1. Chapter 19
Magnetism

2. Magnets
In each magnet there are two poles present (the ends where objects are most strongly attracted): north and south
Like (unlike) poles repel (attract) each other (similar to electric charges)
Magnetic poles cannot be isolated – if a permanent magnet is cut in half, you will still have a north and a south pole (unlike electric charges)
There is some theoretical basis for monopoles, but none have been detected

3. Magnetism
An unmagnetized piece of iron can be magnetized by stroking it with a magnet (like stroking an object to charge an object)
Magnetism can be induced – if a piece of iron, for example, is placed near a strong permanent magnet, it will become magnetized
Soft magnetic materials (such as iron) are easily magnetized and also tend to lose their magnetism easily
Hard magnetic materials (such as cobalt and nickel) are difficult to magnetize and they tend to retain their magnetism

4. Magnetic Fields
The region of space surrounding a moving charge includes a magnetic field (the charge will also be surrounded by an electric field)
A magnetic field surrounds a properly magnetized magnetic material
A magnetic field is a vector quantity symbolized by B
Its direction is given by the direction a north pole of a compass needle pointing in that location
Magnetic field lines can be used to show how the field lines, as traced out by a compass, would look

5. Magnetic Field Lines
A compass can be used to show the direction of the magnetic field lines

6. Magnetic Field Lines
Iron filings can also be used to show the pattern of the magnetic field lines
The direction of the field is the direction a north pole would point
Unlike poles (compare to the electric field produced by an electric dipole)

7. Magnetic Field Lines
Iron filings can also be used to show the pattern of the magnetic field lines
The direction of the field is the direction a north pole would point
Unlike poles (compare to the electric field produced by an electric dipole)
Like poles (compare to the electric field produced by like charges)

8. Earth’s Magnetic Field
The Earth’s geographic north (south) pole corresponds to a magnetic south (north) pole – a north (south) pole should be a “north- (south-) seeking” pole
The Earth’s magnetic field resembles that achieved by burying a huge bar magnet deep in the Earth’s interior
The most likely source of the Earth’s magnetic field – electric currents in the liquid part of the core

9. Earth’s Magnetic Field
The magnetic and geographic poles are not in the same exact location – magnetic declination is the difference in directions to the geographic north pole and the magnetic south pole
The amount of declination varies by location on the earth’s surface
The direction of the Earth’s magnetic field reverses every few million years (the origin of these reversals is not understood)

10. Earth’s Magnetic Field
If a compass is free to rotate vertically as well as horizontally, it points to the earth’s surface
The angle between the horizontal and the direction of the magnetic field is called the dip angle
The farther north the device is moved, the farther from horizontal the compass needle would be
The compass needle would be horizontal at the equator and the dip angle would be 0°
The compass needle would point straight down at the south magnetic pole and the dip angle would be 90°

11. Magnetic Fields
Nikola Tesla
1856 – 1943
When moving through a magnetic field, a charged particle experiences a magnetic force
This force has a maximum (zero) value when the charge moves perpendicularly to (along) the magnetic field lines
Magnetic field is defined in terms of the magnetic force exerted on a test charge moving in the field with velocity v
The SI unit: Tesla (T)

12. Magnetic Fields Conventional laboratory magnets: ~ 2.5 T
Superconducting magnets ~ 30 T
Earth’s magnetic field ~ 5 x 10-5 T

13. Direction of Magnetic Force
Experiments show that the direction of the magnetic force is always perpendicular to both v and B
Fmax occurs when v is perpendicular to B and F = 0 when v is parallel to B
Right Hand Rule #1 (for a + charge): Place your fingers in the direction of v and curl the fingers in the direction of B – your thumb points in the direction of F
If the charge is negative, the force points in the opposite direction

14. Direction of Magnetic Force
The blue x’s indicate the magnetic field when it is directed into the page (the x represents the tail of the arrow)
Blue dots would be used to represent the field directed out of the page (the • represents the head of the arrow)

15. Force on a Charged Particle in a Magnetic Field
Consider a particle moving in an external magnetic field so that its velocity is perpendicular to the field
The force is always directed toward the center of the circular path
The magnetic force causes a centripetal acceleration, changing the direction of the velocity of the particle

16. Force on a Charged Particle in a Magnetic Field
This expression is known as the cyclotron equation
r is proportional to the momentum of the particle and inversely proportional to the magnetic field
If the particle’s velocity is not perpendicular to the field, the path followed by the particle is a spiral (helix)

17. Chapter 19 Problem 6
A proton moves perpendicularly to a uniform magnetic field at 1.0 × 107 m/s and exhibits an acceleration of 2.0 × 1013 m/s2 in the +x-direction when its velocity is in the +z-direction. Determine the magnitude and direction of the field.

18. Chapter 19 Problem 40
A particle with charge q and kinetic energy KE travels in a uniform magnetic field of magnitude B. If the particle moves in a circular path of radius R, find expressions for (a) its speed and (b) its mass.

19. Magnetic Force on a Current Carrying Wire
The current is a collection of many charged particles in motion
The magnetic force is exerted on each moving charge in the wire
The total force is the sum of all the magnetic forces on all the individual charges producing the current
Therefore a force is exerted on a current-carrying wire placed in a magnetic field:

20. Magnetic Force on a Current Carrying Wire
The direction of the force is given by right hand rule #1, placing your fingers in the direction of I instead of v

21. Chapter 19 Problem 21
Consider the system pictured in the figure. A 15-cm length of conductor of mass 15 g, free to move vertically, is placed between two thin, vertical conductors, and a uniform magnetic field acts perpendicular to the page. When a 5.0-A current is directed as shown in the figure, the horizontal wire moves upward at constant velocity in the presence of gravity. (a) What forces act on the horizontal wire, and under what condition is the wire able to move upward at constant velocity? (b) Find the magnitude and direction of the minimum magnetic field required to move the wire at constant speed. (c) What happens if the magnetic field exceeds this minimum value? (The wire slides without friction on the two vertical conductors.)

22. Torque on a Current Loop

23. Torque on a Current Loop
Applies to any shape loop
Torque has a maximum value when q = 90°
Torque is zero when the field is perpendicular to the plane of the loop

24. Magnetic Moment The vector is called the magnetic moment of the coil
Its magnitude is given by μ = IAN
The vector always points perpendicular to the plane of the loop(s)
The equation for the magnetic torque can be written as τ = BIAN sinθ = μB sinθ
The angle is between the moment and the field

25. Chapter 19 Problem 30
A copper wire is 8.00 m long and has a cross-sectional area of 1.00 × 10−4 m2. The wire forms a one-turn loop in the shape of square and is then connected to a battery that applies a potential difference of V. If the loop is placed in a uniform magnetic field of magnitude T, what is the maximum torque that can act on it? The resistivity of copper is 1.70 × 10−8 Ω · m.

26. Electric Motor
An electric motor converts electrical energy to mechanical energy (rotational kinetic energy)
An electric motor consists of a rigid current-carrying loop that rotates when placed in a magnetic field
The torque acting on the loop will tend to rotate the loop to smaller values of θ until the torque becomes 0 at θ = 0°

27. Electric Motor
If the loop turns past this point and the current remains in the same direction, the torque reverses and turns the loop in the opposite direction
To provide continuous rotation in one direction, the current in the loop must periodically reverse
In ac motors, this reversal naturally occurs
In dc motors, a split-ring commutator and brushes are used

28. Electric Motor
Just as the loop becomes perpendicular to the magnetic field and the torque becomes 0, inertia carries the loop forward and the brushes cross the gaps in the ring, causing the current loop to reverse its direction
This provides more torque to continue the rotation
The process repeats itself
Actual motors would contain many current loops and commutators

29. Magnetic Fields – Long Straight Wire
A current-carrying wire produces a magnetic field
The compass needle points in the direction of the magnetic field produced by the current (tangential to the circle)
Right Hand Rule #2: Grasp the wire in your right hand and point your thumb in the direction of the current
Your fingers will curl in the direction of the field

30. Magnetic Fields – Long Straight Wire
The magnitude of the field at a distance r from a wire carrying a current of I is
µo = 4  x 10-7 T.m / A: permeability of free space

31. Ampère’s Law
Ampère’s Circuital Law: a procedure for deriving the relationship between the current in an arbitrarily shaped wire and the magnetic field produced by the wire
Choose an arbitrary closed path around the current and sum all the products of B|| Δℓ around the closed path
 B|| Δℓ = µo I

32. Ampère’s Law for a Long Straight Wire
Use a closed circular path
The circumference of the circle is 2 r
 B|| Δℓ = µo I
B  Δℓ = B 2 r = µo I

33. Chapter 19 Problem 51
A wire carries a 7.00-A current along the x-axis, and another wire carries a 6.00-A current along the y-axis, as shown in the figure. What is the magnetic field at point P, located at x = 4.00 m, y = 3.00 m?

34. Magnetic Force Between Two Parallel Conductors

35. Magnetic Force Between Two Parallel Conductors
The force (per unit length ) on wire 1 due to the current in wire 1 and the magnetic field produced by wire 2:
Parallel conductors carrying currents in the same direction attract each other
Parallel conductors carrying currents in the opposite directions repel each other

36. Chapter 19 Problem 57
A wire with a weight per unit length of N/m is suspended directly above a second wire. The top wire carries a current of 30.0 A and the bottom wire carries a current of 60.0 A. Find the distance of separation between the wires so that the top wire will be held in place by magnetic repulsion.

37. Magnetic Field of a Current Loop
The strength of a magnetic field produced by a wire can be enhanced by forming the wire into a loop
All the segments, Δx, contribute to the field, increasing its strength

38. Magnetic Field of a Current Loop
The magnitude of the magnetic field at the center of a circular loop with a radius R and carrying current I is
With N loops in the coil, this becomes

39. Magnetic Field of a Solenoid
If a long straight wire is bent into a coil of several closely spaced loops, the resulting device is called a solenoid
It is also known as an electromagnet since it acts like a magnet only when it carries a current
The field inside the solenoid is nearly uniform and strong – the field lines are nearly parallel, uniformly spaced, and close together
The exterior field is nonuniform, much weaker, and in the opposite direction to the field inside the solenoid

40. Magnetic Field of a Solenoid
The field lines of the solenoid resemble those of a bar magnet
The magnitude of the field inside a solenoid is approximately constant at all points far from its ends
B = µo n I
n = N / ℓ : the number of turns per unit length
The same result can be obtained by applying Ampère’s Law to the solenoid

41. Magnetic Field of a Solenoid
A cross-sectional view of a long tightly wound solenoid
If the solenoid is long compared to its radius, we assume the field inside is uniform and outside is zero
Apply Ampère’s Law to the blue dashed rectangle

42. Magnetic Effects of Electrons – Orbits
An individual atom should act like a magnet because of the motion of the electrons about the nucleus
Each electron circles the atom once in about every seconds; this would produce a current of 1.6 mA and a magnetic field of about 20 T at the center of the circular path
However, the magnetic field produced by one electron in an atom is often canceled by an oppositely revolving electron in the same atom
The net result is that the magnetic effect produced by electrons orbiting the nucleus is either zero or very small for most materials

43. Magnetic Effects of Electrons – Spins
Electrons also have spin (it is a quantum effect)
The classical model is to consider the electrons to spin like tops
The field due to the spinning is generally stronger than the field due to the orbital motion
Electrons usually pair up with their spins opposite each other, so their fields cancel each other, hence most materials are not naturally magnetic

44. Magnetic Effects of Electrons – Domains
In some materials – ferromagnetic – the spins do not naturally cancel
Large groups of atoms in which the spins are aligned are called domains
When an external field is applied, it causes the material to become magnetized: the domains that are aligned with the field tend to grow at the expense of the others

45. Domains and Permanent Magnets
In hard magnetic materials, the domains remain aligned after the external field is removed
The result is a permanent magnet
In soft magnetic materials, once the external field is removed, thermal agitation causes the materials to quickly return to an unmagnetized state
With a core in a loop, the magnetic field is enhanced since the domains in the core material align, increasing the magnetic field

46. Answers to Even Numbered Problems
Chapter 19:
Problem 2
toward the left; into the page; out of the page; toward top of page; into the page; out of the page
the answers for part (b) are reversed from those given in part (a)

47. Answers to Even Numbered Problems
Chapter 19:
Problem 26
4.33 × 10−3 N m

48. Answers to Even Numbered Problems
Chapter 19:
Problem 44
toward the left
out of the page
lower left to upper right

49. Answers to Even Numbered Problems
Chapter 19:
Problem 48
40.0 μT into the page
5.00 μT out of the page
1.67 μT out of the page