Ch. 28: Sources of Magnetic Fields

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1 Ch. 28: Sources of Magnetic Fields Electric Currents Create Magnetic Fields A long, straight wire A current loop A solenoid Slide 24-14

2 Biot-Savart Law Current produces a magnetic field The Biot-Savart law gives the magnetic field arising from an infinitesimal current element: ~ db = µ 0 4 I dl ~ ˆr r 2 Permeability of free space: µ 0 = Tm/A Integrating along the wire: ~B = Z µ0 4 I ~ dl ˆr r 2

3 ~B = Z µ0 4 I ~ dl ˆr r 2 At location P, what is the direction of the infinitesimal contribution db created by the current in dl? A) Up the page B) Directly away from dl (in the plane of the page) C) Into the page D) Out of the page E) Some other direction ~r

4 ~B = Z µ0 4 I ~ dl ˆr r 2 What is the magnitude of ~dl ˆr r 2? a) dl sin b) r 2 dl cos Origin c) d) e) r 2 dl sin r dl cos r 3 P ~r dl r 2 θ dl

5 Magnetic field of long straight wire sin( ) =sin( ) = x p x2 + y 2 ~B = Z µ0 4 I ~ dl ˆr r 2 B z = µ 0I 4 Z a as a goes to infinity: a sin r 2 dx sin =sin( ) =d/r ) B z = µ 0I 4 B z! µ 0I 2 d B z = µ 0I 4 Z a a ddx (d 2 + x 2 ) 3/2 2a d p d 2 + a 2

6 Clicker Question CT 32.14b What is the direction of the Force acting on the Red Wire? I I A) Up B) Right C) Left D) Into the Page E) Out of the Page University of Colorado, Boulder (2008)

7 CT 32.7a A long straight wire carries current I out of the page. An electron travels towards the wire from the right. Which way is the force on the electron? I A: B: v e- C: D: E: 0 University of Colorado, Boulder (2008)

9 B field due to a current loop Only component of db in x direction doesn t cancel out ~B = Z µ0 4 I ~ dl ˆr r 2 db x = µ 0I 4 dl x 2 + a 2 cos B = µ 0 Ia 2 2( x 2 + a 2 ) 3 2 For large distances (x >> a), this reduces to B = µ 0 Ia2 2x 3

11 CT What is the direction of the B- Field at point P? A: B: C: 0 D: E: Other University of Colorado, Boulder (2008)

13 Magnetic dipoles A small current loop constitutes a magnetic dipole. Magnitude of B decreases with distance as 1/x 3 Its dipole moment is µ = IA, with A the loop area. For an N-turn loop, µ = NIA. The direction of the dipole moment vector is perpendicular to the loop area. The fields of electric and magnetic dipoles are similar far from their sources, but differ close to the sources.

14 Clicker Question CT 32.14c Two loops of wire have current going around in the same direction. The magnetic forces on the loops are: A: Attractive B: Repulsive C: Net force is zero. i 2 i 1 University of Colorado, Boulder (2008)

16 Ampère s law From the Biot-Savart law, Ampere s law can be derived I ~B = Z µ0 4 I dl ~ ˆr r 2 ~B ~dl = µ 0 I enc For steady currents The integral is taken around any closed loop, and I enc is the current encircled by that loop. Useful only for current distributions with symmetry

17 Field due to a long cylindrical conductor outside conductor: I encl =I 2 Br = µ 0 I =) B = µ 0 2 inside conductor: I encl = J r 2, J = I R 2 2 Br = µ 0 J r 2 = µ 0 I r2 R 2 =) B = µ 0 2 I r Ir R 2

18 Solenoids A solenoid is a long, tightly wound coil of wire. When a solenoid s length is much greater than its diameter, the magnetic field inside is nearly uniform except near the ends, and the field outside is very small..

19 Clicker Question

20 In the ideal limit of an infinitely long solenoid, the field inside the solenoid is uniform everywhere, and the field outside is zero. Application of Ampère s law shows that the field of an infinite solenoid is B = µ 0 ni, where n is the number of turns per unit length I ~B ~dl = BL = µ 0 nil B = µ 0 ni

Classical picture of magnetic dipole moment arising from orbiting electron

24 Magnetic Materials Magnetism in matter arises from atomic current loops associated with valence electrons having orbital angular momentum and spin. Classical picture of magnetic dipole moment arising from orbiting electron Electron s also have spin an intrinsic angular momentum that causes a magnetic moment.

25 Paramagnetism External magnetic fields exert a torque on individual atomic magnetic moments. This torque tends to align the atomic magnetic moments in the direction of the external B field. Within the material, the B field due to the aligned magnetic moments adds to the external B field. ~B = ~ B ext + µ 0 ~ M M is the magnetization, which is the combined magnetic dipole moment per unit volume. For many materials, over a modest range of temperatures, M is proportional to B ext B field is increased by a factor K m (relative permeability) K m typically ranges from to 1.003

26 Diamagnetism In some materials, the B field is actually weaker than the external field. The means that the magnetic moments tend to be oriented in the opposite direction as the external field. The effect is very weak, with K m -1 roughly equal to Atoms associated with diamagnetic materials tend to have no permanent magnetic moments. The magnetic moment is created in response to the external magnetic field. Most textbooks give an overly simplified explanation using Faraday s law of induction, but the actual mechanism is purely quantum-mechanical.

27 Ferromagnetism In ferromagnetic materials like iron, a small magnetic field can produce a large magnetization. Permanent magnets large magnetization even in absence of an external B field. Interactions between neighboring atoms is very important. Quantum-mechanical effect (called the exchange force) makes it energetically favorable for neighboring spins to line up if spacing of atoms is large enough. Individual domains (containing atoms) have all magnetic moments aligned (if above Curie temperature)

28 Hysterisis fff

29 Magnetism in matter Paramagnetic materials exhibit much weaker magnetism. Diamagnetic materials respond oppositely, and are repelled by magnets. B = B vac (1 + )

30 Inducing a Magnetic Moment in a Piece of Iron

31 Electromagnet Use current to induce magnetic moment in iron

32 Electromagnetic Train

33 Consider again a charged particle moving near a long wire with current I. Q v I Recall the Lorentz force is given by ~F = q~v ~ B What frame of reference of reference do we use to determine the velocity? Answer: Same frame in which the magnetic field was determined. What if we switch frames to a frame where the velocity is zero?

Chapter 27 Sources of Magnetic Field

Chapter 27 Sources of Magnetic Field In this chapter we investigate the sources of magnetic of magnetic field, in particular, the magnetic field produced by moving charges (i.e., currents). Ampere s Law