lect24 - Physics 227 Lecture 24 Light • Lecture 23 review...

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Unformatted text preview: Physics 227: Lecture 24 Light • Lecture 23 review: • • RLC circuits with impedance: • ZR = R, ZL = jωL, ZC = 1/jωC = -j/ωC. Transformers: • • • Monday, December 5, 2011 V1/N1 = V2/N2 Current reduced by same factor voltage increased. Load resistance also transformed. Light iclicker Which of the following is true about the nature of light. There may be more than one correct answer. A. Light is an electromagnetic wave. B. Light is made of particles called photons. C. The speed of light in vacuum is c ≈ 3x108 m/s. D. The direction of motion of light is ⊥ its E and B fields. E. Light can be produced by accelerating charged particles. All answers are correct! Monday, December 5, 2011 Some Math • • • • We want to find a solution to Maxwell’s Equations in a region far, far away from any sources. Following our usual technique - knowing the answer in a dvance - I propose the following as a possible solution: ￿ E = E0 cos(kz − ω t)ˆ x ￿ = B0 cos(kz − ω t)ˆ B y Recall (Physics 124) that this is a transverse traveling wave • • • Fields are in x and y directions. Movement is to the +z direction. Wave speed is ω/k. Let’s check that this solution is consistent with Maxwell’s Equations. Monday, December 5, 2011 Math I • Divergence of E = 0? (No charges.) ￿￿ ∇·E = • • ￿ ￿ ∂ ∂ ∂ x+ ˆ y+ ˆ z · E0 cos(kz − ω t)ˆ = 0 ˆ x ∂x ∂y ∂z Yes! The derivatives are zero except for the d/dz, and the dot products are 0 except for the x-component, so the divergence is 0. Similarly, the divergence of the B-field is zero. Monday, December 5, 2011 Math II • Curl of E = -dB/dt? ￿ ￿ ∇×E = ￿ ￿ ∂B ￿ ￿ ∇×E =− ∂t ￿ ∂ ∂ ∂ x+ ˆ y+ ˆ z × E0 cos(kz − ω t)ˆ ˆ x ∂x ∂y ∂z ￿ ￿ ∇ × E = −kE0 sin(kz − ω t)ˆ y ￿ ∂B ∂ − = − B0 cos(kz − ω t)ˆ = −ω B0 sin(kz − ω t)ˆ y y ∂t ∂t • Yes, as long as kE0 = ωB0! Monday, December 5, 2011 Math III • Curl of B = μ0ε0dE/dt? (No currents.) ￿ ￿ ∇×B = ￿ ￿ ∂E ￿ ￿ ∇ × B = µ0 ￿0 ∂t ￿ ∂ ∂ ∂ x+ ˆ y+ ˆ z × B0 cos(kz − ω t)ˆ ˆ y ∂x ∂y ∂z ￿ ￿ ∇ × B = kB0 sin(kz − ω t)ˆ x ￿ ∂E ∂ µ0 ￿0 = µ0 ￿0 E0 cos(kz − ω t)ˆ = µ0 ￿0 ω E0 sin(kz − ω t)ˆ x x ∂t ∂t • Yes, as long as kB0 = μ0ε0ωE0! Monday, December 5, 2011 Math IV: Summary • The proposed solution to Maxwell’s Equation far away from charges is a transverse E+B traveling wave: ￿ E = E0 cos(kz − ω t)ˆ B = B0 cos(kz − ω t)ˆ x￿ y • • • • The divergences of E & B are 0. The curl of E = -dB/dt as long as kE0 = ωB0! • Denoting the traveling wave velocity as c = ω/k, E0 = cB0. • • Using E0 = cB0: kB0 = μ0ε0ωcB0 → 1 = μ0ε0c2 → c2 = 1/μ0ε0. Since there are no currents, the curl of B = μ0ε0dE/dt holds as long as kB0 = μ0ε0ωE0! c = 1/√(μ0ε0) = 1/√(4πx10-7 x 8.854x10-12) = 3.0x108 m/s Conclusion: one solution to M.E. is a plane wave of E & B fields traveling at speed c = 1/√(μ0ε0). This is light. Monday, December 5, 2011 Key points about light: • • • • • A transverse E+B traveling wave - there is no oscillating medium like for a mechanical wave. Definite ratio between E & B: E0 = cB0. Fixed speed: c = 1/√(μ0ε0) = 3.0x108 m/s. E ⊥ B, E & B ⊥ direction of motion. While our “demonstration” has been for plane waves, other solutions are possible that have these features. Monday, December 5, 2011 A traveling EM Wave my x = their y y = their = my z Monday, December 5, 2011 Backward Wave iclicker What would change in the picture if the wave moved to the left - “kz+ωt” instead of “kz-ωt”? There may be more than one correct answer. A. E would change sign. B. B would change sign C. E and B would change sign. D. k would change sign. E. The wave cannot travel to the left. Monday, December 5, 2011 With a +ωt, in the evaluation of the curls of E and B, there is an additional “-” sign that indicates E and B have o pposite signs. See slides 5 and 6. A traveling EM Wave If we go back to o ur solution to Maxwell’s Equations, and change (kz-ωt) to (kz+ωt) -assume k and ω > 0 -- you will see that for M.E. to hold, one of E0 and B0 has to change sign, to be < 0. Monday, December 5, 2011 How do you generate such a traveling wave? • • • • Monday, December 5, 2011 Consider a line of charges along the y axis, at x = z = 0. Let them oscillate up and down along the y axis. As you look from large x, there w ill be kinks in the electric field in the y direction resulting from the acceleration. With the current moving in the y direction, you see B-field kinks in the z direction. Methods of Generating Light • • • • • Classically: accelerate / oscillate particles along a line, light moves out in a dipole pattern ⊥ the line. • dipole antennas, synchrotron radiation Classical thermodynamics: black body radiation (really quantum mechanical) results from interactions (and accelerations) of moving particles. (Incandescent lights.) Quantum mechanical: atomic electrons moving from highenergy atomic states to lower energy ones emit photons. Also occurs in nuclear and particle systems. (Fluorescent lights, lasers, LEDs) Interaction of charged particles with matter: bremsstrahlung radiation, Cerenkov radiation, transition radiation. Subatomic physics: particle and anti-particle annihilate generating a photon. Monday, December 5, 2011 Methods of Generating Light lots of light in this direction θ θ a⊥ = asinθ a • • a little light in this direction For a dipole antenna, the component of acceleration a you see is asinθ. Expect E-field to vary as sinθ, and power to vary as sin2θ. Monday, December 5, 2011 The EM Spectrum • • • • Why is visible light visible? About peak in solar power generation, and About peak in atmospheric transmission of light Recall from earlier wave physics: λf = c. Monday, December 5, 2011 Light iclicker In vacuum, red light has a wavelength of 700 nm and violet light has a wavelength of 400 nm. That means that in vacuum, red light The speed of light in vacuum is independent of the wavelength. The longer wavelength red light has to have a lower frequency. A. has a higher frequency and moves faster than violet light. B. has a higher frequency and moves slower than violet light. C. has a lower frequency and moves slower than violet light. D. has a higher frequency and the same speed as violet light. E. has a lower frequency and the same speed as violet light. Monday, December 5, 2011 Thursday: More light, last lecture of the term except for the review Some summer job opportunites - internships with Federal government: www.orau.gov/dhsinternships www.orau.org/ornl http://science.energy.gov/wdts/suli/ http://www.fossil.energy.gov/education/lelandfellowships/ http://www.nsf.gov/crssprgm/reu Monday, December 5, 2011 ...
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This note was uploaded on 01/03/2012 for the course PHYSICS 750:227 taught by Professor Ronaldgilman during the Fall '11 term at Rutgers.

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