Lecture 14.1

1. Lecture 14.1 Finishing up chapter 34: RL circuit Chapter 35: Maxwell equations and electromagnetic waves

2. Review of Last Lecture • Faraday’s Law: change of magnetic flux induces electric field: • Lenz’s Law: induced field (EMF) reacts against the flux changeI • Inductor and Inductance:

3. The potential at a is higher than the potential at b. Which of the following statements about the inductor current I could be true? • I is from b to a and is steady. • I is from b to a and is increasing. • I is from a to b and is steady. • I is from a to b and is increasing. • I is from a to b and is decreasing.

4. The potential at a is higher than the potential at b. Which of the following statements about the inductor current I could be true? • I is from b to a and is steady. • I is from b to a and is increasing. • I is from a to b and is steady. • I is from a to b and is increasing. • I is from a to b and is decreasing.

5. LR circuits

6. LR circuits

7. Rank in order, from largest to smallest, the time constants τa, τb, and τc of these three circuits. • τa > τb > τc • τb > τa > τc • τb > τc > τa • τc > τa > τb • τc > τb > τa

8. Rank in order, from largest to smallest, the time constants τa, τb, and τc of these three circuits. • τa > τb > τc • τb > τa > τc • τb > τc > τa • τc > τa > τb • τc > τb > τa

9. Chapter 35. Electromagnetic Fields and Waves To understand a laser beam, we need to know how electric and magnetic fields change with time. Examples of time-dependent electromagnetic phenomena include high-speed circuits, transmission lines, radar, and optical communications. Chapter Goal: To study the properties of electromagnetic fields and waves.

10. Chapter 35. Electromagnetic Fields and Waves Topics: • E or B? It Depends on Your Perspective • The Field Laws Thus Far • The Displacement Current • Maxwell’s Equations • Electromagnetic Waves • Properties of Electromagnetic Waves • Polarization

11. E or B? It Depends on Your Perspective

12. E or B? It Depends on Your Perspective Whether a field is seen as “electric” or “magnetic” depends on the motion of the reference frame relative to the sources of the field.

13. E or B? It Depends on Your Perspective • In the frame S, a moving charge q generates magnetic field in the space. • On the other hand, in the frame S’ that is moving with the charge. The charge is at rest, and does not generate any magnetic field. • The value of magnetic field depends on the inertial frame. • You can not really separate electric field from magnetic field  electromagnetic field

14. The Laws of Electromagnetism so far • Electric Gauss’ Law (electric charge  E field) • Magnetic Gauss’ Law (No magnetic dipole) • Faraday’s Law (Changing B field  E field) • “Ampere’s Law” (Moving charge  B field) Maxwell noticed an inconsistency in Ampere’s law.

15. Ampère’s law Whenever total current Ithrough passes through an area bounded by a closed curve, the line integral of the magnetic field around the curve is The figure illustrates the geometry of Ampère’s law. In this case, Ithrough = I1− I2 .

16. Ampère’s law • Choose surface S1 • Choose surface S2

17. Displacement Current • Between two plates: no current, but there is an increasing E field

18. The Displacement Current The electric flux due to a constant electric field E perpendicular to a surface area A is The displacement current is defined as Maxwell modified Ampère’s law to read Changing E field also induces B field

19. EXAMPLE 35.3 The fields inside a charging capacitor

20. Maxwell’s Equations Lorentz Force Law

21. The Fundamental Ideas of Electromagnetism

22. Lecture 14.2 Electromagnetic waves

23. Maxwell’s Equations Lorentz Force Law

24. Maxwell Equations (Differential Forms) • Electric Gauss’ Law • Magnetic Gauss’ Law • Faraday’s Law • Ampere-Maxwell’s Law

25. Electromagnetic Waves • Maxwell, using his equations of the electromagnetic field, was the first to understand that light is an oscillation of the electromagnetic field. Maxwell was able to predict that • Electromagnetic waves can exist at any frequency, not    just at the frequencies of visible light. This prediction    was the harbinger of radio waves. • All electromagnetic waves travel in a vacuum with the    same speed, a speed that we now call the speed of    light.

27. Properties of Electromagnetic Waves Any electromagnetic wave must satisfy four basic conditions: •  The fields Eand B and are perpendicular to the direction of propagation vem.Thus an electromagnetic wave is a transverse wave. • Eand B are perpendicular to each other in a manner such that E× B is in the direction of vem. •  The wave travels in vacuum at speed vem = c • E = cBat any point on the wave.

28. Energy Density and Energy Flow Energy density The energy flow of an electromagnetic wave is described by the Poynting vector defined as The magnitude of the Poynting vector is Energy transferred by the wave per unit area and per unit time The intensity of an electromagnetic wave whose electric field amplitude is E0 is

29. Radiation Pressure Momentum transferred to an object that absorbs EM waves The radiation pressure on an object that absorbs all the light is where I is the intensity of the light wave. The subscript on prad is important in this context to distinguish the radiation pressure from the momentum p.

30. Science Fiction: Solar Sailing

31. Light reflected by flat surfaces are usually horizontally polarized: Sun glasses are designed to filtered out waves with this polarization.

33. Malus’s Law Suppose a polarized light wave of intensity I0 approaches a polarizing filter. θ is the angle between the incident plane of polarization and the polarizer axis. The transmitted intensity is given by Malus’s Law:

34. Malus’s Law If the light incident on a polarizing filter is unpolarized, the transmitted intensity is In other words, a polarizing filter passes 50% of unpolarized light and blocks 50%.

35. Two Polarizers

36. Electromagnetic Spectrum

37. Chapter 35. Summary Slides

38. General Principles

39. General Principles

40. General Principles

41. Important Concepts

42. Important Concepts

43. Applications

44. Chapter 35. Clicker Questions

45. Which diagram shows the fields in frame S´?

46. Which diagram shows the fields in frame S´?

47. The electric field in four identical capacitors is shown as a function of time. Rank in order, from largest to smallest, the magnetic field strength at the outer edge of the capacitor at time T. • Ba = Bb > Bc = Bd • Bd > Bc > Ba = Bb • Ba > Bb > Bc > Bd • Ba = Ba > Bc > Bd • Bc > Ba > Bd > Bb

48. The electric field in four identical capacitors is shown as a function of time. Rank in order, from largest to smallest, the magnetic field strength at the outer edge of the capacitor at time T. • Ba = Bb > Bc = Bd • Bd > Bc > Ba = Bb • Ba > Bb > Bc > Bd • Ba = Ba > Bc > Bd • Bc > Ba > Bd > Bb

49. An electromagnetic wave is propagating in the positive x-direction. At this instant of time, what is the direction of at the center of the rectangle? • In the positive x-direction • In the negative x-direction • In the positive z-direction • In the negative z-direction • In the positive y-direction

50. An electromagnetic wave is propagating in the positive x-direction. At this instant of time, what is the direction of at the center of the rectangle? • In the positive x-direction • In the negative x-direction • In the positive z-direction • In the negative z-direction • In the positive y-direction