## The Magnetic Dipoles

**The Magnetic Dipoles** Assignment Help | **The Magnetic Dipoles** Homework Help

# The Magnetic Dipoles

Consider a circular loop of wire of radius a, situated at the origin in a plane perpendicular to the z-axis and carrying a current I. We shall first calculate the vector potential A at a point P on the x-axis z-plane.The vector potential dA at this point due to an element dI is

dA = μ

_{0}/4ππ I dI/r’

Since dI is in x-plane, dA is in x-y plane.

For any given value of r’, we can consider two symmetrical dI elements whose y-components add and whose x-components cancel. Then we need only calculated the y, or azimuthally components of A. Thus total vector potential due to the whole loop is

A = μ

_{0}/4π I ʃ dI/r’ cos Φ

= μ

_{0}/4π I ʃ (adΦ) cos ɸ/r’ Φ1.

Here, Φ

_{1}is the azimuthally unit vector in spherical polar coordinates.

0.

We must now express r’ in terms of r and of ψ.

r’

^{2}= r

^{2 }+ a

^{2}– 2ar cos ψ.

Or 1/r’ = 1/r [1 + a

^{2}/r2 – (2a/r) cos ψ]1/2 = 1/r [1 – a

^{2}/2r

^{2}+ a/r cosψ]

We have assumed here that a << r. To replace cos ψ, let us write r.a in two ways and compare (xi + zk).(a cos Φi + a sin Φj) = ra cos ψ or x cos Φ = r cos ψ.

. : 1/r’ = 1/r [1 – a

^{2}/2r

^{2}+ ax/r

^{2}cos Φ

and A = μ

_{0}I/4π ʃ a cos ɸ/r [1 – a

^{2}2r

^{2}+ ax cos ɸ/r

^{2}]dɸ

= μ

_{0}/4π Iπa

^{2}r

^{3}x = μ

_{0}4π Iπa

^{2}r

^{2}sinθ = μ

_{0}4π / mr

^{2}sin θ.

Or A = μ

_{0}/4π m x r

^{1}/r

^{2}= μ

_{0}/4π ∇ x m/r,

Here m is the dipole moment, given by m = I(πa

^{2})k = IS.

r

^{1}is a unit vector.

To find B, we simply compute ∇ x A.

The magnetic induction B is given by

B = ∇ x A μ

_{0}/4π ∇ [m.∇(1/r)].

Here we have used relations ∇x ∇ x A = ∇ (∇.A) - ∇2 A and ∇2 (1/r) = 0- for r ≠ 0.

The relation for B is of the same form as the expression for the electgric field due to an electric dipole of dipole moment P:

E = 1/4πε

_{0}∇ [p. ∇ (1/r)].

Thus the electric and magnetic dipoles give rise to similar field.

**The Magnetic Dipoles**click the button below to submit your homework assignment