Maxwell's Equations: Understanding Electromagnetic Waves in Vacuum and Matter, Study notes of Physics

Download Maxwell's Equations: Understanding Electromagnetic Waves in Vacuum and Matter and more Study notes Physics in PDF only on Docsity! Announcements • HWK 5 due today. • Online participation available at the website • Browse Chap. 3 TZ&D Atoms (you know it already). • Reading for Fri.: TZ&D Chap. 4.1-4.3 • Check out the PhET site (link is on our website) and play with some sims. First comment: 0 E ρ ε ∇ =i 0 0 0 EB J t μ μ ε ∂∇× = + ∂ BE t ∂ ∇× = − ∂ 0B∇ =i OK, so maybe panic a bit, but these eqs. are not that bad: Each of these is a single partial differential eq. Total of 8 coupled PDEs to solve for E and B, given ρ and J. Each of these is a set of three coupled partial differential eqs. Tool kit item #1: complex exponential solutions Constant vector prefactors tell the direction and maximum strength of E and B ( ) ( )0, , , exp x y zE x y z t E i k x k y k z tω⎡ ⎤= + + −⎣ ⎦ ( ) ( )0, , , exp x y zB x y z t B i k x k y k z tω⎡ ⎤= + + −⎣ ⎦ Complex exponentials for the sinusoidal spatial and time oscillation of the waves Remember: [ ]exp cos sinii e iθθ θ θ≡ = + Or, ( ) ( )0, , , expE x y z t E i k r tω⎡ ⎤= −⎣ ⎦i Euler’s Equation. The electric field for a plane wave is given by: ( ) ( )0, , , expE x y z t E i k r tω A) The direction of the electric field vector. B) The direction of the magnetic field vector. C) The direction in which the wave is not varying. D) The direction the plane wave moves. E) None of these. ⎡ ⎤= −⎣ ⎦i The vector, k, tells you: Maxwell’s Equations in vacuum: “In vacuum” are code words. They really mean “No charges and no currents.” Then, the equations become: 0E∇ =i 0 0 EB t μ ε ∂∇× = ∂ BE t ∂ ∇× = − ∂ 0B∇ =i 12 0 8.85 10 /farad mε −= × 7 0 4 10 /henries mμ π −≡ × Maxwell’s Equations in vacuum: 0 0ik E =i 0 0 0 0ik B i Eωμ ε× = − 0 0ik E i Bω× = 0 0ik B =i • Transverse waves The complex exponentials reduce Maxwell to a set of vector algebraic relations between the three vectors k, E0, and B0, and the angular frequency, ω: . E0 and B0 are perpendicular to k. • E0 and B0 are perpendicular to each other. Maxwell’s Equations in vacuum: 0 0ik E =i 0 0 0 0ik B i Eωμ ε× = − 0 0ik E i Bω× = 0 0ik B =i The complex exponentials reduce Maxwell to a set of vector algebraic relations between the three vectors k, E0, and B0, and the angular frequency, ω: 2 2 2 0 0 1 c k ω μ ε = ≡ 0 0 E c B = Maxwell’s Equations describe EM radiation in vacuum: k B E E, B, and k form a ‘right-handed system’, with the wave traveling in the direction of k at speed, c. Notice: We never mentioned the wave equation. Wave or Particle? Question arises often throughout course: • Is something a wave, a particle, or both? How do we know? • When best to think of as a wave? as a particle? In classical view of light, EM radiation viewed as a wave (after lots of debate in 1600-1800’s). How to decide it is a wave EXPERIMENTALLY? What is most definitive observation we can make that tells us something is a wave? EM radiation is a wave What is most definitive observation we can make that tells us something is a wave? Ans: Observe interference. Constructive interference: c c (peaks are lined up and valleys are lined up) EM radiation is a wave What is most definitive observation we can make that tells us something is a wave? Ans: Observe interference. Destructive interference: c c (peaks align with valleys – cancel) Wave Interference Sim ElectronsTest metal Two metal plates in vacuum, adjustable voltage between them, shine light on one plate. Measure current between plates. Photolelectric effect experiment apparatus. 10 V A B Potential difference between A and B = +10 V Measure of energy an electron gains going from A to B.+- I. Understanding the apparatus and experiment. Photolelectric effect experiment apparatus. 10 V A B Potential difference between A and B = a. 0 V, b. 10 V, c. infinite volts +- Uniform E-field between plates 10Volts0V 10V 0V Constant force on electron constant acceleration E F + + + + + Potential difference between A and B = a. 0 V, b. 10 V, c. inf. V ans. b. 10 V. No electrons can get across gap, Note: if stuck one in space at plate A, would move to B and pick up energy equivalent to 10 V. Electron feels electric field, accelerates to + plate, picks up energy = q(10V) = 1 electron charge x 10 V = 10 eV A B Good place for a break. Additional material about how EM Waves interact with matter. Not covered in class 10/1/08 Comments on the response of matter: Matter is typically composed of charge neutral atoms. Behavior of the atoms is critical: +e-e +e-e E atom perturbed atom +e-e B Δv perturbed atom p qd= Induced electric dipole moment Induced magnetic dipole moment m Ia= P E Eχ= M Hχ=M Induced electric dipole moment per unit volume Induced magnetic dipole moment per unit volume ? Maxwell’s Equations in matter: ( ) 0Eε∇ =i ( )EB t ε μ ∂ ∇× = + ∂ BE t ∂ ∇× = − ∂ 0B∇ =i Now we have the full equations that include the effects of bound charge and associated currents: SO: Maxwell’s Equations in matter: Matter introduces currents and charge densities. However, for linear, isotropic systems, the equations are quite similar to vacuum: ( ) 0Eε∇ =i EB t με ∂∇× = ∂ BE t ∂ ∇× = − ∂ 0B∇ =i ε and μ depend on the particular material. More later, but notice that what we learn for vacuum is easy to transfer to matter. Maxwell’s Equations in matter: 0 0ik E =i 0 0ik B i Eωμε× = − 0 0ik E i Bω× = 0 0ik B =i • Transverse waves The complex exponentials reduce Maxwell to a set of vector algebraic relations between the three vectors k, E0, and B0, and the angular frequency, ω: . E0 and B0 are perpendicular to k. • E0 and B0 are perpendicular to each other. Maxwell’s Equations in meta-materials: 2 2 2 1 mc k ω με = ≡ However, the energy transport is the wrong way unless k has the opposite sign: 0 0m cn c με μ ε ≡ = − Here ε <0 and μ <0. Signs cancel here. Looks OK. 1S E B μ ≡ × ‘Negative phase velocity’ in meta-materials. Use the negative root so energy transport is still the right direction. An electromagnetic plane wave propagates to the right. Four antennas are labeled 1-4. The antennas are oriented vertically. Antennas 1,2, and 3 lie in the x-y plane. Antennas 1,2, and 4 have the same x- coordinate, but antenna 4 is located further out in the z-direction. Which choice below is the best ranking of the time-averaged signals received by each antenna? (Hint: the time-averaged signal in the signal averaged over several cycles of the wave.) CT34-44 A)1=2=3>4 B)3>2>1=4 C)1=2=4>3 D)1=2=3=4 E)3>1=2=4 Main points so far • EM waves are based on sinusoidal solutions of Maxwell’s Equations and travel in vacuum (the absence of matter) with a wavelength and frequency that multiply to yield the speed of light. • The differential approach allows E and B to be related to each other locally, without reference to the remote sources. Very nice. Matter introduces local charges and currents, which modify the EM waves. More fun for Tutorial 2.