The magnetic circuits and fields in materials Lecture 14 1 Linear current above a magnetic block In this example, assume a current density J above an infinite slab of linear magnetic material, with permeability, µ, Figure 1. The field created by J polarizes the material, and because the material is linear, only a surface current is supported; K = M ˆn. H = (1/µ 0 ) M = (1/µ) M = µ µ 0 µµ 0 = µ µ 0 µ 0 H J v = M = ( µ µ 0 µ 0 ) H = 0 Then since there are no free currents H = 0 = M For a surface current K we find the magnetic field by Ampere s law. s = µ 0 I c /2 = µ 0 M/2 To this field we add the field due to the current, J. Just above the surface this results in the vector potential equation; A = µ 0 4π dσ M ˆn r r + µ 0 4π dτ J r r The first term is obtained from s above. The magnetic induction just below the surface is ; = A = µ 0 M/2 + µ 0 4π dτ Substitute for in terms of M. This gives; µ + µ 0 M 2(µ µ 0 ) = µ 0 4π dτ J r r The magnetic induction for the medium is then; m = 2µµ2 0 µ + µ 0 dτ J r r J r r 1
z J x n^ M Surface K y Figure 1: The fields due to a current density above a clab of magnetic material 2 The field inside a torus The geometry of this problem is illustrated by Figure 2. The field can be obained from Ampere s law. The flux inside the magnetic material is assumed to capture the field so that there is no leakage of flux. For a tightly wound coil we have that; H d l = µifree In this expression, I free is the total current that passes through the Amperian loop. Suppose a current, I, and N turns through the loop. Then; = µni 2πr Then as H = (1/µ 0 ) M ; M = µ µ 0 µµ 0 = µ µ 0 2πµ 0 r I The magnetic flux in the torus is ; Flux = d area Given the geometry, it is not so easy to evaluate the integral. To approximate we take the field at the center of the torus, 2
I b a r Figure 2: The field inside a torus = µni π(a + b) and multiply by the cross sectional area, π(a b) 2 /4 µ(a b)2 Flux = NI 4(a + b) µ(a b)2 In the above NI is called the magnetomotance and the factor is the permeance. 4(a + b) The inverse of the permeance is the reluctance. This equation can be put in the same form as Ohm s law. If there is time changing flux (voltage as we will see later) which induces an electromotance in the magnetic circuit then (here NI I); V = I µ(a b)2 4(a + b) Electric and and magnetic circuits are similar, differing in that the magnetic flux does not follow a prescribed path as does an electric current, as only part of the magnetic flux follows a magnetic path in material, although with high values of µ the flux leakage is small. 3
3 Force and Torque As previously, the force due to a magnetic field acting on a current element is; df = Id l which can be put in the form; F = dτ J Then the torque about some axis for a current element displaced by r is dn = r Id l and N = dτ [ r ( J )] Then expand (r) at the point, r = a, where the torque is applied; (r) = (a) + ( r ) (a) + Substitution gives; F = dτ [ J (a)] + dτ[ J ( r ) (a)] The first term vanishes as dτ J = 0 Thus one can write; F i ǫ ijk dτ [ Ji ( R ) k ] We developed the magnetic moment using an expression; r dτ r J i = (1/2) [ r dτ ( r J)] i Use this in the expression for F. dτ ( r ) b (a) J l = (1/2)[ k (a) dτ( r J)] l F i = ǫ ijk [( m ) j k (a)] This results the vector equation; 4
F = ( m ) = ( m ) m( ) = ( m ) If we define the energy as W = m then; F = W For the torque; N = dτ r ( J ) = dτ [( r ) J ( r J) ] It can be shown that the second term vanishes leaving; N = dτ( r ) J The leading term, when is expanded as above; N m (a) 4 Example of the torque on a current loop Suppose a planar, square current loop of side length, a, and carring current, I, as shown in Figure 3. The force on the sides parallel to the (y, z) plane lie parallel to ˆx and are in opposite directions. The force on the sides parallel to the (x, y) plane lie parallel to ŷ and also point in opposite directions. However, these later forces exert a torque on the current loop as seen in the figure. This torque is given by; N = 2( a/2) a I N = [Ia 2 sin(θ)] ˆx = m Now note that the magnetic moment of the current loop can be expressed as; d m = r Id l/2 M = 2[ a/2 a] = Ia 2 [cos(θ) ẑ + sin(θ) ŷ] N = m 5
F I out a/2 z I a/2 I in y F Component of F producing a torque Figure 3: An example of the torque on a loop of current Increasing F F I I Figure 4: An example of Lenz s Law and diamagnetism 5 Microscopic description of magnetic phenomenon Magnetic phenomenon have atomic origin, and to be precise they should be treated quantum mechanically. However, we can describe the qualitative features in classical terms. There are 3 ways that materials respond to magnetic fields; 1) diamagnetism, 2) paramagnetism, 3) ferromagnetism. We begin the discussion by introducing Lenz s law. This law is really a descriptive version of Faraday s law which we will introduce in electrodynamics. Here, it serves the purpose of providing a basis for the descriptive behavior of diamagnetic materials. 5.1 Lenz s Law Lenz s law states that if there is a changing magnetic flux through a current loop, a current will flow attempting to keep the flux through the loop constant. Look at left figure in Figure 5. Assume the magnetic field increases from zero. The initial flux was zero so a current is induced to keep the flux zero. It does this by generating a magnetic field in opposition to the increasing, applied field. If the flux decreases, an induced current attempts to increase the magnetic field to keep the flux constant. 6
Increasing F F I I Figure 5: An example of Lenz s Law and diamagnetism 5.2 Diamagnetism Diamagnetism is the result of Lenz s Law operating on the atomic scale. Consider an electron orbiting an atom. This orbital motion of charge creates a magnetic moment, m = Iarea = (1/2)q( r v) m = (er/2m e ) L In the above M e is the electron mass, r, its obital radius, q its charge, and L its angular momentum. A normal material has no residual magnetic moment because the orbits of the electrons are randomly oriented. Suppose a magnetic field is applied to the material. The electron orbit acts as a current loop and the electrons either speed up or slow down to change the magnetic flux through the current loop. For increasing flux, the electron with orbital moment along the field slows decreasing its orbital dipole moment. An electron anti-parallel to the field will increase in velocity and thus increase its dipole moment. oth tend to create and enhance a dipole moment in opposition to the applied field. If the applied field is not spatially constant, ie it has a non-zero gradient, then there will be force tending to push the dipole in a direction away from a stronger field. This is illustrated in the figure on the right side of Figure 5. Since all materials have orbiting electrons, all materials experience diamagnetism. 5.3 Paramagnetism Electrons not only have orbital angular momentum but they possess and intrinsic spin. Classically a spinning charge distribution has a magnetic moment as observed in the equation given above, m = q S/2M e where S is the intrinsic spin. Quantum mechanically, this moment is intrinsic to the particle and is constant. In a classical sense, since the electronic charge and spin are quantized, the moment cannot change. Now because the electrons have a magnetic moment they experience a torque in a magnetic field as discussed above. This torque tends to align the moments with the applied field. In an atom, the electronic spins are usually arranged in pairs such that their moments cancel(ie the lowest energy state is attained be- 7
Saturation H Saturation Figure 6: An example of a feromagnetic material showing the magnetic domains cause of the Pauli principle by placing electrons in unoccupied quantum states). If there is an odd number of electrons then there is one unpaired moment, which can be aligned by the application of an external field. An aligned moment is opposite to the induced moments discussed in the diamagnetic case. While the induced moments aligned anti-parallel to the field have a repulsive force, aligned moments experience an attractive force. When the attractive force dominates the material becomes paramagnetic. 5.4 Ferromagnetism In special cases, certain atoms the last 2 electrons have a lower energy configuration in certain crystals when their moments are aligned rather than anti-aligned. These materials are called ferromagnetic. Ferromagnetic materials exist in permanently magnetized macroscopic crystals called domains. However the polarization of each domain is randomly oriented so that in general, a section of ferromagnetic material is unpolarized. y applying an external field, these domains can be expanded and oriented to produce a large magnetic field throught the material. Figure 6 illustrates the domain structure and polarization of a ferromagnetic material. Ferromagnetic materials are temperature dependent. elow a critical temperature (Curie temperature) the material is ferromagnetic. Above this temperature the material is usually paramagnetic. 5.5 Summary Materials that are diamagnetic experience a weak repulsive force when placed in a magnetic field that is spatially diverging. Paramagnetic materials when placed in a diverging magnetic field have weak attraction. Whether a material is diamagnetic or paramagnetic is an interplay between the attractive alignment of electron spins (moments) and the repulsive induced moments due to orbital motion. Ferromagnetic materials have large magnetization 8
due to the coherent alignment of a number of electon moments in certain crystals. Examples of magnetic susceptabilities of different materials is given in Table 1. Table 1: Examples of diamagnetic and paramagnetic materials material χ m Aluminum 1.75 10 6 Copper -0.76 Lithium 1.90 Sodium 0.68 Uranium 4.86 Graphite -7.9 Nickel Chloride 150 Sodium Chloride -1.09 6 Magnetic susceptibility As we previously defined an electric susceptibility as P = χ e E we define a magentic susceptibility, M = χ m H In both cases the assumption is that the materials are linear. However in the magnetic field case, this holds for diamagnetic (χ m < 0) and paramagnetic (χ m > 0) materials. In these cases we have; = µ 0 ( H + M) = µ 0 ( H + χ m H) and = µ H. In the case of ferromagnetic materials the linear relation no longer is valid, although the use of = µ H is sometimes applied, but µ is not a constant. Note that H and are not necessarily in the same direction. A plot of vs H is shown in Figure 7. This is a hysterisis curve. It shows a saturation of the magnetization M when and H have a linear relation. 9
Figure 7: A hysteresis curve showing the nonlinear behavior of the relationship between and H due to the magnetization of the material 10