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ΑΒΓΔΕΖΗΘΙΚΛΜΝΞΟΠΡΣΤΥΦΧΨΩ ∇∙E=0 Homework Assignment #8 due Friday 10/24 αβγδεζηθικλμνξοπρςστυφχψω ∇ × E = −∂B /∂t 1. Problem 16.1 +<=>|~±×÷′″⁄⁒←↑→↓⇒⇔ ∂Δ∇∈∏∑ 2. Problem 16.2 ∇∙B=0 (ε1 3. e1 + Problem ε 2 e2 ) 16.3 ∇ × B = + μ0ε0 ∂E /∂t 4. Problem 16.10 ∓∔⁄∗∘∙√∞∫∮∴ 5. Problem 16.13 ≂≃≄≅≆≠≡≪≫≤≥ 6. (a) Explain briefly (1 paragraph with one figure) how to produce linearly polarized electromagnetic waves. (b) Explain briefly (1 paragraph with one figure) how to e−iωt produce circularly polarized waves. ε Recitation Session Tuesday at 6:30. − Chapter 16 : Waves in vacuum MAXWELL’S EQUATIONS IN VACUUM A linearly polarized electromagnetic plane wave; polarized in the x direction and propagating in the z direction. ∇∙E=0 ∇ × E = −∂B /∂t ∇∙B=0 ∇ × B = + μ0ε0 ∂E /∂t These are infinite planes; the fields E(r,t) and B(r,t) do not depend on x or y on these planes. There are neither charges nor currents in vacuum. x y z Harmonic solutions (i.e., harmonic in time). The problem is to find a complete set of solutions that are harmonic in time. “Harmonic” means that there is a fixed angular frequency ω. We’ll use complex fields to solve the field equations. Then the physical fields are the real parts of the complex fields complex fields physical fields E = Re E B = Re B Do not confuse the complex fields (E,B) and the physical fields (E,B) although we’ll use the same symbols for both (E, B). E(r,t) = E(r) e−iωt B(r,t) = B(r) e−iωt ∂ /∂t is the same as multiplication by −iω so ∇ ∙ E(r) = 0 ∇ × E(r) = +iω B(r) ∇ ∙ B(r) = 0 ∇ × B(r) = −iω μ0ε0 E(r) The equations for E(r) and B(r) are linear. Therefore the natural solutions are exponentials. {{ dy /dx = y implies y(x) = C exp(x) }} ∇ × (∇ × E ) = − ∇ 2 E = i ω ∇ × B = ω 2 μ 0ε 0 E − ∇ 2 E = (ω2 /c2) E − ∇ 2 E = (ω2 /c2) E Solve by separation of variables E(r) = E0 e i k∙r and B(r) = B0 e i k∙r Solving the Maxwell equations ... ∇∙E=0 Note: ∇ is the same as multiplication by ik. Requires k∙E0 = 0 ∇∙B=0 Requires k∙B0 = 0 ∇ × E = −∂B /∂t Requires k × E0 = ω B0 ∇ × B = + μ0ε0 ∂E /∂t Requires k × B0 = −ω/c2 E0 Result: E(r) = E0 e i k∙r Also, recall Faraday’s law, iω B(r) = i k × E(r) ; thus B(r) = B0 e i k∙r where B0 = k/ω × E0 The transverse plane Both E0 and B0 are perpendicular to k; i.e., they lie in the transverse plane. Polarization What is the direction of E(r,t) (the physical field) as a function of r and t ? We have this complex field E(r,t) = ( ε1 e1 + ε2 e2 ) eiφ(r,t) , where φ(r,t) = k∙r − ωt. Let e1 and e2 be orthogonal basis vectors for the transverse plane. (These are not uniquely determined; but once they have been chosen we’ ll keep them fixed.) Then E0 = ε1 e1 + ε2 e2 B0 = k/ω × (ε1 e1 + ε2 e2 ) = (−ε2 e1 + ε1 e2 )/c Take k to be real (for some applications, k might be complex); also, ω = ck. Now write ε1 = A exp(i δ1) and ε2 = B exp(i δ2) where A, B, δ1 and δ2 are real. The electric field is E(r,t) = Re [ ( ε1 e1 + ε2 e2 ) eiφ(r,t) ] ● Linear Polarization If δ = 0, then this is a linearly polarized wave. The direction of the electric field alternates between +a and −a : ● Circular Polarization If δ = π/2 or −π/2, and A = B, then this is a circularly polarized wave. The direction of the electric field rotates around the wave vector (k), with constant magnitude in the transverse plane. ● Elliptical Polarization This is the general case, with no conditions on A, B, δ1 and δ2 . (Both linear and circular polarization are special cases of elliptical polarization.) The direction of the electric field rotates around k, and the magnitude varies according to the position on an ellipse. ΑΒΓΔΕΖΗΘΙΚΛΜΝΞΟΠΡΣΤΥΦΧΨΩ ∇∙E=0 Homework Assignment #8 due Friday 10/24 αβγδεζηθικλμνξοπρςστυφχψω ∇ × E = −∂B /∂t 1. Problem 16.1 +<=>|~±×÷′″⁄⁒←↑→↓⇒⇔ ∂Δ∇∈∏∑ 2. Problem 16.2 ∇∙B=0 (ε1 3. e1 + Problem ε 2 e2 ) 16.3 ∇ × B = + μ0ε0 ∂E /∂t 4. Problem 16.10 ∓∔⁄∗∘∙√∞∫∮∴ 5. Problem 16.13 ≂≃≄≅≆≠≡≪≫≤≥ 6. (a) Explain briefly (1 paragraph with one figure) how to produce linearly polarized electromagnetic waves. (b) Explain briefly (1 paragraph with one figure) how to e−iωt produce circularly polarized waves. ε Recitation Session Tuesday at 6:30. −