Write down a comprehensive note on “The Magnetic Dipole Moment”.

Question

Write down a comprehensive note on “The Magnetic Dipole Moment”. 

Answer 1

A rectangular loop carries a current I in the presence of a uniform magnetic field B that is in the plane of the loop. There are no forces on the sides of the rectangle of length "a" that are parallel to the magnetic field;

ds×B is then 0 for these sides. For the sides of the loop of length "b" that are not parallel to the magnetic field, the magnitude of the forces will be F₁ = F₂ = IbB.

From an end view of the loop, one would see a side of length "a" with its left end driven upward by F₁ and its right end driven downward by F₂, which translates to a clockwise rotation of the entire rectangular loop. The torque produced by F₁ and F₂ that causes the loop's rotation about a centerline running perpendicular to the "a" sides is given by τ_max = F₁(a/2) + F₂(a/2) or IbB(a/2) + IbB(a/2) or IabB where ab or A is the area enclosed by the rectangular loop. Note that this result is only valid when B is parallel to the plane of the loop.

Let B make an angle θ with respect to a line perpendicular to the plane of the loop and assume that B is perpendicular to the sides of length "b". The magnetic forces F₃ and F₄ on the sides of length "a" would then cancel each other and produce no torque since they pass through a common origin. F₁ and F₂, however, form a couple while acting on the sides of length "b" and thus produce a torque about any point. Again taking a bottom view of an "a" side would show that the moment arm of F₁ about a centerline running perpendicular to the "a" sides is equal to (a/2)sin θ; the moment arm of F₂ about this same centerline is also (a/2)sin θ.

From F₁ = F₂ = IbB comes the net torque about a centerline running perpendicular to the "a" sides with magnitude τ equal to F₁(a/2)sin θ + F₂(a/2)sin θ or 2IbB(a/2)sin θ or IabBsin θ or IABsin θ with A the area of the loop equal to ab. IABsin θ is maximized when θ = 90° or B is parallel to the plane of the loop and is equal to 0 when B is perpendicular to the plane of the loop. The loop will tend to rotate to smaller values of θ (that is, such that the normal to the plane of the loop rotates toward the direction of the magnetic field).

A vector expression for the torque is τ = IA × B; A is a vector perpendicular to the plane of the loop with magnitude of A equal to ab, again the area of the current loop. Curling the four fingers of the right hand in the direction of I will orient the right thumb in the direction of A. IA is by definition the Magnetic Dipole Moment µ of the current loop or µ = IA; the unit for this moment is the Ampère-Dot-Square Meter (or A•m²).