Does a bar magnet exert a torque on itself due to its own field ? Does one element of a curren-carrying wire exert a force on another element of the same wire ?

Explanation

No. There is no force or torque on an element due to the field produced by that element itself. But there is a force (or torque) on an element of the same wire. (for the special case of a straight wire, this force is zero).