Magnetic Dipoles and Magnetic Materials 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 p^ (1) where qm is the magnitude of the magnetic charges (pole strengths), d is the distance separating them, andp ^ 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 jdj ! 0 while qm ! 1 so that jpmj 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) Vm1(r) = 3 : (3) 4πµ0jr − rp)j 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) Vm1(r) = 3 dVp; (5) 4πµ0 jr − rp)j 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 · n^ 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 jr − r0j 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 r · J = 0, r · 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 jr0j << jrj. (In the following, use r = jrj and r0 = jr0j.) Thus one may factor out r from the denominator in eq.(8) to obtain µ I I dl A(r) = 0 : (9) 4πr jr^ − r0=rj EECE557 Magnetic Dipoles { supplement to text - Fall 2008 4 Next, recall that jr^ − r0=rj = p(^r − r0=r) · (^r − r0=r) = p1 − 2^r · r0=r + (r0=r)2 (10) and expand the reciprocal of p1 − 2^r · r0=r + (r0=r)2 by the binomial theo- rem, to obtain in eq.(9) µ I I A(r) = 0 [1 +r ^ · (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 (^r · r0)dl: (12) 4πr2 Now there is a theorem in vector analysis, that is similar to Stoke's the- orem, 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 ^ · r0. Now one may evaluate r0Φ using Cartesian components. Letr ^ = ax^ + by^ + cz^. Then r0Φ = r0(ax0 + by0 + cz0) =r: ^ (14) Finally, µ I ZZ µ ZZ µ A(r) = 0 (−r^) × dS = − 0 r^ × I dS = 0 (m × r^): (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 jr − rmj One may now find the magnetic flux density due to the magnetic moment, m, of the infinitesimal current loop by B = r × 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 = −µ0rVm1 using the Vm1 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 jr − rmj 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) rm = 3 ; (19) jr − rmj jr − rmj 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], r × ( F ) ≡ (r × F ) + (r ) × F (20) and ZZ ZZZ −F × dS ≡ r × 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 r0 × M(r0) A(r) = 0 + 0 dv0; (22) 4π jr − r0j 4π jr − r0j 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 jr − r0j one finds that one may replace the magnetization in a material, M, by equiv- alent magnetization current densities, J m = r × M and J sm = M × n;^ (24) where the volume magnetization current density is the curl of the magneti- zation 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 ^. 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 r · B = 0 and r × 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 + r × M (26) then one may write the Maxwell curl equation as B r × − 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 r × 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 macro- scopically 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.