Chapter 33. The Magnetic Field

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Aubrey Bell
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1 Chapter 33. The Magnetic Field Digital information is stored on a hard disk as microscopic patches of magnetism. Just what is magnetism? How are magnetic fields created? What are their properties? These are the questions we will address. Chapter Goal: To learn how to calculate and use the magnetic field.

2 What is the shape of the trajectory that a charged particle follows in a uniform magnetic field? A. Helix B. Parabola C. Circle D. Ellipse E. Hyperbola

3 What is the shape of the trajectory that a charged particle follows in a uniform magnetic field? A. Helix B. Parabola C. Circle D. Ellipse E. Hyperbola

4 The magnetic field of a straight, current-carrying wire is A. parallel to the wire. B. inside the wire. C. perpendicular to the wire. D. around the wire. E. zero.

5 The magnetic field of a straight, current-carrying wire is A. parallel to the wire. B. inside the wire. C. perpendicular to the wire. D. around the wire. E. zero.

6 The Earth and Sun are magnetic

7 Connections to current A magnetic field can be sensed with a magnetic material (compass) and is associated with a current. The earth s field results from large scale internal currents. The field of a permanent magnet results from atomic scale currents.

The earth s field results from large scale internal currents.

8 The field appears only if there is current. It is associated with moving charge.

9 The Source of the Magnetic Field: Moving Charges The magnetic field of a charged particle q moving with velocity v is given by the Biot-Savart law: where r is the distance from the charge and θ is the angle between v and r. (Valid if v<<c.) The Biot-Savart law can be written in terms of the cross product as

where r is the distance from the charge and θ is the angle between v and r.

10 The field of an element of circuit The average B field db due to the moving charge in an element of circuit of vector length ds carrying current I follows from superposing the fields of the moving charges: db r di=ids db = µ o Ids r ˆ 4π r 2

length ds carrying current I follows from superposing the

11 The field of straight wire All current elements produce B out of page db = µ o 4π Ids ˆ r r 2 = µ o 4π r sinθ 2 = µ o 4π I I a r 2 r = µ o 4π I a ( x 2 + a 2 ) 3 / 2 a Add them all up: B = µ oia 4π r dx ( ) = 3 / 2 x 2 + a 2 r = x 2 + a 2 apple µ o I 2πa x 11

12 EXAMPLE 33.4 The magnetic field strength near a heater wire

13 EXAMPLE 33.4 The magnetic field strength near a heater wire Twice B(Earth).

14 Magnetic field lines Magnetic field lines close upon themselves they circulate rather them emanate from charges. Although a current loop can appear like a dipole pair of charges, there is no magnetic charge.

15 The magnetic dipole moment of a current loop enclosing an area A is defined as Magnetic Dipoles The SI units of the magnetic dipole moment are A m 2. The on-axis field of a magnetic dipole is

16 EXAMPLE 33.7 The field of a magnetic dipole

17 EXAMPLE 33.7 The field of a magnetic dipole

18 The superposition of the fields of a stack of loops is a field like that of a bar magnet. The bar magnetic field results from alignment of many atomic scale electronic currents/ magnetic dipoles. Magnetic materials

19 Line integrals Magnetic field lines close upon themselves they circulate rather them emanate from poles. The line integral of B around a closed loop is a measure of the strength in circulation.

20 Ampère s law Whenever total current I through passes through an area bounded by a closed curve, the line integral of the magnetic field around the curve is given by Ampère s law:

21 Ampère s law example Could have used Ampere s law to calculate B B ds = Bds = B ds = B2πr= µ o I B = µ oi 2πr B ds B constant on path path length = 2πr Circular path r B(r) Surface bounded by path I

22 The strength of the uniform magnetic field inside a solenoid is where n = N/l is the number of turns per unit length.

23 Gauss s law for magnetism Net magnetic flux through any closed surface is always zero B da = 0 Compare to Gauss law for electric field E da = Q enclosed ε o No magnetic charge, so right-hand side=0 in the case of magnet field.

24 General laws of electromagnetism Gauss law Ampere s law Magnetostatics B da = 0 B ds = µ o I Electrostatics E da = Q enclosed ε o E ds =?0 Integral of E-field around closed loop is is the change in electric potential going around = zero. x

25 The Magnetic Force on a Moving Charge The magnetic force on a charge q as it moves through a magnetic field B with velocity v is where α is the angle between v and B.

26 Motion in a uniform constant magnetic field The magnetic force on a moving charge q is perpendicular to B and to v and results in helical motion. (Circular motion if there is no velocity component along the field.) Beam of electrons moving in a circle. Lighting is caused by excitation of atoms of gas in a bulb. wiki/magnetic_field

27 Derive force by adding forces on charges constituting the current.

28 Magnetic Forces on Current-Carrying Wires Consider a segment of wire of length l carrying current I in the direction of the vector l. The wire exists in a constant magnetic field B. The magnetic force on the wire is where α is the angle between the direction of the current and the magnetic field.

29 EXAMPLE Magnetic Levitation

30 EXAMPLE Magnetic Levitation

31 EXAMPLE Magnetic Levitation

32 Interaction between a field and a dipole An current loop in a magnetic field is subject to a net torque aligning the dipole moment with the field. τ = r F τ = 2 2 F sinθ F = IB τ = AIBsinθ A = 2 =loop area I F r B apple F I B

33 Interaction between electromagnet and a magnetic substance An electromagnet can be used to pick up a ferromagnetic material. The field of the electromagnet induces an alignment of the atomic scale dipoles resulting in a net force of attraction.

34 Does the compass needle rotate clockwise (cw), counterclockwise (ccw) or not at all? A. Clockwise B. Counterclockwise C. Not at all

35 Does the compass needle rotate clockwise (cw), counterclockwise (ccw) or not at all? A. Clockwise B. Counterclockwise C. Not at all

36 The magnetic field at the position P points A. Into the page. B. Up. C. Down. D. Out of the page.

37 The magnetic field at the position P points A. Into the page. B. Up. C. Down. D. Out of the page.

38 The positive charge is moving straight out of the page. What is the direction of the magnetic field at the position of the dot? A. Left B. Right C. Down D. Up

39 The positive charge is moving straight out of the page. What is the direction of the magnetic field at the position of the dot? A. Left B. Right C. Down D. Up

40 What is the current direction in this loop? And which side of the loop is the north pole? A. Current counterclockwise, north pole on bottom B. Current clockwise; north pole on bottom C. Current counterclockwise, north pole on top D. Current clockwise; north pole on top

41 What is the current direction in this loop? And which side of the loop is the north pole? A. Current counterclockwise, north pole on bottom B. Current clockwise; north pole on bottom C. Current counterclockwise, north pole on top D. Current clockwise; north pole on top

42 An electron moves perpendicular to a magnetic field. What is the direction of? A. Left B. Into the page C. Out of the page D. Up E. Down

43 An electron moves perpendicular to a magnetic field. What is the direction of? A. Left B. Into the page C. Out of the page D. Up E. Down

44 Which magnet or magnets produced this induced magnetic dipole? A. a or d B. a or c C. b or d D. b or c E. any of a, b, c or d

45 Which magnet or magnets produced this induced magnetic dipole? A. a or d B. a or c C. b or d D. b or c E. any of a, b, c or d

Force on a square loop of current in a uniform B-field.

Force on a square loop of current in a uniform B-field. F top = 0 θ = 0; sinθ = 0; so F B = 0 F bottom = 0 F left = I a B (out of page) F right = I a B (into page) Assume loop is on a frictionless axis