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Chapter 32: Maxwell’s Equation and EM Waves Slide 29-1 Equations of electromagnetism: a review • We’ve now seen the four fundamental equations of electromagnetism, here listed together for the first time. • But one is incomplete: Ampère’s law needs refining Slide 29-2 Maxwell’s Adjustment to Ampere’s Law I ~ = µ0 Ienc ~ · dr B • For the situation on right, Ampere’s law predicts that the B field depends on which Amperian loop is used. • Can’t have contradictory results – either there is a B field or there isn’t! • Maxwell postulated that a changing electric flux acts as a source of magnetic fields (in addition to currents) Slide 29-3 Maxwell’s Adjustment to Ampere’s Law • Need to add a term to the right side of Ampere’s law to account for the changing electric flux. This is the displacement current. • Notice, for a parallel plate capacitor, the electricZflux is E = ~ = EA = ~ · dA E Q A= ✏0 ✏0 d E I = dt ✏0 • So Maxwell added a displacement current (Ampere-Maxwell term to Ampere’s law: • The the rate of change of the flux is I ✓ ⇤ · d⇤⇥ = µ0 (I + Id )enc = µ0 I + B • Corresponding displacement current density: d E 0 dt ◆ Law) enc ~ d E J~d = ✏0 dt Slide 29-4 • Changing B field induces E Field. • Changing E field induces B field I d E ~ ~ B · dl = µ0 ✏0 dt I ~ = ~ · dl E d B dt Slide 29-5 Consider a large parallel plate capacitor as shown, charging so that Q = Q0+βt on the positively charged plate. Assuming the edges of the capacitor and the wire connections to the plates can be ignored, what is the magnitude of the magnetic field B halfway between the plates, at a radius r? s z a r I Q I -Q d Slide 29-6 I ⇤ · d⇤⇥ = µ0 B d E 0 dt r<R: I 2⇡rB = µ0 ✏0 ⇡r ✏0 ⇡R2 r B = µ0 I (r < R) 2⇡R2 2 Slide 29-7 I ⇤ · d⇤⇥ = µ0 B d E 0 dt r>R: I 2⇡rB = µ0 ✏0 ⇡R ✏0 ⇡R2 2 µ0 I B= (r > R) 2⇡r Slide 29-8 Maxwell’s equations • These equations form a complete description of electric and magnetic fields • Combined with the Lorentz force, F = q(E + vxB), they form the complete theory of electromagnetism (classically). Slide 29-9 Maxwell’s equations in vacuum • In a vacuum there’s no electric charge density and also no current density. Maxwell’s equations in vacuum • A changing electric field is a source for a magnetic field, and a changing magnetic field is a source for an electric field. • These equations infer the possibility of electromagnetic waves! Slide 29-10 The Wave Equation • Maxwell’s equations can be manipulated to show that, for situations where E and B only vary in the x direction: 2⇤ ⇥ E = µ0 2 ⇥x ⇤ ⇥2B = µ0 2 ⇥x 2⇤ ⇥ E 0 ⇥t2 ⇤ ⇥2B 0 ⇥t2 • more generally, 2⇤ E ⇥ 2⇤ r E = µ0 0 2 ⇥t r2 = 2 x2 x̂ + 2 y2 2⇤ B ⇥ 2⇤ r B = µ0 0 2 ⇥t ŷ + 2 z2 ẑ Slide 29-11 Waves on a String • Let’s consider first mechanical waves (Ch. 15): • Consider a string with tension T and linear mass density µ • Application of Newton’s 2nd law gives the wave equation for the string: @ 2 y(x, t) µ @ 2 y(x, t) = 2 @x T @t2 • Sinusoidal Solutions: y(x, t) = A cos 2⇡ ✓ x t P ◆ Slide 29-12 Waves on a String @ 2 y(x, t) µ @ 2 y(x, t) = 2 @x T @t2 ✓ y(x, t) = A cos 2⇡ (2⇡)2 2 µ (2⇡)2 = T P2 phase velocity: P x = t P s ◆ T µ Alternate Expression: y(x, t) = A cos (kx k = 2⇡/ !t) ! = 2⇡f = 2⇡/P Slide 29-13 Clicker Question Two traveling waves 1 and 2 are described by the equations. y1 (x, t) = 2sin(2x − t) y2 (x, t) = 4sin(x − 2t) All the numbers are in the appropriate SI (mks) units. Which wave has the higher speed? A) Wave 1 B) Wave 2 C) Both have the same speed. Slide 29-14 Clicker Question @2E @x2 @2E ✏ 0 µ0 2 = 0 @t • What is the velocity of an electromagnetic wave? 1. ✏ 0 µ0 1 2. ✏ µ 0 0 3. 4. r p 1 ✏ 0 µ0 ✏ 0 µ0 Slide 29-15 Electromagnetic Waves • Maxwell postulated (1865) that light is an electromagnetic wave. • Electromagnetic waves travel in a vacuum with the speed of r light, 1 c= ✏ 0 µ0 = 3 ⇥ 108 m/s • Electromagnetic waves can exist having any frequency, not just at the frequencies of light. • Foreshadow of radio waves! • Confirmed by Heinrich Hertz (1887) Slide 29-16 Plane Waves as Solutions 2 @ E @x2 @ 2 y(x, t) 1 @y(x, t) = 2 2 @x v @t2 2 @ E ✏ 0 µ0 2 = 0 @t E(x, t) = E0 sin(kx !t) (in y direction) B(x, t) = B0 sin(kx !t) (in z direction) k 2 E0 sin(kx ⇥t) = 0 µ0 ( )⇥ 2 E0 sin(kx ⇥t) This is a solution if ⇥ v= = k 1 0 µ0 Also, since kEp = =c=3 108 m/s Bp 1 Bp = Ep c Slide 29-17 Electromagnetic Plane Waves • Plane wave are waves propagating in one direction with one wavelength. (E and B do not vary with respect to the other two dimensions) • E and B are transverse to each-other and to direction of propagation (E ⇥ B gives direction of propagation) • No medium required for wave to propagate! • Example (shown in figure): ~ E(x, t) = Ep sin(kx !t) ĵ ~ B(x, t) = Bp sin(kx !t) k̂ Slide 29-18 Clicker question • At a particular point, the electric field of an electromagnetic wave points in the + y direction, while the magnetic field points in the − z direction. Which of the following describes the propagation direction? A. + x B. − x C. either + x or − x but you can’t tell which D. − y Slide 29-19 Clicker question • A planar electromagnetic wave is propagating through space. Its electric field vector is given by ⇥ = Ep cos(kz E t)î Its magnetic field vector is ⇥ = Bp cos(kz 1) B ⇥ = Bp cos(ky 2) B t)ĵ t)k̂ ⇥ = Bp cos(ky 3) B ⇥ = Bp cos(kz 4) B t)k̂ ⇥ = Bp sin(kz 5) B t)î t)ĵ Slide 29-20 The Electromagnetic Spectrum Slide 29-21 Clicker question • Which type of radiation travels with the highest speed? 1. visible light 2. X-rays 3. Gamma-rays 4. radio waves 5. they all have the same speed Slide 29-22 An Active Galaxy seen in Multiple Wavelengths Slide 29-23 Antennae An electric field parallel to an antenna (electric dipole) will “shake” electrons and produce an AC current. A magnetic dipole antenna (for AM radios) should be oriented so that the B-field passes into and out of the plane of a loop (as the wave passes by), inducing a current in the loop. Slide 29-24 Producing electromagnetic waves • Electromagnetic waves are generated ultimately by accelerating electric charge. • Details of emitting systems depend on wavelength, with most efficient emitters being roughly a wavelength in size. • Radio waves are generated by alternating currents in metal antennas. • Molecular vibration and rotation produce infrared waves. • Visible light arises largely from atomic-scale processes. • X rays are produced in the rapid deceleration of electric charge. • Gamma rays result from nuclear processes. A radio transmitter and antenna Electric fields of an oscillating electric dipole Slide 29-25 Energy in EM Waves • Recall that the energy density due to electric and magnetic fields (in a vacuum) is given by 1 1 2 2 u = uE + uB = 2 ✏0 E + 2µ0 B • The Poynting vector describes the rate of energy flow per unit area (W/m2 in SI). It’s magnitude is given by (B 2 = E 2 /c2 = µ0 ✏0 E 2 ) S = uc = (uE + uB )c = E 2 c 1 ~ • A more general equation is S~ = E~ ⇥ B µ0 • For plane waves (traveling in x direction with E oriented in z direction): 1 ~ S= Ep Bp cos2 (kx µ0 !t)î = ✏0 Ep2 c cos2 (kx !t) î Slide 29-26 Energy in EM waves • Electromagnetic waves transport energy • Averaging over the time variations of the oscillating fields gives the average value, also called the intensity: 1 I =< S >= ✏0 Ep2 c 2 • For an isotropic source of radiation having a luminosity (output power) L: L I= 4⇡r2 • Notice this implies that the electric field goes as 1/r (rather than 1/r2 for stationary charges) Slide 29-27 Example: The luminosity of the Sun is L=3.8 x 1026 W. Earth is a distance r=1.5 x 1011 m away. What is the intensity of sunlight here on Earth? Slide 29-28 Flux and Solar Heating Flux is the rate of energy incident on a surface per unit area Slide 29-29 • The US uses about 4 million GWh of energy per year. Roughly what surface area would be needed to supply this energy solely using solar farms? Slide 29-30 Seasons Why is it hotter in summer than in winter? 1. The Earth is closer to the Sun. 2. The Earth is tilted towards the Sun, causing half the Earth to be closer to the Sun 3. The Earth is tilted towards the Sun, causing the sunlight to be more direct (more intense). 4. The Sun becomes more luminous. Slide 29-31