Chapter 26

1. Chapter 26 Magnetic Fields

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), and the force between two poles varies as the inverse square of the distance between them • Magnetic poles cannot be isolated – if a permanent magnetic 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. Magnets • The poles received their names due to the way a magnet behaves in the Earth’s magnetic field • If a bar magnet is suspended so that it can move freely, it will rotate • The magnetic north pole points toward the Earth’s north geographic pole • This means the Earth’s north geographic pole is a magnetic south pole • Similarly, the Earth’s south geographic pole is a magnetic north pole

4. Magnets • 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 magneticmaterials (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

5. 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

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

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)

8. 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)

9. Nikola Tesla 1856 – 1943 Magnetic Fields • 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)

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

11. 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

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

13. Differences Between Electric and Magnetic Fields • The electric force acts along the direction of the electric field, whereas the magnetic force acts perpendicular to the magnetic field • The electric force acts on a charged particle regardless of whether the particle is moving, while the magnetic force acts on a charged particle only when the particle is in motion • The electric force does work in displacing a charged particle, whereas the magnetic force associated with a steady magnetic field does no work when a particle is displaced (because the force is perpendicular to the displacement)

14. 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

15. 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)

16. Particle in a Nonuniform Magnetic Field • The motion is complex

17. Charged Particles Moving in Electric and Magnetic Fields • In many applications, charged particles move in the presence of both magnetic and electric fields • In that case, the total force is the sum of the forces due to the individual fields:

18. Chapter 26Problem 23 Microwaves in a microwave oven are produced by electrons circling in a magnetic field at a frequency of 2.4 GHz. (a) What’s the magnetic field strength? (b) The electrons’ motion takes place inside a special tube called a magnetron. If the magnetron can accommodate electron orbits with maximum diameter 2.5 mm, what’s the maximum electron energy?

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. Magnetic Force on a Current CarryingWire of an Arbitrary Shape • For a small segment of the wire, the force exerted on this segment is • The total force is

22. Chapter 26Problem 28 A wire with mass per unit length 75 g/m runs horizontally at right angles to a horizontal magnetic field. A 6.2-A current in the wire results in its being suspended against gravity. What’s the magnetic field strength?

23. Félix Savart 1791 – 1841 Jean-Baptiste Biot 1774 – 1862 Biot-Savart Law • Biot and Savart arrived at a mathematical expression that gives the magnetic field at some point in space due to a current • The magnetic field is dB at some point P; the length element is ds; the wire is carrying a steady current of I

24. Biot-Savart Law • Vector dB is perpendicular to both ds and to the unit vector directed from ds toward P • The magnitude of dB is inversely proportional to r2, where r is the distance from ds to P • The magnitude of dB is proportional to the current and to the magnitude ds of the length element

25. Biot-Savart Law • The magnitude of dB is proportional to sinq, where q is the angle between the vectors ds and • The observations are summarized in the mathematical equation called the Biot-Savart law (magnetic field due to the current-carrying conductor): • µo = 4  x 10-7 T.m / A:permeability of free space

26. Biot-Savart Law • To find the total field, sum up the contributions from all the current elements

27. Biot-Savart Law • The magnitude of the magnetic field varies as the inverse square of the distance from the ds element • The electric field due to a point charge also varies as the inverse square of the distance from the charge • The electric field created by a point charge is radial in direction • The magnetic field created by a current element is perpendicular to both the length element and the unit vector • The current element producing a magnetic field is part of an extended current distribution

28. A Long, Straight Conductor • The thin, straight wire is carrying a constant current

29. A Long, Straight Conductor • The thin, straight wire is carrying a constant current • If the conductor is an infinitely long, straight wire, θ1= π/2 and θ2= – π/2 , and the field becomes

30. The magnetic field lines are circles concentric with the wire The field lines lie in planes perpendicular to the wire The magnitude of the field is constant on any circle of radius a Right Hand Rule #2: Grasp the wire in your right hand and point your thumb in the direction of the current and your fingers will curl in the direction of the field A Long, Straight Conductor

31. Find the field at point O due to the wire segment (I,a are constants) The field at the center of the full circle loop A Curved Wire Segment

32. Magnetic Field of a Current Loop

33. Magnetic Field of a Current Loop • The field contribution from a current element I dl= I dx • For large distances (x >> a), this reduces to

34. Chapter 26Problem 30 A single-turn wire loop is 2.0 cm in diameter and carries a 650-mA current. Find the magnetic field strength (a) at the loop center and (b) on the loop axis, 20 cm from the center.

35. Torque on a Current Loop

36. 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

37. Magnetic Moment • The vector is called the magnetic dipole 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

38. Potential Energy • The potential energy of the system of a magnetic dipole in a magnetic field depends on the orientation of the dipole in the magnetic field • Umin = – μB and occurs when the dipole moment is in the same direction as the field • Umax = + μB and occurs when the dipole moment is in the direction opposite the field

39. Chapter 26Problem 35 A single-turn square wire loop 5.0 cm on a side carries a 450-mA current. (a) What’s the loop’s magnetic dipole moment? (b) If the loop is in a uniform 1.4-T magnetic field with its dipole moment vector at 40° to the field, what’s the magnitude of the torque it experiences?

40. 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°

41. 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

42. 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

43. Magnetic Force Between Two Parallel Conductors

44. 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

45. Chapter 26Problem 63 A long, straight wire carries 20 A. A 5.0-cm by 10-cm rectangular wire loop carrying 500 mA is 2.0 cm from the wire, as shown in the figure. Find the net magnetic force on the loop.

46. 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 (put the thumb of your right hand in the direction of the current through the loop and your fingers curl in the direction you should integrate around the loop)

47. Ampère’s Law for a Long Straight Wire • Use a closed circular path • The circumference of the circle is 2 r

48. Ampère’s Law for a Long Straight Wire

49. Magnetic Field of a Solenoid

50. 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