Magnetic Dipoles and Magnetic Materials

Home , Current density, Dipole, Magnetic dipole, Magnetic moment, Magnetization

Kenneth H. Carpenter Department of Electrical and Computer Engineering Kansas State University October 17, 2008

1 Magnetic dipoles as pairs of magnetic charges

The analogy of static magnetism and electrostatics makes it possible to deﬁne magnetic dipoles as a pair of opposite signed, equal magnitude magnetic charges, located near each other. Such a pair has the dipole moment given by

pm = qm d pˆ (1) where qm is the magnitude of the magnetic charges (pole strengths), d is the distance separating them, andp ˆ is a unit vector pointing from the negative charge of the pair to the positive charge. z

When qm is at position r+ and −qm is at r− q then r m + −q m p = q (r −r ) = q d, for d = (r −r ) r− m m + − m + − r (2) y the vector distance between charges. In the

limit |d| → 0 while qm → ∞ so that |pm| remains ﬁnite, the result is the ideal point x dipole.

1 EECE557 Magnetic Dipoles – supplement to text - Fall 2008 2

1.1 The magnetic scalar potential due to a single dipole One may deﬁne a magnetic scalar potential in the same way as an electric scalar potential, just as long as the current density is zero in the region of in- terest: J = 0. The same derivation[1] may be followed as for the electrostatic potential of an electric dipole to obtain the potential Vm(r) due to dipole pm,

pm · (r − rp) Vm∞(r) = 3 . (3) 4πµ0|r − rp)|

1.2 Dipole density A magnetic material contains large numbers of magnetic dipoles. Thus one may deﬁne a dipole moment density, in a similar manner to electric polar- ization, as P i pmi P m = lim . (4) ∆V →δ>0 ∆V One may now use this dipole moment density to ﬁnd the magnetic scalar potential it produces: ZZZ 1 P m(rp) · (r − rp) Vm∞(r) = 3 dVp, (5) 4πµ0 |r − rp)| where the dipole moment density P m is a function of the coordinates Rp over which the integration is performed.

1.3 Using the scalar magnetic potential and magnetic dipole density All the derivations[1] made to ﬁnd equivalent bound charge densities for the electrostatic dielectric materials may be repeated to ﬁnd a set of equations for magnetic materials as

ρmsb = P m · nˆ and ρvmb = −∇ · P m (6) as surface and volume magnetic charge densities. These may be used to ﬁnd the pole strengths of permanent magnets. EECE557 Magnetic Dipoles – supplement to text - Fall 2008 3

2 Magnetic dipoles as inﬁnitesimal current loops

The treatment of magnetic dipoles as pairs of magnetic charges fails to pro- vide an adequate description of the total magnetic ﬁelds when currents are present. Then the magnetic scalar potential cannot be used. Instead one must use the magnetic vector potential, which has current density as its source. Thus one needs to be able to account for the ﬁelds of magnetic dipoles as due to currents. One way to do this is as follows.

2.1 Magnetic potential of an infinitesimal current loop In consideration of the torque on a current loop[2] the relationship ZZ m = I dS (7) defines the magnetic moment, m, of a filamentary current loop having current I. The surface integral is over any surface bounded by the filamentary loop, with the direction of dS given by the right hand rule applied to the current direction on the edge. Now we consider the magnetic flux density produced by such a current loop. Let dl be the line element along the current loop. Then the magnetic vector potential due to the current loop is given by

µ I I dl A(r) = 0 , (8) 4π |r − r0| where r is the position vector at the field point (where A is evaluated) and r0 is the position vector at the source point (where dl is evaluated). Note that this is the correct form for magnetic vector potential for static fields, where ∇ · J = 0, ∇ · A = 0, and A has its zero at infinite distance. Next, assume that the origin for the position vectors is on the surface bounded by the current loop, but that the field point is far from the current loop. Then one has |r0| << |r|. (In the following, use r = |r| and r0 = |r0|.) Thus one may factor out r from the denominator in eq.(8) to obtain

µ I I dl A(r) = 0 . (9) 4πr |rˆ − r0/r| EECE557 Magnetic Dipoles – supplement to text - Fall 2008 4

Next, recall that

|rˆ − r0/r| = p(ˆr − r0/r) · (ˆr − r0/r) = p1 − 2ˆr · r0/r + (r0/r)2 (10) and expand the reciprocal of p1 − 2ˆr · r0/r + (r0/r)2 by the binomial theorem, to obtain in eq.(9) µ I I A(r) = 0 [1 +r ˆ · (r0/r) + O(< r0/r >2)]dl, (11) 4πr where O(< r0/r >2) stands for terms on the order of the square of the ratio of r0/r. These terms will vanish in the limit of the inﬁnitesimal current loop. Further, note that H 1dl = 0, so that µ I I A(r) = 0 (ˆr · r0)dl. (12) 4πr2 Now there is a theorem in vector analysis, that is similar to Stoke’s theorem, which states[3, 4] I ZZ Φ(r0)dl ≡ (−∇0Φ) × dS. (13)

In order to apply this theorem to eq.(12) one must take Φ(r0) =r ˆ · r0. Now one may evaluate ∇0Φ using Cartesian components. Letr ˆ = axˆ + byˆ + czˆ. Then ∇0Φ = ∇0(ax0 + by0 + cz0) =r. ˆ (14) Finally, µ I ZZ µ ZZ µ A(r) = 0 (−rˆ) × dS = − 0 rˆ × I dS = 0 (m × rˆ). (15) 4πr2 4πr2 4πr2 Since the origin of coordinates was taken to be on the surface bounded by the current loop, in the limit of an inﬁnitesimal current loop, the loop is at the origin. When one has a magnetic dipole moment m that is located at rm instead of at the origin, eq.(15) becomes

µ0 m × (r − rm) A(r) = 3 . (16) 4π |r − rm| One may now find the magnetic flux density due to the magnetic moment, m, of the infinitesimal current loop by B = ∇ × A using the A of eq.(16) and EECE557 Magnetic Dipoles – supplement to text - Fall 2008 5

see that it is identical to the magnetic ﬂux density found by B = −µ0∇Vm∞ using the Vm∞ of eq.(3) provided the identiﬁcation is made that

pm = µ0m. (17)

Thus, magnetic dipoles may be equivalently represented as pairs of magnetic charges or current loops. In the following, the current loop form, m, will be called the magnetic dipole moment.

3 Magnetic ﬁelds in the presence of magnetic materials

The form of magnetic vector potential due to the dipoles in a magnetic material may be expressed as an integral over the magnetic dipole moment density M, sometimes called the magnetization,( M = P m/µ0) with eq.(16) as the integrand: ZZZ µ0 M(rm) × (r − rm) A(r) = 3 dvm, (18) 4π |r − rm| where integration is over the component coordinates of rm.

3.1 Conversion from magnetization to magnetization currents In a manner similar to that used with electrostatic dipoles, one may use

1 (r − rm) ∇m = 3 , (19) |r − rm| |r − rm| where the gradient is with respect to the coordinates of rm, in eq.(18) to obtain a form for manipulation with vector analysis identities[4],

∇ × (ψF ) ≡ ψ(∇ × F ) + (∇ψ) × F (20) and ZZ ZZZ −F × dS ≡ ∇ × F dv. (21) h EECE557 Magnetic Dipoles – supplement to text - Fall 2008 6

Using eqs.(19), (20), and (21 in eq.(18), one obtains µ ZZ M(r0) × dS µ ZZZ ∇0 × M(r0) A(r) = 0 + 0 dv0, (22) 4π |r − r0| 4π |r − r0| h where r0 is the dummy variable of integration. On comparing eq.(22) with the form for vector magnetic potential in static ﬁelds in terms of currents: µ ZZZ J(r0) A(r) = 0 dv0 (23) 4π |r − r0| one ﬁnds that one may replace the magnetization in a material, M, by equivalent magnetization current densities,

J m = ∇ × M and J sm = M × n,ˆ (24) where the volume magnetization current density is the curl of the magnetization and the surface magnetization current density, on the surface of the material (or at a discontinuity in material properties) is the cross product of the magnetization and the outward normal from the material volume,n ˆ.

3.2 Modification of the magnetic field equations in the presence of magnetic materials The Maxwell equations for magnetic flux density for static fields are

∇ · B = 0 and ∇ × B = µ0J total (25) where the “total” subscript is used to indicate that if magnetic materials are present the the magnetization currents must be included. If one explicitly separates the magnetization current density to write

J total = J + ∇ × M (26) then one may write the Maxwell curl equation as B ∇ × − M = J. (27) µ0 Now, deﬁne the magnetic ﬁeld intensity in the presence of magnetic material as B H = − M (28) µ0 EECE557 Magnetic Dipoles – supplement to text - Fall 2008 7 and one has the new Maxwell equation

∇ × H = J. (29)

One often writes eq.(28) as B = µ0H + µ0M to appear more like the macro- scopic electric field equation D = 0E + P . But one should note that D is the macroscopically defined quantity for electric fields while H is the macroscopically defined quantity for magnetic fields.

3.3 Magnetization in terms of applied ﬁelds

In an isotropic magnetic material, one may write M = χmH in a manner similar to P = χe0E for dielectrics. Then one may deﬁne µr = 1 + χm and write B = µ0(H + M) = µ0(1 + χm)H = µrµ0H = µH. (30)

4 Problems

1. Given a magnetic dipole moment m = 3( A/m)ˆy is located at (1 m, 2 m, −3 m)

(a) ﬁnd the expression for A(r) at general coordinate r. (b) ﬁnd B as ∇ × A.

2. A material has µr = 1.0002. Find M at a point where H = 3( A/m)ˆx. 3. A long, straight ﬁlamentary conductor lies along the z axis and carries current 5 A in the +z direction. Region ρ < 0.1 m is vacuum. Region 0.1 m < ρ < 0.5 m has µr = 300. Region 0.5 m < ρ is vacuum. (Here ρ is the radial coordinate in cylindrical coordinates.)

(a) Find H everywhere. (b) Find B everywhere.

(d) Find the magnetization surface current density J sm on the surfaces of the magnetic material at ρ = 0.1 m and ρ = 0.5 m. EECE557 Magnetic Dipoles – supplement to text - Fall 2008 8

References

[1] Kenneth H. Carpenter, “Dipoles and Dielectrics,” Department of Elec- trical and Computer Engineering, Kansas State University, September 22, 2008.

[2] Kenneth H. Carpenter, “Torque on a Current Loop,” Department of Electrical and Computer Engineering, Kansas State University, October 5, 1992.

[3] Chester H. Page, Physical Mathematics, D. Van Nostrand Co., Prince- ton, N.J., 1955, p 18.

[4] Murray R. Spiegel, “Theory and Problems of Vector Analysis,” Schaum’s Outline Series, McGraw-Hill, New York, 1959.