M130; Magnetic Dipoles

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MISN-0-130

MAGNETIC DIPOLES by MAGNETIC DIPOLES Kirby Morgan

1. Introduction ...... 1 2. Fundementals of Magnetic Dipoles a. Deﬁning the Magnetic Dipole Moment ...... 1 b. A Typical Magnetic Dipole ...... 1 3. Magnetic Field due to a Dipole ...... 2 4. Existence of Magnetic Dipoles a. The Simplest Magnetic Structure is the Dipole ...... 3 b. A Current Loop is a Magnetic Dipole ...... 4 c. A Bar Magnet Consists of Tiny Current Loops ...... 5

5. Dipole In External Field a. Torque on a Magnetic Dipole ...... 5 b. Work Done on a Magnetic Dipole ...... 5 c. Potential Energy of a Magnetic Dipole ...... 6

6. Electric/Magnetic Dipole Analogy ...... 7

Acknowledgments...... 7

ProjectPHYSNET·PhysicsBldg.·MichiganStateUniversity·EastLansing,MI

2 ID Sheet: MISN-0-130

THIS IS A DEVELOPMENTAL-STAGE PUBLICATION Title: Magnetic Dipoles OF PROJECT PHYSNET Author: Kirby Morgan, HandiComputing, 319 E. Henry, Charlotte, MI The goal of our project is to assist a network of educators and scientists in 48813 transferring physics from one person to another. We support manuscript Version: 11/6/2001 Evaluation: Stage 0 processing and distribution, along with communication and information systems. We also work with employers to identify basic scientific skills Length: 1 hr; 20 pages as well as physics topics that are needed in science and technology. A Input Skills: number of our publications are aimed at assisting users in acquiring such skills. 1. Vocabulary: electric dipole, electric dipole moment, point dipole (MISN-0-120). Our publications are designed: (i) to be updated quickly in response to 2. Calculate the torque, potential energy, and work for an electric field tests and new scientific developments; (ii) to be used in both class- dipole in an external electric field (MISN-0-120). room and professional settings; (iii) to show the prerequisite dependen- 3. Calculate the torque on a current loop in an external magnetic cies existing among the various chunks of physics knowledge and skill, field (MISN-0-123). as a guide both to mental organization and to use of the materials; and (iv) to be adapted quickly to specific user needs ranging from single-skill Output Skills (Knowledge): instruction to complete custom textbooks. K1. Define the magnetic dipole moment (vector) for a system of N New authors, reviewers and field testers are welcome. (fictitious) magnetic monopoles. K2. Define the magnetic dipole moment for a magnetic dipole. PROJECT STAFF K3. Explain the existence of magnetic dipoles in spite of the apparent non-existence of magnetic monopoles. Andrew Schnepp Webmaster K4. Write expressions for the torque, work, potential energy, and field Eugene Kales Graphics of magnetic dipoles. Peter Signell Project Director Output Skills (Problem Solving): ADVISORY COMMITTEE S1. Determine the torque on a magnetic dipole in an external magnetic field and the work done on it in changing its orientation. D. Alan Bromley Yale University S2. Determine the potential energy of a given magnetic dipole or sys- E. Leonard Jossem The Ohio State University tem of dipoles in an external field. A. A. Strassenburg S. U. N. Y., Stony Brook S3. Determine the magnetic field at some given point in space due to a given magnetic dipole. Views expressed in a module are those of the module author(s) and are Post-Options: not necessarily those of other project participants. 1. “The Search for Magnetic Monopoles” (MISN-0-140). c 2001, Peter Signell for Project PHYSNET, Physics-Astronomy Bldg., ° 2. “Magnetic Fields in Bulk Matter: Magnets” (MISN-0-141). Mich. State Univ., E. Lansing, MI 48824; (517) 355-3784. For our liberal use policies see: http://www.physnet.org/home/modules/license.html.

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MAGNETIC DIPOLES m1 m3 by r r 3 1 m Kirby Morgan 2 r2 1. Introduction The magnetic dipole moment is deﬁned in a manner similar to the rN electric dipole moment. This analogy can be made even though the magnetic monopole has never been observed in an experiment. mN

Figure 1. Magnetic monopoles mi located at positions ~ri. 2. Fundementals of Magnetic Dipoles poles taken as +m and m, the dipole moment of this magnet is found 2a. Defining the Magnetic Dipole Moment. The magnetic dipole from the general definition,− Eq. (1). It is: moment can be defined in a manner analogous to that for the electric 12 dipole moment. The magnetic quantity that corresponds to an electric µ~ = m~r1 m~r2 = m(~r2 ~r1), (2) charge is the magnetic monopole (or, for short, magnetic pole).3 Although − − 4 ~ a magnetic monopole has never been observed, it nevertheless is used to where ~r1 ~r2 = l is the vector separation of the two poles. The magnetic − define the general expression for the magnetic dipole moment. dipole moment can thus be written: ~ Consider a collection of N magnetic monopoles, m1, m2, . . . mN . µ~ = m`, (3) Relative to a fixed coordinate system, each pole is located at a point ~ given by a vector ~r; ~r for m , ~r for m , etc. (see Fig. 1). The magnetic where by definition l points from the South pole to the North pole ( m 1 1 2 2 ~ − dipole moment µ~ (or, for short, the magnetic moment) is a vector defined to m, see Fig. 2). Eq. (3) is analogous to the expression p~ = ql for the as: electric dipole moment. As in that case, the magnetic dipole moment of N two equal but opposite poles is independent of the coordinate system used µ~ m1~r1 + m2~r2 + . . . + mN ~rN mi~ri. (1) ≡ ≡ to describe it: it is a property of the system. Xi=1 2b. A Typical Magnetic Dipole. The most common dipole moment 3. Magnetic Field due to a Dipole vector is the magnetic dipole: two equal but opposite magnetic poles separated by a distance `. For example, most of the external effects of a The electric dipole analogy is not strictly correct but it can provide us small bar magnet, such as a compass needle, can be viewed as resulting with the correct expression for the magnetic field produced by a magnetic 5 from equal but opposite poles at its two ends. With the strength of the dipole. By the same method used for the electric dipole, the magnetic field due to a point magnetic dipole (i.e. a dipole whose size, `, is negligible 1See “Electric Dipoles” (MISN-0-120). 2 compared with the distance r to the point where the field of the dipole is Observing a magnetic monopole would be like finding only a “North” pole or only 6 a “South” pole of a magnet. In Nature, North poles and South poles are always found observed) is: together in pairs. 3 (3µ~ ~r)~r µ~ See “Magnetic Monopole” (MISN-0-140). B~ (~r) = km · , (` r). (4) 4Observing a magnetic monopole would be like finding only a “North” pole or only · r5 − r3 ¸ ¿ a “South” pole of a magnet. In nature, North poles and South poles are always found 5 together. See “Electric Dipoles” (MISN-0-120). 6 −7 2 −7 2 km ≡ 10 N/A = 10 kg m/C Help: [S-1]

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+m r1-r2=l

-m r1

Figure 2. A common magnetic dipole with dipole moment µ~ = m~l.

The field lines for a dipole oriented along the z-axis are indicated in Fig. 3. Figure 4. Iron filings placed of a sheet of paper over a bar magnet help in visualizing the magnetic field lines. 4. Existence of Magnetic Dipoles produce a pattern that suggests the dipole may be viewed as two magnetic 4a. The Simplest Magnetic Structure is the Dipole. In electric- poles, one at each end of the magnet (see Fig. 4). However, all attempts ity, the isolated charge q can exist by itself; but in magnetism, isolated to isolate these poles fail. If the bar magnet is cut in half, the halves also magnetic poles have never been observed. The simplest magnetic struc- turn out to be magnetic dipoles and not isolated poles. ture is the magnetic dipole, which is characterized by the magnetic dipole moment µ~. The most familiar example of a magnetic dipole is a bar mag- 4b. A Current Loop is a Magnetic Dipole. The explanation for net. Iron filings sprinkled on a sheet of paper placed over a bar magnet the behavior of the bar magnet can be approached by first examining the magnetic field of a circular loop of current. The magnetic field produced by a circular loop of current is identical to that produced by a point a) z b) z r z r +m y l y l u = m z -m y I x l0 x Figure 5. The dipole moment of a circular current loop is perpendicular to the plane of Figure 3. (a) A magnetic dipole oriented in the z-direction; x the loop. (b) Magnetic field lines around this dipole.

7 8 MISN-0-130 5 MISN-0-130 6 magnetic dipole (see Eq. (4) and Fig. 3), so the loop is also a dipole. q Consider a current loop in the x-y plane of area A carrying a current B u I in the direction indicated in Fig. 5. The loop has a dipole moment along the normal (nˆ) to the plane of the loop given by µ~ = IA nˆ. The direction of the normal is given by a right hand rule: when the fingers of the right q qA hand curl in the direction of the current, the thumb points along the ß normal. ß 4c. A Bar Magnet Consists of Tiny Current Loops. The fact that a current loop is a magnetic dipole suggests that the real sources of the dipole field produced by a bar magnet are tiny currents at the atomic Figure 6. A dipole in an exter- Figure 7. A dipole is rotated level. These currents are produced by orbital electrons in the atoms of nal magnetic field. in a magnetic Field. the bar magnet. In a bar magnet, the tiny magnetic dipole moments are aligned such that they add together to produce a strong magnetic field.7 Now it is apparent why the magnet always has two poles: no matter how Let us calculate how much work is done by the field when rotating it is cut, it will always still contain the tiny current loops which give it the dipole from angle θA to θB as shown in Fig. 7. Since the forces vary, its dipole field. we must integrate F~ d~s, which can be simplified to8 · θB WA,B = τ dθ (8) 5. Dipole In External Field ZθA θB 5a. Torque on a Magnetic Dipole. A magnetic dipole placed in an = µB sin θ dθ (9) external magnetic field experiences a torque. The torque on a magnetic ZθA ~ dipole µ~ placed in an external magnetic field B is = µB(cos θB cos θA). (10) − − ~τ = µ~ B~ . (5) This can be written as:9 × ~ ~ WA,B = ( µ~ B) θ=θB ( µ B) θ=θA . (11) Compare this expression to that for an electric dipole in an electric field: − · | − − · | ~τ = p~ E~ . (6) 5c. Potential Energy of a Magnetic Dipole. A magnetic dipole × has potential energy since the work depends only upon the initial and If µ~ makes an angle θ with the external magnetic field as shown in Fig. 6, final positions, not the path taken. As was done for the electric dipole, then: we will set the potential energy Ep equal to zero when the dipole is at τ = uB sin θ. (7) right angles to the field, i.e., when θ = 90◦. Thus the magnetic potential energy of a magnetic dipole is Thus the torque on the current loop (or any magnetic dipole) is a maxi- ◦ ◦ ◦ mum for θ = 90 and zero for θ = 0 . Ep(θ) = W90◦,θ = uB(cos 90 cos θ). − − 5b. Work Done on a Magnetic Dipole. Since a magnetic dipole or: placed in an external magnetic field experiences a torque, work (positive Ep = µB cos θ = µ~ B~ . (12) or negative) must be done by an external agent in order to change the − − · orientation of the dipole. 8See “Electric Dipoles” (MISN-0-120) for the steps in this simplification. 7See “Magnetic Fields in Bulk Matter: Magnets” (MISN-0-141). 9This shows that the units of µ~ are J/T [S-2].

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6. Electric/Magnetic Dipole Analogy The table below summarizes some relationships that are mathemati- PROBLEM SUPPLEMENT cally similar for electric and magnetic dipoles. In addition, the dimensions of the quantities (in terms of the fundamental dimensions M, L, T , Q) Note: Problems 9 and 10 also occur in this module’s Model Exam. are given as reminders of both the similarities and the diﬀerences.

1. A point dipole, with magnetic moment µ~, is in an external magnetic Quantity/Property Electric Magnetic filed B0yˆ where B0 is a constant. −1 −1 Monopole Charge q [Q] m [LT Q ] a. Determine the torque on the dipole for each of these conditions: − − Dipole Moment p~ = q~` [QL] µ~ = m~` [L2T 1Q 1] (i) µ~ = µxˆ. Field at ~r due to E~ = B~ = (ii) µ~ = µyˆ. point dipole at origin (3p~ ~r)~r p~ (3µ~ ~r)~r µ~ (iii) µ~ = µxˆ. ke · km · − · r5 − r3 ¸ · r5 − r3 ¸ (iv) µ~ = µyˆ. [MLT−2Q−1] [MT−1Q] − b. Determine the potential energy of the dipole in each of the four Force on monopole at F~ = qE~ , F~ = mB~ , orientations given in part (a). ~ ~ ~r due to dipole at ori- using above E using above B c. Determine the work done in turning the dipole: gin [MLT−2] [MLT−2] (i) from xˆ to yˆ. ~ ~ Torque exerted on ~τ = p~ Eext ~τ = µ~ Bext (ii) from xˆ to xˆ. dipole placed in ex- × [ML2T−2] × [ML2T−2] − (iii) from yˆ to xˆ. ternal field − ~ ~ 2. Consider a point dipole, at the origin, with magnetic moment µ~ = Potential Energy of Ep = p~ Eext Ep = µ~ Bext −2 2 −1 − − ( 10 m s C) xˆ in a uniform external magnetic field B~ = 50 T yˆ. dipole in external − · [ML2T 2] − · [ML2T 2] Determine:− field a. the potential energy of the dipole. b. the torque on the dipole. Acknowledgments 3. A magnetic dipole µ~ = µxˆ is located at the origin of a coordinate This module was based on an earlier module by P. Sojka, J. Kovacs, system. Determine the magnetic field(magnitude and direction) at and P. Signell. Preparation of this module was supported in part by each of the points a, b, c, d shown in the sketch: the National Science Foundation, Division of Science Education Devel- opment and Research, through Grant #SED 74-20088 to Michigan State University.

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7. A point dipole with magnetic moment µ~ is in an external ﬁeld y B~ = B0xˆ. Since torque and potential energy have the same units, determine the direction(s) in which the dipole must be pointing if (0,y0) a its torque and potential energy are to be “equal.” Determine the work that must now be done to reorient the dipole so it points in the direction of the ﬁeld. d b x 8. A dipole µ~ 1 = µ1yˆ is located at the origin, while another, µ~ 2 = µ2yˆ, (-x0,0) (x0,0) is located at the point x = x0, y = 0.

a. Write down the expression for the magnetic field at the site of µ~ 2 due to µ~ 1. (0,-y0) c b. Treating the field due to µ~ 1 as a field external to µ~ 2 (as it is), find the potential energy of µ~ 2 in this field, hence finding the interaction potential energy of these two dipoles. 4. Determine the magnetic field at each of the four points in Problem y −1 (3) if µ~ = 2 (J T ) xˆ and x0 = 0.01 m, y0 = 0.01 m. 5. The magnetic moment µ~ of a point dipole in a constant magnetic field B~ makes an angle of 30◦ with the field. 0 u u a. Determine the potential energy of the dipole. 1 2 x b. Determine the torque on the dipole. x 6. Determine the magnetic field at points a, b, c, d shown below if the 0 magnetic dipole makes an angle of 45◦ with the positive x and y axes in the x-y plane. −1 c. Find the interaction potential energy if x0 = 0.05 m, µ1 = 3 J T −1 y and µ2 = 2 J T . Put in the proper units of all quantities and assure yourself that all of the units combine properly to give the (0,y0) a appropriate units for your answer.

9. A point dipole with magnetic moment µ~ = µxxˆ+µyyˆ is in an external ~ 45° magnetic ﬁeld B = B0xˆ. d u b x a. What is the torque on the dipole? (-x0,0) (x0,0) b. Find the potential energy of the dipole. c. How much work must be done to reorient the dipole so it points in the direction of the ﬁeld? (0,-y0) c 10. A point dipole µ~ 1 = µ1yˆ is at the origin, while at x = 0, y = y0 is located another dipole µ~ 2 = µ2xˆ.

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−2 2 −1 a. What is the magnetic ﬁeld at the site of µ~ 2 due to µ~ 1? b. ~τ = µ~ B~ = ( 10 m s C) xˆ (50 T)yˆ = 0.5 N m zˆ, × − −1× −1 − −1 where we have used: T = kg s C . b. Find the energy of the system using y0 = .04 m, µ1 = .5 J T and −1 µ2 = .6 J T . 3µxˆ rˆ µxˆ 3. B~ = km · . · r5 − r3 ¸

3µy0(xˆ yˆ)y0yˆ µxˆ µ At a: ~r = y0yˆ; B~ = km · = km xˆ. · y5 − y3 ¸ − y3 u2 0 0 0 ~ 3µx0(xˆ xˆ)x0xˆ µxˆ µ At b: ~r = x0xˆ; B = km 5· 3 = 2km 3 xˆ. y0 · x0 − x0 ¸ x0

~ 3µy0(xˆ yˆ)( y0yˆ) µxˆ µ At c: ~r = y0yˆ; B = km − ·5 − − 3 = km 3 xˆ. − · y0 − y0 ¸ − y0 3µx0(xˆ xˆ)( x0xˆ) µxˆ µ u1 ~ At d: ~r = x0xˆ; B = km − ·5 − 3 = km 3 xˆ. − · x0 − x0 ¸ x0 4. At a: −7 −1 2 −1 ~ µxˆ (10 T m C s)(2 m s C) xˆ B = km 3 = 3 = 0.2 xˆ T. − y0 − (0.01 m) − −7 −1 2 −1 Brief Answers: ~ µxˆ (10 T m C s)(2 m s C) xˆ At b: B = km 3 = 3 = 0.4xˆ T. x0 (1/2)(0.01 m) ~ 1. a. ~τ = µ~ B At c: B~ = 0.2 xˆ T. × − (i) ~τ = µxˆ B0yˆ = µB0zˆ. × At d: B~ = +0.4xˆ T. (ii) ~τ = µyˆ B0yˆ = 0. × ◦ (iii) ~τ = µxˆ B yˆ = µB zˆ. 5. a. Ep = µ~ B~ = µB0 cos 30 = 0.866µB0. 0 0 − · − − − × − ◦ (iv) ~τ = µyˆ B0yˆ = 0. ~ − × b. ~τ = µ~ B = µB0 sin 30 = 0.5µB0. ~ | | | × | b. Ep = µ~ B. xˆ + yˆ − · 6. µ~ = µ (i) Ep = µB0xˆ yˆ = 0. √ − · 2 (ii) Ep = µB0yˆ yˆ = µB0. − · − At a: (iii) Ep = µB0xˆ yˆ = 0. 3µy0yˆ µ xˆ + yˆ √2µ · B~ = km = km ( xˆ + 2yˆ) (iv) Ep = µB0yˆ yˆ = µB0. · √ 5 − y3 √ ¸ 2y3 − · 2y0 0 2 0 ~ ~ c. W = ( µ~ B)θ=θB ( µ~ B)θ=θA . 7. τ = µB0 sin θ; Ep = µB0 cos θ − · − − · − (i) W = µB0 0 = µB0. − − − so sin θ = cos θ which means θ = 135◦ or 315◦ (ii) W = 0 0 = 0. − − ◦ ◦ (iii) W = 0 ( µB0) = µB0. W = µB0(cos 0 cos 135 ) = 1.707µB0. − − − − − −2 2 −1 ◦ ◦ 2. a. Ep = µ~ B~ = ( 10 m s C) xˆ (50 T) yˆ = 0. W = µB0(cos 0 cos 315 ) = 0.239µB0. − · − − · − − −

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8. a. µ~ = µyˆ; ~r = x0xˆ ~ (3µ~ ~r)~r µ µyˆ µ SPECIAL ASSISTANCE SUPPLEMENT B = km ·5 5 = km 0 3 = km 3 yˆ. · r − r ¸ · − x0 ¸ x0 ~ µ1 µ1µ2 b. Ep = µ~ B = µ2yˆ km 3 yˆ = km 3 . S-1 (from TX-3) − · − · ·− x0 ¸ x0 −7 −2 2 −1 2 −1 m kg C−2 (10 m kg C )(2 m s C)(3 m s C) −3 −1 −1 c. Ep = = 4.8 10 J. T = tesla = −1 = kg s C . (5 10−2 m)3 × m C s × T is the SI unit for magnetic ﬁelds. 9. a. ~τ = µyB0zˆ.

b. Ep = µxB0. − S-2 (from TX-5b) c. W = µB0 + µxB0, − W = µB(cos θB cos θA) 2 2 − − µ = µx + µy. J = (units of µ) T q units of µ = J T−1 = ( kg m2 s−2)/( kg s2 C−1) = m2 s−1 C. ~ µ1 10. a. B = 2km 3 yˆ. y0 b. Zero.

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Brief Answers: MODEL EXAM 1. See this module’s text. 1. Explain how there can be magnetic dipoles in nature while there don’t 2. See this module’s text. seem to be magnetic monopoles. 3. See Problem 9 in this module’s Problem Supplement. 2. For both electric and magnetic dipoles, write the corresponding expressions for torque, work, potential energy and ﬁeld. 4. See Problem 10 in this module’s Problem Supplement.

3. A point dipole with magnetic moment µ~ = µxxˆ + µyyˆ is in an external magnetic ﬁeld B~ = B0xˆ. a. What is the torque on the dipole? b. Find the potential energy of the dipole. c. How much work must be done to reorient the dipole so it points in the direction of the ﬁeld?

4. A point dipole µ~ 1 = µ1yˆ is at the origin, while at x = 0, y = y0 is located another dipole µ~ 2 = µ2xˆ.

a. What is the magnetic ﬁeld at the site of µ~ 2 due to µ~ 1? −1 b. Find the energy of the system using y0 = 0.04 m, µ1 = 0.5 J T −1 and µ2 = 0.6 J T .

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