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Magnetic dipole - Moving charges and magnetism

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Question

Calculate the magnetic moment (in Am2 ) of a thin wire with a current I=8 A, wound tightlly on a half a tore (see figure). The diameter of the cross section of the tore is equal to d=5 cm, and the number of turns is N=500.

Moderate
Solution

The magnetic moment of circular current is given by Al, being the circulating current and A rs the area of cross-section; the direction is perpendicular to the plane of current. Now, for an element of toroid of length rdq, its magnetic moment is along the direction of arrow as shown, of magnitude (perpendicular to its cross-section) = current x area x number of turns in the length rdq.

dM=Iπd24Nπrrdθ=N4d2Idθ

Resolving dM into components along x and !, we get dM sin θ and dM cos θ; components along y from neighbouring elements cancel out to zero, and components along x are added. So

M=dMsinθ=0πN4·d2Isinθdθ=Nd2I2

Putting the given values we get M = 5 Am2


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