# How torque acting on a dipole?

1
by sachuz

2015-01-13T01:19:23+05:30

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See diagram.

let us say the question is about an electric dipole or about a magnetic dipole.  It is very similar for magnetic or electric dipole.

Let an electric dipole of charges +q at point A and -q at point B be attached by a thin weightless dielectric rod AB of length  2a.  The electric dipole moment is 2 a q = p.  The thin rod id held fixed at a pivot O in the center of AB.  It can rotate freely in the plane of electric field E and the dipole axis.

Let there be an electric field E uniform in that region.  Let the electric dipole line joining them make an angle with the electric field E.  Then there is a force F on +q in the direction of E, and F = + qE.  There is a force on the charge  -q (coulomb's force) F2 = - F = -q E.

Since these two forces are not collinear, they do not cancel. There is a torque around the fixed point O.  Because F and -F try to rotate the rod AB in the same direction.

magnitude of  Torque = sum of  arm length *  force
=  a Sin θ * q E + a Sin θ * q E
= 2 a q E Sin θ =  p E  Sin θ

In vector notation, we have
T = Sigma  r X F    =  a q E Sin θ  k + (- a) ( - q E)  k
= 2 a q E Sin θ  = p X E

The dipole and field are in x-y plane and torque is perpendicular to them is parallel to the z axis, passing through O.