Electricity and Magnetism Current Loops and Magnetic Dipoles Magnetism in Matter

Transcription

1 Electricity and Magnetism Current Loops and Magnetic Dipoles Magnetism in Matter Lana Sheridan De Anza College Mar 5, 2018

2 Last time magnetic field inside a solenoid forces between current-carrying wires

3 Overview magnetic field around a current loop more about magnetic dipoles magnetism of matter

4 Current loops and Magnetic dipoles We are now going to return to current loops and see why they have associated magnetic dipole moments, and how these behave.

5 ral result discussed in Example Magnetic field from a circular loop of wire In the last lecture, we considered the magnetic moment of a loop of wire, now we look at the magnetic field along a line through the the Axis center of of a Circular the loop. Current Loop located in the yz Figure Calt P a distance x S d s y u ˆr a db S xample 23.8 for d B O igure 30.5 shows r z e to a single curld vector can be u x I P e axis of the ring db x x about the magment at Each the botof the situation, little segment Figure 30.5 of wire (Example ds with 30.3) current Geometry Ifor contributes calculating the a field db. magnetic field at a point P lying on the axis of a current loop. By symmetry, the total field S B is along this axis. due to By elements symmetry, we can see the the components that are parallel to his cancellation the plane of the ring will cancel. ring, so we can ignore the perpendicular component of the field and focus

6 Magnetic field from a circular loop of wire e Axis of a Circular Current Loop ated in the yz ure Cala distance x S d s y a db S mple 23.8 for d B O re 30.5 shows r z o a single curvector can be u x I P xis of the ring db x x out the magnt at the botthe situation, Figure 30.5 (Example 30.3) Geometry for calculating the magnetic field at a point P lying on the axis of a current loop. By symmetry, the total db field = µ S 0 I ds ˆr B is along this axis. e to elements 4π r 2 s cancellation This time r ds g, so we can ignore the perpendicular component of the field and focus ly add. ds ˆr = ds (cos θi + sin θj) c field due to a simple current distribution, so this example is a typical u ˆr

7 Magnetic field from a circular loop of wire (y-comp. cancels) B = µ 0 cos θ 4π I r 2 ds i Notice cos θ = a r, and r = a 2 + x 2. They are independent of the integration variable! B = µ 0 Ia 4π (a 2 + x 2 ) 3/2 ds i

8 Magnetic field from a circular loop of wire (y-comp. cancels) B = µ 0 cos θ 4π I r 2 ds i Notice cos θ = a r, and r = a 2 + x 2. They are independent of the integration variable! B = µ 0 Ia 4π (a 2 + x 2 ) 3/2 ds i = µ 0 Ia 4π (a 2 + x 2 (2πa) i ) 3/2 = µ 0 Ia 2 2(a 2 + x 2 ) 3/2 i Similar to the E-field of an electric dipole...

9 Magnetic field from a circular loop of wire Very far from the wire, B = µ 0Ia 2 2πx 3 i In terms of the magnetic moment µ = Iπa 2 B = µ 0 µ 2π x 3 i Far from an electric dipole, along the axis of the dipole: E = 1 2πɛ 0 p x 3 i

10 a current loop; one side of the loop acts as a north pole (in the direction of ) and the other side as a south pole, as suggested by the lightly drawn magnet in the figure. If we were to place a current-carrying coil in an external magnetic Current Loop Question Consider field, it would thetend four to arrangements rotate just like a bar of magnet circular would. loops of radius r or 2r, centered on vertical axes (perpendicular to the loops) and carrying identical CHECKPOINT currents 3 in the directions indicated. Rank the The figure here shows four arrangements of circular loops of radius r or 2r, centered on arrangements according to the magnitude the net magnetic field vertical axes (perpendicular to the loops) and carrying identical currents in the directions theindicated. dot, midway Rank the arrangements between the according loops to the onmagnitude the central of the net axis, magnetic greatest at field at the dot, midway between the loops on the central axis, greatest first. first. (a) (b) (c) (d) A a, b, c, d B b, c, d, a C d, a, (b and c) D (b and c), d, a 1 Halliday, Resnick, Walker, pg 779.

11 a current loop; one side of the loop acts as a north pole (in the direction of ) and the other side as a south pole, as suggested by the lightly drawn magnet in the figure. If we were to place a current-carrying coil in an external magnetic Current Loop Question Consider field, it would thetend four to arrangements rotate just like a bar of magnet circular would. loops of radius r or 2r, centered on vertical axes (perpendicular to the loops) and carrying identical CHECKPOINT currents 3 in the directions indicated. Rank the The figure here shows four arrangements of circular loops of radius r or 2r, centered on arrangements according to the magnitude the net magnetic field vertical axes (perpendicular to the loops) and carrying identical currents in the directions theindicated. dot, midway Rank the arrangements between the according loops to the onmagnitude the central of the net axis, magnetic greatest at field at the dot, midway between the loops on the central axis, greatest first. first. (a) (b) (c) (d) A a, b, c, d B b, c, d, a C d, a, (b and c) D (b and c), d, a 1 Halliday, Resnick, Walker, pg 779.

12 Magnetic Moment for a Current Loop For a current loop, we can define the magnetic moment of the loop as µ = IA And for a coil N turns (loops) of wire carrying a current: µ = NIA Then the expression for the torque can be written τ = µ B

13 Reminder: Electric Dipole Moment that z d.at such larg proximation, we can neg Recall our definition for the Electric dipole moment: dipole moment: p = q d z where d is a vector pointing from the negative charge to the positive charge, and its magnitude d is the separation of the Up here the +q charges and each charge in the dipole has magnitude q. field dominates. + The product qd, w dipole, is the magnitude p : of the dipole. (The un Torque on a electric dipole in an electric field: Dipole center p τ = p E Potential energy: U = p E p : The direction of is t dipole, as indicated in

14 Current Loop vs Bar Magnet RRYING COIL AS A MAGNETIC DIPOLE 779 A loop of wire with a current in it produces a magnetic field somewhat similar to a bar magnet. PART 3 etic dipole. space? The seful; so we sider only a central axis, e magnetic i N µ i (29-26) S B the point in f the magoment : of Fig A current loop produces a

15 Magnetic Dipole Moment magnetic dipole moment, µ The quantity relating an external magnetic field that a magnet or coil of wire is in to the torque on the magnet or coil due to that field. τ = µ B For a magnet, it is a vector pointing from the south pole of a magnet to the north pole, that is proportional to the strength of the B-field produced by the magnet itself. For a coil, it is defined according the the right hand rule for current in a wire loop and is proportional to the coil area and current.

16 Potential Energy of a Dipole in a B-Field The magnetic moment vector attempts to align with the magnetic field. τ = µ B B µ Us netic fi in whic W i µ Highest energy The energy Fig. can be foundthe by integrating orientations the torque of highest over the angle of rotation, and choosing lowest energy U(π/2) = of 0. a (See magnetic Lecturedipole 13 for derivation.) (here a coil carrying U = current) µ B in an external magnetic field. The direction of the cur- B : Lowest energy i which exerte In each the vec A

17 are shown in Table Question CHECKPOINT 5 The figure shows four orientations, at angle θ, of a magnetic dipole : The figure moment shows µ in four a magnetic orientations, field. at angle Rank u, the of orientations a magnetic dipole according moment to in a magnetic field. Rank the orientations according to (a) the magnitude of the torque on the magnitude of the torque on the dipole, greatest first. the dipole and (b) the orientation energy of the dipole, greatest first. 1 µ µ 2 θ θ θ θ B 4 µ µ 3 (A) (1 and 2), (3 and 4) (B) (1 and 4), (2 and 3) (C) 3, 2, 1, 4 (D) all the same 1 Halliday, Resnick, Walker, 9th ed, page 745.

18 are shown in Table Question CHECKPOINT 5 The figure shows four orientations, at angle θ, of a magnetic dipole : The figure moment shows µ in four a magnetic orientations, field. at angle Rank u, the of orientations a magnetic dipole according moment to in a magnetic field. Rank the orientations according to (a) the magnitude of the torque on the magnitude of the torque on the dipole, greatest first. the dipole and (b) the orientation energy of the dipole, greatest first. 1 µ µ 2 θ θ θ θ B 4 µ µ 3 (A) (1 and 2), (3 and 4) (B) (1 and 4), (2 and 3) (C) 3, 2, 1, 4 (D) all the same 1 Halliday, Resnick, Walker, 9th ed, page 745.

19 are shown in Table Question CHECKPOINT 5 The figure shows four orientations, at angle θ, of a magnetic : The figure dipole shows moment four µ orientations, in a magnetic at angle field. u, Rank of a magnetic the orientations dipole moment in a magnetic field. Rank the orientations according to (a) the magnitude of the torque on according to the orientation energy of the dipole, greatest first. the dipole and (b) the orientation energy of the dipole, greatest first. 1 µ µ 2 θ θ θ θ B 4 µ µ 3 (A) (1 and 2), (3 and 4) (B) (1 and 4), (2 and 3) (C) 3, 2, 1, 4 (D) all the same 1 Halliday, Resnick, Walker, 9th ed, page 745.

20 are shown in Table Question CHECKPOINT 5 The figure shows four orientations, at angle θ, of a magnetic : The figure dipole shows moment four µ orientations, in a magnetic at angle field. u, Rank of a magnetic the orientations dipole moment in a magnetic field. Rank the orientations according to (a) the magnitude of the torque on according to the orientation energy of the dipole, greatest first. the dipole and (b) the orientation energy of the dipole, greatest first. 1 µ µ 2 θ θ θ θ B 4 µ µ 3 (A) (1 and 2), (3 and 4) (B) (1 and 4), (2 and 3) (C) 3, 2, 1, 4 (D) all the same 1 Halliday, Resnick, Walker, 9th ed, page 745.

21 Electric Dipole and Magnetic Dipole electric dipole magnetic dipole torque τ τ = p E τ = µ B potential energy U U = p E U = µ B

22 Magnetism in Matter: Magnetic Moment of Atoms Atoms and subatomic particles also have magnetic moments! The electron has an angular S Why? Consider a classical momentum model of alhydrogen in one atom. direction One electron orbits the nucleus. m S electrons model, an g charge), al motion. e in good quantum and a magnetic moment the opposite direction. µ = IA S L in m circular about the ctron is its n uniform I m S r e

23 Magnetic Moment of Atoms The current is the rate of charge flow with time: I = e T = e v 2πr assuming an orbital radius of r, speed v. µ = IA = e v 2πr (πr 2ˆn) = evr 2 ˆn

24 Magnetic Moment of Atoms The current is the rate of charge flow with time: I = e T = e v 2πr assuming an orbital radius of r, speed v. µ = IA = e v (πr 2ˆn) 2πr = evr 2 ˆn Recall that for a particle of mass m orbiting at a radius r, velocity v, the angular momentum is: L = mvr µ = e 2m e L

25 Magnetism in Matter: Magnetic Moment of Atoms electrons The electron has an angular S Orbital magnetic moment momentum L in one direction and a magnetic µ = e moment m S in L the opposite 2m e direction. model, an g charge), al motion. e in good quantum S L m circular about the ctron is its n uniform I m S r e Figure An electron mov-

26 Electron Spin Angular Momentum Electrons also 872have another CHAPTER kind of 32 angular MAXWELL S momentum: EQUATIONS; intrinsic angular momentum. This is also called spin. Spin is an inherent property For an electron, of all electrons. the spinit cannot be Substituting understood with classical is opposite mechanics, the magnetic but also contributes a magnetic moment. dipole moment. B S where the pl to the z axis, The qua : µ s Spin magnet be expressed component o

27 particle is meaningless. Rotation about its axis a applies only to a rigid object, with sical interpreta an extent in space, as in Chapter spin is on the 10. Spin angular momentum is You might imagine an electron as a rigid charge sphere spinning tum S actually a relativistic effect. L on due to an axis through its center... of an electron Electron Spin Angular Momentum S m S spin The magnetic the value This combinat Figure Classical model of...but really, it s not. a spinning electron. We can adopt this model to remind ourselves that electrons have an intrinsic

28 For an electron, the spin is opposite the magnetic dipole moment. Electron Spin Angular Momentum B S where the pl to the z axis, The qua Electron s Fig. spin magneticthe dipole spin moment:, spin magnetic dipole moment, and magnetic dipole field B : of an electron µ s = represented g e S as a microscopic 2m e sphere. where g 2. : s S : µ s Spin magnet be expressed component o (The quantu veals that m s,z When an be associated :

29 Magnetic Moment of Atoms In atoms with many electrons, the electrons tend to cancel out each other s magnetic moments, but outer-shell, unpaired electrons can contribute a significant magnetic moment. The particles in the nucleus also have magnetic moments, but they are much smaller. Most of an atom s magnetic moment comes from unpaired electons. These tiny magnetic moments add up to big effects in bulk materials.

30 Three Types of Bulk Magnetism ferromagnetism paramagnetism diamagnetism

31 Ferromagnetism Atoms of ferromagnetic materials have non-zero magnetic moments. Interactions between outer electrons in different atoms causes alignment of each atom s magnetic moment. Magnetic moments reenforce each other and will tend to spontaneously align within domains.

32 Ferromagnetism Atoms of ferromagnetic materials have non-zero magnetic moments. Interactions between outer electrons in different atoms causes alignment of each atom s magnetic moment. Magnetic moments reenforce each other and will tend to spontaneously align within domains. Examples of ferromagnetic materials: iron nickel cobalt gadolinium dysprosium

33 the atomic magnetic dipoles are Ferromagnetism randomly oriented. ple as c, the very etiza- agita-. itical agned the ation tance ances No external B-field a

34 larger, giving the sample a net Ferromagnetism magnetization. interof an ernal ment mize Applied external B-field weak sing gnegnet- b da S S B

35 c netic the external field become very Ferromagnetism small. tion, n an magstatic antinetic elechat is Strong external B-field S B S B

36 Paramagnetism Atoms of paramagnetic materials have non-zero dipole moments, but electrons of different atoms do not interact with each other. They can interact with a strong magnetic field, and will align with the field. Paramagnetic effects tend to be much smaller than ferromagnetic ones.

37 Paramagnetism Atoms of paramagnetic materials have non-zero dipole moments, but electrons of different atoms do not interact with each other. They can interact with a strong magnetic field, and will align with the field. Paramagnetic effects tend to be much smaller than ferromagnetic ones. Examples of paramagnetic materials: Tungsten Cesium Aluminium Lithium Magnesium Sodium

38 Paramagnetism Liquid oxygen stream deflected in a strong magnetic field. The stream collects in the field. 1 Image created by Pieter Kuiper.

39 Diamagnetism Diamagnetism occurs in all materials, but is a weak effect, so it is drowned out if a material is ferro- or paramagnetic. It is the dominant (but weak) effect when the net magnetic moment of a material s atoms is zero. The field magnetizes the atoms and the resulting magnetic moments oppose the external magnetic field.

40 Diamagnetism Diamagnetism occurs in all materials, but is a weak effect, so it is drowned out if a material is ferro- or paramagnetic. It is the dominant (but weak) effect when the net magnetic moment of a material s atoms is zero. The field magnetizes the atoms and the resulting magnetic moments oppose the external magnetic field. Examples of diamagnetic materials: Pyrolytic carbon Bismuth Mercury Silver diamond (form of Carbon) water Also superconductors can be said to exhibit extreme diamagnetism.

41 Diamagnetism 1 Levitating pyrolytic carbon on neodymium magnets. Image by Splarka.

42 Diamagnetism 1 Magnet photo by Mai-Linh Doan, Wikipedia; Frog photo by Lijnis Nelemans/High Field Magnet Laboratory/Radboud University Nijmeg.

43 Summary B-field near a current loop more about magnetic dipoles magnetism of matter 3rd Test Friday, March 9. Homework Serway & Jewett: PREVIOUS: Ch 29, Problems: 47, 53 (dipole energy) NEW: Ch 30, Problems: 7, 49