Magnetism

1. Magnetism November 8, 2014

2. Magnets and Magnetic Fields • Magnets cause space to be modified in their vicinity, forming a “magnetic field”. • The magnetic field caused by magnetic “poles” is analogous to the electric field caused by electric “poles” or “charges”. • The north pole is where the magnetic field lines leave the magnet, and the south pole is where they reenter. • Magnetic field lines differ from electric field lines in that they are continuous loops with no beginning or end.

3. Magnetic Field, B

4. Magnetic “Monopoles” do not exist • This is another way that magnetic fields differ from electric fields. • Magnetic poles cannot be separated from each other in the same way that electric poles (charges) can be.

5. Units of Magnetic Field • Tesla (SI) – N/(C m/s) – N/(A m) • Gauss – 1 Tesla = 104 gauss

6. Magnetic Force on Particles • Magnetic fields cause the existence of magnetic forces. • A magnetic force is exerted on a particle within a magnetic field only if: • the particle has a charge. • the charged particle is moving with at least a portion of its velocity perpendicular to the magnetic field.

7. Magnetic Force on a Charged Particle • magnitude: F = q v B sin θ • q: charge in Coulombs • v: speed in meters/second • B: magnetic field in Tesla • θ: angle between v and B • direction: Right Hand Rule

8. The Right Hand Rule • Align your hand along the first vector. • Orient your wrist so that you can “cross” your hand into the second vector. • Your thumb gives you the direction of the third vector (which is the result).

9. Sample Problem: Calculate the magnitude force exerted on a 3.0 μC charge moving north at 300,000 m/s in a magnetic field of 200 mT if the field is directed North. South. East. West.

10. Magnetic forces… • are always orthogonal (at right angles) to the plane established by the velocity and magnetic field vectors. • can accelerate charged particles only by changing their direction. • can cause charged particles to move in circular or helical paths.

11. Magnetic forces cannot... • change the speed or kinetic energy of charged particles. • do work on charged particles.

12. Magnetic Forces… • …are centripetal. • Remember that centripetal acceleration is v2/r. • Centripetal force is therefore mv2/r.

13. Sample Problem: Draw the free body diagram and draw the path that the charged particle will take as it moves through the B-field. Ignore gravitational effects.

14. Sample Problem: An electric field of 2000 N/C is directed to the south. A proton is traveling at 300,000 m/s to the west. What is the magnitude and direction of the force on the proton? Describe the path of the proton? Ignore gravitational effects.

15. Sample Problem: A magnetic field of 2000 mT is directed to the south. A proton is traveling at 300,000 m/s to the west. What is the magnitude and direction of the force on the proton? Describe the path of the proton? Ignore gravitational effects.

16. Sample Problem: Calculate the force and describe the path of this electron.

17. Sample problem: Calculate the force and describe the path of this electron.

18. Sample Problem: How would you arrange a magnetic field and an electric field so that a charged particle of velocity v would pass straight through without deflection?

19. Sample Problem: It is found that protons traveling at 20,000 m/s pass undeflected through the velocity filter below. What is the magnitude and direction of the magnetic field between the plates?

20. Magnetic Force on Current -Carrying Wire • F = I L B sin θ • I: current in Amperes • L: length in meters • B: magnetic field in Tesla • q: angle between current and field

21. Sample Problem: What is the force on a 100 m long wire bearing a 30 A current flowing north if the wire is in a downward-directed magnetic field of 400 mT?

22. Sample Problem: What is the magnetic field strength if the current in the wire is 15 A and the force is downward and has a magnitude of 40 N/m? What is the direction of the current?

23. Magnetic Fields… • Affect moving charge • F = q v B sin θ • F = I L B sin θ • Hand rule is used to determine direction of this force. • They are also caused by moving charge – slightly paradoxical (Lenz’s Law)

24. Right Hand Rule for Straight Currents: • Curve your fingers • Place your thumb (which is presumably pretty straight) in direction of current. • Curved fingers represent curve of magnetic field. • Field vector at any point is tangent to field line.

25. Magnetic Field produced by straightcurrents.

26. Magnitude of Magnetic Field for straight currents. • μo: 4π × 10-7 T m / A • magnetic permeability of free space • I: current (A) • r: radial distance from center of wire (m)

27. Sample Problem: What is the magnitude and direction of the magnetic field at point P, which is 3.0 m away from a wire bearing a 13.0 Amp current?

28. Sample problem: what is the magnitude and direction of the force exerted on a 100 m long wire that passes through point P which bears a current of 50 amps in the same direction?

29. Principle of Superposition • When there are two or more currents forming a magnetic field, calculate B due to each current separately and then add them together using vector addition.

30. Sample Problem: What is the magnitude and direction of the electric field at point P if there are two wires producing a magnetic field at this point?

31. Sample Problem: Where would the magnetic field be zero?

32. Solenoid • A solenoid is a coil of wire. • When current runs through the wire, it causes the coil to become an “electromagnet”. • Air-core solenoids have nothing inside of them. • Iron-core solenoids are filled with iron to intensify the magnetic field.

33. Right Hand Rule for Curved Wires • Curve your fingers. • Place them along wire loop so that your fingers point in direction of current. • Your thumb gives the direction of the magnetic field in the center of the loop, where it is straight. • Field lines curve around and make complete loops.

34. Sample Problem: What is the direction of the magnetic field produced by the current I at A? At B?

35. Magnetic Flux • The product of magnetic field and area. • Can be thought of as a total magnetic “effect” on a coil of wire of a given area.

36. Maximum Flux • The area is aligned so that a perpendicular to the area points parallel to the field

37. Magnetic Flux • FB = B A cos θ • FB: magnetic flux in Webers (Tesla meters squared) • B: magnetic field in Tesla • A: area in meters squared. • θ: the angle between the area and the magnetic field.

38. Sample Problem: Calculate the magnetic flux through a rectangular wire frame 3.0 m long and 2.0 m wide if the magnetic field through the frame is 4.2 mT. a) Assume that the magnetic field is perpendicular to the area vector. b) Assume that the magnetic field is parallel to the area vector. c) Assume that the angle between the magnetic field and the area vector is 30-degrees.

39. Sample Problem: Assume the angle is 40o, the magnetic field is 50 mT, and the flux is 250 mWb. What is the radius of the loop?

40. Induced Electric Potential • A system will respond so as to oppose changes in magnetic flux. • A change in magnetic flux will be partially offset by an induced magnetic field whenever possible. • Changing the magnetic flux through a wire loop causes current to flow in the loop. • This is because changing magnetic flux induces an electric potential.

41. Faraday’s Law of Induction – e: induced potential (V) – N: # loops – FB: magnetic flux (Webers, Wb) – t: time (s) • To generate voltage, change B, change A, change q

42. Sample Problem: A coil of radius 0.5 m consisting of 1000 loops is placed in a 500 mT magnetic field such that the flux is maximum. The field then drops to zero in 10 ms. What is the induced potential in the coil?

43. Sample Problem: A single coil of radius 0.25 m is in a 100 mT magnetic field such that the flux is maximum. At time t = 1.0 seconds, field increases at a uniform rate so that at 11 seconds, it has a value of 600 mT. At time t = 11 seconds, the field stops increasing. What is the induced potential A) at t = 0.5 seconds? B) at t = 3.0 seconds? C) at t = 12 seconds?

44. Lenz’s Law • The current will flow in a direction so as to oppose the change in flux. • Use in combination with hand rule to predict current direction.

45. Sample Problem: The magnetic field is increasing at a rate of 4.0 mT/s. What is the direction of the current in the wire loop?

46. Sample Problem: The magnetic field is increasing at a rate of 4.0 mT/s. What is the direction of the current in the wire loop?

47. Sample Problem: The magnetic field is decreasing at a rate of 4.0 mT/s. The radius of the loop is 3.0 m, and the resistance is 4 W. What is the magnitude and direction of the current?

48. Motional emf ε = BLv – B: magnetic field (T) – L: length of bar moving through field – v: speed of bar moving through field. • Bar must be “cutting through” field lines. It cannot be moving parallel to the field.

49. Sample Problem: How much current flows through the resistor, and in what direction? How much power is dissipated by the resistor?

50. Sample Problem: How much force is required to keep the rod moving at the constant speed shown?