Why ?

Heating coils work perfectly well in AC - which shows that wires do heat up when AC passes through them.

Some electrical phenomena depend directly on the current, so they grow and fall and, indeed, reverse direction when AC is used. Often, our senses, or measuring instruments can not keep up with the rapid variation and show the average effect. For instance, the pointer of a table (analog) galvanometer will swing back and forth when subjected to a low frequency AC, but will usually stay put at the zero mark (or, at best, tremble slightly) when higher frequencies are used. In fact, the AC supplied by the mains has a large enough frequency for this to happen (in my country, the line frequency is 50 Hz, which means that one complete cycle takes only 20 milliseconds).

Heating is different! That's because the rate at which heat is produced in a wire is directly proportional not to the current, but to the square of the current. This means that while the heat generated in successive small intervals will grow and fall during a cycle, it will never be negative. Thus the heating will always add up - never cancel!

It is true that the average rate at which a wire heats up when AC passes through it is less than when a DC of strength equal to the peak current is used. On the other hand, alternating currents and voltages are rated using their RMS (or root mean square) values. A 120 Volt AC will produce heat, on average, at the same rate that a 120 Volt DC supply will, when connected across the same wire - but of course this means that the peak value if the AC voltage will have to be higher (by about 40%).