# II. Electromagnetic Waves A.Displacement Current 1.Recall Ampere’s Law: 2.As we’ve learned it, AL is incomplete. We need to add an additional current,

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II. Electromagnetic Waves A.Displacement Current 1.Recall Ampere’s Law: 2.As we’ve learned it, AL is incomplete. We need to add an additional current, called the displacement current, I D. ID arises from time-varying electric fields (not present in a steady current along an infinite wire): (II.A.1,2)

II. Electromagnetic Waves A.Displacement Current 3.General form of Ampere’s Law includes terms due to “conduction current” and “displacement current”: (II.A.3)

II. Electromagnetic Waves B.MAXWELL’S EQUATIONS 1.Unified description of E, B: (II.B.1-4) (Gauss’s Law) (Gauss’s Law for B) (Faraday’s Law for B) (Ampere’s Law for B)

II. Electromagnetic Waves B.MAXWELL’S EQUATIONS 2.Plane Wave a)As we shall see, the solution to Maxwell’s Equations is a wave of Electric and Magnetic Fields. b)Plane Wave Definition: Wave in which the transverse components are uniform on a plane perpendicular to the direction of propagation. V

II. Electromagnetic Waves B.MAXWELL’S EQUATIONS 2.Plane Wave a)As we shall see, the solution to Maxwell’s Equations is a wave of Electric and Magnetic Fields. b)Plane Wave Definition: Wave in which the transverse components are uniform on a plane perpendicular to the direction of propagation. E B

II. Electromagnetic Waves B.MAXWELL’S EQUATIONS 2.Plane Wave a)As we shall see, the solution to Maxwell’s Equations is a wave of Electric and Magnetic Fields. b)Plane Wave Definition: Wave in which the transverse components are uniform on a plane perpendicular to the direction of propagation. E B E B

II. Electromagnetic Waves B.MAXWELL’S EQUATIONS 2.Plane Wave a)As we shall see, the solution to Maxwell’s Equations is a wave of Electric and Magnetic Fields. b)Plane Wave Definition: Wave in which the transverse components are uniform on a plane perpendicular to the direction of propagation. E B E B E B

II. Electromagnetic Waves B.MAXWELL’S EQUATIONS 3.Electromagnetic Wave Properties a) Transverse wave b) Ratio between E,B: E/B = c.(II.B.5) c) Constant speed d) No medium required: E and B reinforce each other. E B E B E B E B

II. Electromagnetic Waves B.MAXWELL’S EQUATIONS 4.Derivation of Solution: Plane Wave a)Consider a plane wave with B z, E y propagating in the x-direction with speed v. After time t, the two wave fronts are separated by a distance  x. b) Apply Faraday’s Law to a rectangle in the xy-plane: E B E B x x  x y z B, A a x y

II. EM Waves B.Maxwell’s Equations 4.Derivation of Solution: Plane Wave b) Apply Faraday’s Law to a rectangle in the xy-plane: Assume  x is small enough that B z ~ uniform over surface. E B E B x x  x y z B, A a x y xx

II. EM Waves B.Maxwell’s Equations 4.Derivation of Solution: Plane Wave b) Apply Ampere’s Law to a rectangle in the zx-plane: Assume  x is small enough that E y ~ uniform over surface. E B E B x x  x y z E, A a x z xx

II. EM Waves B.Maxwell’s Equations 4.Derivation of Solution: Plane Wave b) Apply Ampere’s Law to a rectangle in the zx-plane: Assume  x is small enough that E y ~ uniform over surface. E B E B x x  x y z E, A a xx x z

II. EM Waves B.Maxwell’s Equations 4.Derivation of Solution: Plane Wave c)Now take partial time and space derivatives of both equations: E B E B x x  x y z z This is the wave equation with v = (     ) -1/2 = c!(II.B.7) (II.B.6)

II. EM Waves B.Maxwell’s Equations 5.Sinusoidal Waves a)A more accurate representation of EM Waves b)Plane waves can be a good approximation c)For wave propagating in the +x-direction: d)E,B in phase, follow RHR: c, E, B (II.B.8)

C.The Production of Electromagnetic Waves 3.Antennae a)Accelerating charges radiate energy as EM waves b)Oscillating voltage => accelerates charge => EM radiation ++++++ t = 0: Charge placed on metal rods connected to an AC generator. ------ E VV II. EM Waves

F.The Production of Electromagnetic Waves 3.Antennae a)Accelerating charges radiate energy as EM waves b)Oscillating voltage => accelerates charge => EM radiation ++++ t = 0 to T/4: Rods neutralize, and E decreases to 0. Note: Initial E propagates away from array at speed c. ---- E II. EM Waves

C.The Production of Electromagnetic Waves 3.Antennae a)Accelerating charges radiate energy as EM waves b)Oscillating voltage => accelerates charge => EM radiation + t = 0 to T/4: Rods neutralize, and E decreases to 0. Note: Initial E propagates away from array at speed c. - E II. EM Waves

C.The Production of Electromagnetic Waves 3.Antennae a)Accelerating charges radiate energy as EM waves b)Oscillating voltage => accelerates charge => EM radiation t = 0 to T/4: Rods neutralize, and E decreases to 0. Note: Initial E propagates away from array at speed c. E = 0 at t = T/4. II. EM Waves

C.The Production of Electromagnetic Waves 3.Antennae a)Accelerating charges radiate energy as EM waves b)Oscillating voltage => accelerates charge => EM radiation - t = T/4 to T/2: E reverses direction and grows. + II. EM Waves

C.The Production of Electromagnetic Waves 3.Antennae a)Accelerating charges radiate energy as EM waves b)Oscillating voltage => accelerates charge => EM radiation ---- t = T/4 to T/2: E reverses direction and grows. ++++ II. EM Waves

++++++ C.The Production of Electromagnetic Waves 3.Antennae a)Accelerating charges radiate energy as EM waves b)Oscillating voltage => accelerates charge => EM radiation ------ t = T/4 to T/2: E reverses direction and grows. II. EM Waves

C.The Production of Electromagnetic Waves 3.Antennae a)Accelerating charges radiate energy as EM waves b)Oscillating voltage => accelerates charge => EM radiation ------ ++++++ II. EM Waves

C.The Production of Electromagnetic Waves 3.Antennae c)Oscillating E => Oscillating B wave ------ ++++++ v = c. B End result: A transverse wave of E propagating at speed v = (  0  0 ) -1/2 = c. II. EM Waves

C.The Production of Electromagnetic Waves 3.Antennae c)Oscillating E => Oscillating B wave Top View - B II. EM Waves

C.The Production of Electromagnetic Waves 3.Antennae c)Oscillating E => Oscillating B wave Top View - II. EM Waves

C.The Production of Electromagnetic Waves 3.Antennae c)Oscillating E => Oscillating B wave Top View - II. EM Waves

C.The Production of Electromagnetic Waves 3.Antennae c)Oscillating E => Oscillating B wave Top View II. EM Waves

C.The Production of Electromagnetic Waves 3.Antennae c)Oscillating E => Oscillating B wave Top View + II. EM Waves

C.The Production of Electromagnetic Waves 3.Antennae c)Oscillating E => Oscillating B wave Top View II. EM Waves

C.The Production of Electromagnetic Waves 3.Antennae c)Oscillating E => Oscillating B wave Top View - c * E and B perpendicular to each other. * E and B perpendicular to v. * E and B in phase. II. EM Waves

D.Properties of EM Waves 1.Field strengths of EM wave E/B = c.(II.D.1) 2.Poynting Vector: Energy Flow Rate Vector 3.Power and Intensity: P= S per unit area, I = S(avg) I= E max B max /(2  0 ), (II.D.2) = E 2 max /(2  0 c) = B 2 max (c/2  0 ).

D.Properties of EM Waves 4.Radiation Pressure p = I/c (complete absorption)(II.D.3) p = 2I/c(complete reflection)(II.D.4) 5.EM waves in matter n = c/v = “index of refraction”(II.D.5)      II. EM Waves

E.The Electromagnetic Spectrum 1.Units a)Angstrom (Å) = 10 -10 m b)Nanometer (nm) = 10 -9 m c)Micron (  m) = 10 -6 m 2.Radio, Microwave, Infrared, Visible, Ultraviolet, X-rays, Gamma rays 3.VISIBLE: “ROYGBIV” = Red, Orange, Yellow, Green, Blue, Indigo, and Violet (large wavelength to small) II. EM Waves

E.The Electromagnetic Spectrum 1.Units a)Angstrom (Å) = 10 -10 m b)Nanometer (nm) = 10 -9 m c)Micron (  m) = 10 -6 m 2.Radio, Microwave, Infrared, Visible, Ultraviolet, X-rays, Gamma rays 3.VISIBLE: “ROYGBIV” = Red, Orange, Yellow, Green, Blue, Indigo, and Violet (large wavelength to small) II. EM Waves

A.Working Definitions 1.Diffraction occurs when light source is not a perfect point source and wave encounters a sharp edge. 2.Diffraction is essentially an example of interference between a large (continuous) distribution of sources. 3.Limits resolution of instruments—but also can be used to separate multi-chormatic light. III. Diffraction

4.Spreading of wave from its initial line of travel No diffraction III. Diffraction

4.Spreading of wave from its initial line of travel Diffraction III. Diffraction

5.Diffraction occurs when light passes through a narrow opening, around obstacles and at sharp edges. a)Application: Calculating stellar diameters by lunar occultation Unresolved point of light III. Diffraction

5.Diffraction occurs when light passes through a narrow opening, around obstacles and at sharp edges. a)Application: Calculating stellar diameters by lunar occultation Unresolved point of light III. Diffraction

5.Diffraction occurs when light passes through a narrow opening, around obstacles and at sharp edges. a)Application: Calculating stellar diameters by lunar occultation Unresolved point of light III. Diffraction

5.Diffraction occurs when light passes through a narrow opening, around obstacles and at sharp edges. a)Application: Calculating stellar diameters by lunar occultation III. Diffraction Resolved diffraction pattern: spacing of fringes => width of star

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