Torque by magnetic field on a current carrying wire loop ?

[mohamed el teir]

as u know torque is given by NI(A X B) where N and I are number of loop turns and current magnitude respectively, A is area in vector and B is magnetic field in vector. regarding the direction of area : what normal to take as a direction, is at that which makes acute angle with magnetic field (supposing that magnetic field has a specific direction) ? and if the normal to the area is perpendicular to the field so which normal to choose then ? and does the equation of the torque indicate that its direction is independent from the direction of current ?

 

[Philip Wood]

This is a very good question. The handling of this sort of thing causes a lot of confusion, largely because there are at least two approaches.

Approach A (used in more elementary work)

The current, I, is treated as always positive, and the direction of the area vector is taken as the direction in which a right handed screw would advance if turned in the direction of circulation of the current.

Approach B (preferred especially in more advanced work)

The direction of the area vector is defined to be that in which a right handed screw would advance if turned in the same sense as that of an arbitrary agreed sense around the loop (e.g. ABCDA, if A, B, C and D are points in cyclic order around the loop). This is a purely geometric convention, independent of currents etc.

The current, I, although not a vector, can be positive or negative and we take it as positive if it's circulating in the sense ABCDA and negative if in the sense ADCBA.

This is all you need for the vector relationship you quote to deliver the goods, with the direction of the current taken care of by the sign of I. You'll find it gives you the same direction of torque whichever way round you put A, B, C and D in setting up the geometrical convention.

General points for handling cross products of vectors. (1) Imagine rotating the first vector like a spanner (wrench) through the smaller angle to make it lie in the same direction as the second. Direction of product vector is the direction in which a right handed bolt would advance when gripped by the spanner. (2) a minus sign in front of a vector (that is multiplication of the vector by the scalar –1) gives a vector of the same magnitude in the reverse direction.

Hope this helps.