Waves & Oscillations
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(2) Introduction to Optics • If we can identify any system that is described by the wave equation, then we know everything! 1 = • Except that… – – – –. The geometry might be complicated Boundary conditions might not be simple Initial conditions might not be known The solution is the superposition of incident waves and all possible reflected waves. • So we develop new ways of looking at this problem and call it optics..
(3) Electromagnetism • Geometric optics overlooks the wave nature of light. – Light inconsistent with longitudinal waves in an ethereal medium – Still an excellent approximation when feature sizes are large compared with the wavelength of light. • But geometric optics could not explain – Polarization – Diffraction – Interference. • A unified picture was provided by Maxwell c. 1864.
(4) Forces on Charges • Coulomb’s law of electrostatic force: $#. . Charles-Augustin de Coulomb (1736 - 1806). $. • The magnitude of the attractive/repulsive force is where. = . . ̂. 1 = = 8.99 × 10 ∙ ∙ 4. and therefore. !. = 8.85 × 10!# ∙ !# ∙ ! (This constant is called the “permittivity of free space”).
(5) Electric Field • An electric charge changes the properties of the space around it. – It is the source of an “electric field”. – It could be defined as the “force per unit charge”:. # . = 5% %= ̂. &'() – Quantum field theory provides a deeper description…. • Gauss’s Law:. 1. $012034 *, + ∙ % -. = /. Electric “flux” through surface S.
(6) Magnetic Field • Moving charges (ie, electric current) produce a magnetic field: 7 8 -ℓ × ̂ * 6= 4 A moving charge in a magnetic field experiences a force:. = 5 × 6. Lorentz force law:. = 5% + 5 × 6.
(7) Gauss’s Law for Magnetism • Electric charges produce electric fields: $012034 , ∙ % -. = *+ /. • But there are no “magnetic charges”:. 2. , ∙ 6 -. = *+ /. 0. Magnetic “flux” through surface S.
(8) Ampere’s Law • An electric current produces a magnetic field that curls around it:. ;. ;. (Close, but not quite the whole story).
(9) Faraday’s Law of Magnetic Induction • Magnetic flux: <= = >/ 6 ∙ +? -. • Faraday observed that a changing magnetic flux through a wire loop induced a current – It transferred energy to the charge carriers in the wire -<= ℇ = − - B % ∙ -ℓ = − * +? ∙ 6-. - / ;. 3.
(10) The Problem with Ampere’s Law Current through C# is 8 Current through C is 0!. The current through C is zero, but the electric flux is not zero. The electric flux changes as charge flows onto the capacitor..
(11) Maxwell’s Displacement Current • We can think of the changing electric flux through C as if it were a current: -<4 83 = = * % ∙ +? -. - - / B D ∙ Eℓ = GH I + IE F. E , EQ = GH J + GH KH * M ∙ N EL OP. 4.
(12) Maxwell’s Equations (1864) RSNTSEU , ∙ M EQ = B N KH O , ∙ D EQ = H B N O. EVW B M ∙ Eℓ = − EL F. EVU B D ∙ Eℓ = GH I + GH XH EL F. 1 2 3 4.
(13) Maxwell’s Equations in Free Space In “free space” where there are no electric charges or sources of current, Maxwell’s equations are quite symmetric: , ∙ M EQ = H B N O. , ∙ D EQ = H B N O. EVW B M ∙ Eℓ = − EL F EVU B D ∙ Eℓ = GH XH EL F.
(14) Maxwell’s Equations in Free Space EVW B M ∙ Eℓ = − EL F EVU B D ∙ Eℓ = GH XH EL F. A changing magnetic flux induces an electric field. A changing electric flux induces a magnetic field. Will this process continue indefinitely?.
(15) Light is an Electromagnetic Wave • Faraday’s Law:. ∮; % ∙ -ℓ =. 3Z[ − 3\. =−. ]^_ ]\. ∆a∆b. B % ∙ -ℓ = %c b Δa − %c (b# )Δa ;. ≈. x. ]hi ∆b∆a ]j. Ex z1. y. By. Δa. Ex Δb. z2. z.
(16) Light is an Electromagnetic Wave • Ampere’s law:. ∮; 6 ∙ -ℓ =. 3Zk 7 3\. =. ]hi 7 ∆l∆b ]\. B 6 ∙ -ℓ = 6m b# Δl − 6m (b )Δl ;. ≈−. ]^_ ]j. x. ∆b∆l. Ex. ∆b z1 z2. y. By. By. ∆l. z.
(17) Putting these together… 6m %c =− b 6m %c − = 7 b Differentiate the first with respect to b: 6m %c =− b b Differentiate the second with respect to : 6m %c − = 7 b . nP M o nP Mo = GH KH P np nLP.
(18) Velocity of Electromagnetic Waves l 1 l = a nP M o nP M o = GH KH P np nLP Speed of wave propagation is 1 = 7 1 = 4 × 10!q /. 8.854 × 10!# . = P. sst × uHt v/w (speed of light). / . ∙ .
(19) Light is an Electromagnetic Wave • Maxwell showed that these equations contain the wave equation: 1 l l = a where l a, is a function of position and time. • General solution: l a, = y# a − + y (a + ) • Harmonic solutions: l a, = l sin a ± ~ • Wavelength, = 2/ and frequency y = ~/2. • Velocity, = ⁄y = ~⁄ ..
(20) The electromagnetic spectrum • In 1850, the only known forms electromagnetic waves were ultraviolet, visible and infrared. • The human eye is only sensitive to a very narrow range of wavelengths:.
(21) Discovery of Radio Waves.
(22) The Electromagnetic Spectrum.
(23) Electromagnetic Waves nP Mo nP Mo = GH K H P np nLP • A solution is %c b, = % sin b − ~ where ~ = = 2/ • What is the magnetic field? 6m %c =− = − % cos b − ~ b. 6m a, = % sin a − ~ ~.
(24) Electromagnetic Waves ax. bz ly. • %, 6 and are mutually perpendicular. • In general, the direction is ̂ = % × 6.
(25) Energy in Electromagnetic Waves Energy stored in electric and magnetic fields: 1 1 4 = % = = 6 2 27. For an electromagnetic wave, 6 = % ⁄ = % 7 1 1 = = 6 = % = 4 27 2 The total energy density is = = + 4 = % .
(26) Intensity of Electromagnetic Waves • Intensity is defined as the average power transmitted per unit area. Intensity = Energy density × wave velocity % 8 = % = 7 7 = 377 Ω ≡ (Impedance of free space).
(27) Poynting Vector • We can construct a vector from the intensity and the direction ̂ = % × 6: %×6 C = 7 % C = =8 • This represents the flow of power in the direction ̂ • Average electric field: %=2 = % / 2 % C = 2 • Units: Watts/m2.
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