Lecture 31 maxwell's equations. em waves.
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This document summarizes Maxwell's equations and describes electromagnetic waves. It shows that Maxwell's equations predict that changing electric fields produce magnetic fields and vice versa, allowing electromagnetic waves to propagate through space without a medium. Plane electromagnetic waves are described with oscillating and perpendicular electric and magnetic fields traveling at the speed of light. The document derives the wave equation for electromagnetic waves and shows they can be described as sinusoidal solutions. It introduces how electromagnetic waves propagate in materials with a refractive index greater than 1.
Lecture 31 maxwell's equations. em waves.
- 2. Maxwell’s equations r r q Ñ ×dA = ε ∫E 0 Gauss’s law for E r r Ñ ×dA = 0 ∫B Gauss’s law for B r r d ΦB Ñ ×dl = − dt ∫E r r d ΦE Ñ ×dl = µ0 I + ε 0 dt ∫B Faraday’s law ÷ ÷ Ampere’s law
- 3. In the absence of sources The symmetry is then very impressive: r r Ñ ×dA = 0 ∫E r r d ΦB Ñ ×dl = − dt ∫E changing B-field induces E-field r r Ñ ×dA = 0 ∫B r r d ΦE Ñ ×dl = µ0ε 0 dt ∫B changing E-field induces B-field
- 4. E produces B, B produces E Example: At some point P in space, the measured electric field as a function of time is: E P t This E field MUST be accompanied by a B field. But if the induced B field also changes with time, it induces an E field! This can be self-sustained! “Perturbation” (E or B field) propagates in space ⇒ wave Very special wave, can propagate in vacuum: no medium!
- 6. Plane electromagnetic wave Let us assume that we have: • E-field in the y direction, uniform along yz plane • B-field in the z direction, uniform along yz plane • propagation in the x direction y E B x z E and B are the same at all points in this plane, but different at a parallel plane further down the x axis.
- 7. y Faraday’s law: Line integral over a rectangular circuit (sides Δx, a) in the xy plane.: r r Ñ ×dl = E y ( x + ∆x ,t ) a − E y ( x ,t ) a ∫E Ey (x + ∆x,t a ) Ey (x,t ) Bz (x,t ) x Bz (x + ∆x,t ) z ∆x If Δx is small ΦB = a ∫ x +∆x x Bz ( x ,t ) dx ~ aBz ( x ,t ) ∆x r r d ΦB E × =− dl Ñ ∫ dt ∂Bz ( x ,t ) ∂t =− ⇒ ∂Bz ( x ,t ) d ΦB = a ∆x dt ∂t E y ( x + ∆x ,t ) − E y ( x ,t ) a = −a ∆x E y ( x + ∆x ,t ) − E y ( x ,t ) ∆x → ∆x → 0 ∂Bz ( x ,t ) ∂Bz ( x ,t ) ∂t ∂t =− ∂E y ( x ,t ) ∂x
- 8. y Ampere’s law: Line integral over a rectangular circuit (sides Δx, a) in the xz plane: Ey (x,t ) Bz (x,t ) z r r d ΦE Ñ ×dl = ε 0 µ0 dt ∫B −Bz ( x + ∆x ,t ) + Bz ( x ,t ) a = ε 0 µ0a ∆x ε 0 µ0 ∂E y ( x ,t ) ∂t =− a ∂E y ( x ,t ) ∂t Bz ( x + ∆x ,t ) − Bz ( x ,t ) ∆x ε 0 µ0 → ∆x → 0 ∂t =− x Bz (x + ∆x,t ) ∆x ∂E y ( x ,t ) E y ( x + ∆ x, t ) ∂Bz ( x ,t ) ∂x
- 9. ε 0 µ0 ∂E y ∂t =− ∂Bz ε 0 µ0 → ∂ ∂x ∂t ∂Bz ∂t ∂2Bz ∂2E y ∂2Bz ∂ = = ∂t ∂x ∂x ∂t ∂x ∂Bz ∂t =− ∂t 2 =− ∂2Bz ∂t ∂x ∂E y ∂x ∂ =− ÷ ÷ ∂x ε 0 µ0 ∂E y ∂x ∂t 2 = ∂2E y ∂x 2 ÷ ÷ Similarly, we can obtain Remember the wave equation? (lecture 5) ∂2E y ∂2y 1 ∂2y = v 2 ∂t 2 ∂x 2 This value is essentially identical to the speed of light measured by Foucault in 1860! (3×108 m/s) Maxwell identified light as an electromagnetic wave. ε 0 µ0 ∂2Bz ∂t = 2 ∂2Bz ∂x 2 This is a wave with speed v = 1 ε 0 µ0
- 10. Sinusoidal solution One of the solutions: E y = E max cos ( kx − ωt ) with ∂Bz ∂t =− ∂E y ∂x k = 2π λ ⇒ ⇒ ω = 2πf ∂Bz ∂t Bz = c = λf = = kE max sin ( kx − ωt ) k E max cos ( kx − ωt ) ω 1 Bmax = E max c ω 1 = k ε 0 µ0
- 11. Harmonic EM waves In general, harmonic EM wave propagating in the x direction: r r E = E max cos ( kx − ωt ) r r B = Bmax cos kx − ωt ( ) with k = 2π λ ω = 2πf c = λf = r r g E is perpendicular to B r r g propagation direction is E × B g wave speed is c = 1 ε 0 µ0 r r g E and B are always in phase g E = cB (magnitudes) al l for s! True shape wave ω 1 = k ε 0 µ0
- 12. ACT: B field This is the E-field shown at a point in space and time for an EM wave that travels in the negative z direction. What is the direction of the Bfield at this point and time? A. +x B. −x C. +z D. −z r r Propagation direction is E × B
- 14. In-class example: radio stations Radio stations broadcast at frequencies that range from 540 kHz (low end of AM band) up to 108 MHz (high end of FM band). What is the range of wavelengths associated with these frequencies? c A. 5.55 cm to 2.78 m λ= f B. 2.78 m to 55.5 m 3 × 108 m/s λmax = = 555 m 3 C. 55.5 m to 278 m 540 × 10 Hz D. 278 m to 555 m E. 2.78 m to 555 m λmin 3 × 108 m/s = = 2.78 m 6 108 × 10 Hz Big range! Very different physics (we’ll come back to this: diffraction and reflection)
- 15. EM waves not in vacuum Phys 221: E field inside a material is characterized by dielectric constant κ or the dielectric permittivity ε = κε 0 Similarly: B field inside a material is characterized by relative permeability κm or the permeability µ = κµ0 EM wave speed in a dielectric: v = 1 εµ = c κκ m c = n n = κκ m (>1 always ) Refraction index