Magnetization (Griffiths Chapter 6: Sections 1-2)

Size: px

Start display at page:

Download "Magnetization (Griffiths Chapter 6: Sections 1-2)"

Kristian Miles
7 years ago
Views:

1 Dr. Alain Brizard Electromagnetic Theory I (PY 30) Magnetization (Griffiths Chapter 6: Sections 1-) Force and Torque on a Magnetic Dipole A magnetic dipole m experiences a torque when exposed to an external magnetic field. We show this by looking, first, at the torque on a rectangular current loop (of sides a and b) in a uniform magnetic field B. Assuming that the magnetic field is directed along the z-axis, B = Bẑ, and the current is oriented in such a way that its magnetic dipole moment is m = Iab(sin θ ŷ + cos θ ẑ), where the rectangular loop carries current I as shown in the Figure below. The forces on the four segments (i), (ii), (iii), and (iv) are, respectively, given as F + b = IbB cos θ x, F + a = IaB ŷ, F b = IbB cos θ x, F a = IaB ŷ, and are shown in the Figure above. Note that, whereas the forces F ± b do not generate torque on the rectangular loop since they are planar forces (i.e., F ± b m = 0), the forces 1

2 F ± a are nonplanar forces and, therefore, generate torque: N = r + a F + a + r a F a [ b = IaB (cos θ ŷ sin θ ẑ) ŷ + b = I (ab) sin θ x = m B. ( cos θ ŷ + sin θ ẑ) ( ŷ) ] Although it is clear that the net force on the current loop is zero in a uniform magnetic field, it is expressed as F net = U B = (m B), where U B = m B denotes the potential energy of a magnetic dipole in an external magnetic field, in analogy with the potential energy U E = p E of an electric dipole in an electric field. Hence, we note that a magnetic dipole m exposed to a magnetic field B experiences a torque N = m B, which causes it to align itself with the magnetic field in order to minimize its magnetic potential energy U B = m B. Effect of a Magnetic Field on Atomic Orbits In the simplest classical model for an electron orbit (with charge e) about a positivelycharged nucleus (with charge Ze), the electron undergoes uniform circular motion with radius r e and tangential velocity v e = ω e r e determined by the centripetal force Ze 4πɛ 0 re = m e v e r e v e = ω e r e, (1) generated by the electrostatic attractive force between the electron and the nucleus, where ω e = Ze. 4πɛ 0 re 3 Assuming that the circular motion of the electron takes place on the (x, y)-plane, the electron orbit can be interpreted as a small magnetic dipole ( m = I e πre ẑ = eω ) e πre π ẑ = e v e r e ẑ, () where the negative sign comes from the electron s negative charge. Note that if we make use of the expression for the orbital angular momentum L = r e m e v e = m e v e r e ẑ

3 for the electron s circular orbit, we may write the magnetic dipole moment as m = e L m e. The introduction of an external magnetic field B = B ẑ changes the centripetal-force equation (1) to become Ze 4πɛ 0 re v e + e v e B = m e e v e B = m ( ) e v r e r e ve, (3) e where we assume that the electron speeds up (v e >v e ) but remains on the same orbit (r e = r e ). Assuming that v e = v e v e v e is small (i.e., Ω = eb/m e ω e ), we easily find from Eq. (3) v e = er eb, m e and, thus, the magnetic dipole moment () changes by an amount m = e v e r e ẑ = e r e 4 m e B. Note that this change is equivalent to a change in orbital angular momentum L = r e ( e A) = e r e B m = e L, m e where A = 1 r e B. Here, we note that the change in magnetic dipole moment is opposite to the direction of B this is the source of diamagnetism. 1 Magnetization & Diamagnetism and Paramagnetism In the presence of a magnetic field B, thus, matter becomes magnetized, i.e., the state of magnetic polarization of matter is represented by a finite magnetic dipole moment per unit volume known as magnetization and denoted as M. Hence, we see that the magnetization vector M of diagmagnetic materials is aligned anti-parallel to the magnetic field B. While diamagnetism is a universal phenomenon, however, it is typically weaker than paramagnetism, in which paramagnetic material acquire a magnetization vector M parallel to B. 1 The classical discussion presented here must be replaced by a fully quantum-mechanical discussion of diamagnetism. 3

4 As can be seen in the Figure below, most atoms are slightly diamagnetic, some atoms are paramagnetic (with magnetization properties far stronger than diamagnetic atoms), and some atoms (the triumvirate group of Iron, Nickel, and Cobalt) are ferromagnetic. In particular, note the relationship between the presence of d orbital electrons and paramagnetism and ferromagnetism. To investigate further this relationship, we write down the valence-electron configuration for Scandium through Manganese: Scandium (Sc) 3 d 4s Titanium (Ti) 3 d 4s Vanadium (V) 3 d 3 4s Chromium (Cr) 3 d 5 4s Manganese (Mn) 3 d 5 4s and note that paramagnetic properties of these atoms is strongest when the number of valence electrons is odd. Much of paramagnetism can be explained classically by the fact that the torque experienced by a magnetic dipole m in an external field B forces the magnetic dipole to 4

5 line up with the magnetic field (so that its potential energy reaches a minimum value U B = m B < 0). Field of a Magnetized Object Since the magnetic vector potential A(r) of a single magnetic dipole m located at the source point r is expressed as A(r; r ) = µ ( ) 0 r r 4π m, 3 the vector potential associated with a magnetized object is expressed as A(r) = µ 0 4π V M(r ) ( r r 3 ) dτ. (4) Using the identity r r ( ) 1 = 3, we obtain after integration by parts A(r) = µ ( ) 0 1 M(r ) dτ 4π V = µ { [ 0 M(r ] ) dτ + 4π V V M(r ) da }. Hence, by defining the volume bound current density J b and the surface bound current density K b as J b = M and K b = M n, (5) the magnetic vector potential of a magnetized object is expressed in terms of volume and surface bound currents as A(r) = µ 0 4π V J b (r ) dτ + µ 0 4π Notice the striking similarity with the electric case: V ρ b = P J b = M K b (r ) da. (6) σ b = P n K b = M n and that the volume bound current J b is explicitly divergenceless: J b = M =0. 5

6 For example, the magnetic field of a uniformly magnetized sphere (with radius R and magnetization M = M ẑ) is calculated from the bound currents J b = M = 0 and K b = M n = M sin θ ϕ. Noting that this surface current density is analogous to the case of a rotating charged sphere, with σrω M, we find the vector potential A(r) = µ 0 3 M r 1 (r<r inside sphere) (R/r) 3 and the magnetic field inside the magnetized sphere (r>r outside sphere) B in = A in = µ 0 3 M is constant, while the magnetic field outside the sphere is that of a pure magnetic dipole. 6

Eðlisfræði 2, vor 2007

[ Assignment View ] [ Pri Eðlisfræði 2, vor 2007 28. Sources of Magnetic Field Assignment is due at 2:00am on Wednesday, March 7, 2007 Credit for problems submitted late will decrease to 0% after the deadline