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Force due to a Solenoidal Magnet
To obtain this we must remember that the work done on a unit pole by the forces of any system in going from a point P_{1} to a second point P_{2}, V_{1}, V_{2} being the potentials at P_{1} and P_{2}, is V_{1}  V_{2}. Let a be the distance between these two points, and let F be the average value of the magnetic force acting from P_{1} to P_{2} resolved along the line P_{1}P_{2}. Then the work done by the force F in moving the pole is Fa. Hence Fa = V_{1}  V_{2} and if the distance a be sufficiently small, F, the average value of the force between P_{1} and P_{2} may be taken as the force in the direction P_{1}P_{2} at either P_{1} or P_{2}.
Denoting it by F we have
when a is very small Let us suppose that P_{1}, P_{2} are two points on the same radius from O, that OP_{1}=r and OP_{2}=r+δ. Then θ is the same for the two points, and we have
neglecting (δ/r)^{2} and higher powers (see p. 42). Also, in this case, a=δ. Thus
We shall denote this by R, so that R is the force outwards, in the direction of the radiusvector, on a unit pole at a distance r from the centre of a small solenoidal magnet of moment M. If the radiusvector make an angle θ with the axis of the magnet, we have
Again, let us suppose that P_{1}P_{2} (fig, 46) is a small arc of a circle with O as centre, so that
OP_{1}=OP_{2}=r Thus a=P_{1}P_{2} = OP_{1} x P_{1}OP_{2} = rφ.
The force, in this case, will be that at right angles to the radius vector, tending to increase θ; if we call it T we have
These two expressions are approximately true if the magnet NS be very small and solenoidal. We may dispense with the latter condition if the magnet be sufficiently small; for, as we have said, any carefully and regularly magnetised bar behaves approximately like a solenoid with its poles not quite coincident with its ends. In such a case 2l will be the distance between the poles, not the real length of the magnet, and 2ml will still be the magnetic moment.


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