8.03 Lecture 12. Systems we have learned: Wave equation: (1) String with constant tension and mass per unit length ρ L T v p = ρ L

Transcription

1 8.03 Lecture 1 Systems we have learned: Wave equation: ψ = ψ v p x There are three different kinds of systems discussed in the lecture: (1) String with constant tension and mass per unit length ρ L T v p = ρ L () Spring with spring constant k, length l, and mass per unit length ρ L kl v p = ρ L (3) Organ pipe with room pressure P 0 and air density ρ γp0 v p = ρ This time, we are doing EM (electromagnetic) waves!

2 E = ρ ɛ 0 B = 0 E = B ( B = µ 0 J + ɛ ) E 0 Gauss Law Gauss Law for magnetism Faraday s Law Ampere s Law In the vacuum: ρ = 0 and J = 0 and we get: E = 0 B = 0 E = B B = µ 0 ɛ 0 E Where in the last two equations we see a changing magnetic field generates an electric field and a changing electric field generates a magnetic field. Can you see the EM wave solution from these equations? Maxwell saw it! We need to use this identity: ( A) = ( A) ( ) A Where is the Laplacian operator. In the vacuum: B/ ( E) = 0 ( E) ( ) E Where we have made replacements based on the vacuum Maxwell equations above. examine the left hand side: ( B ) = ( ) B = µ 0 ɛ 0 E = E E = µ 0 ɛ 0 E Let s first

3 Recall And so we have a wave equation!! ( ) x + y + z ( ) x + y + z E = µ 0 ɛ 0 E This equation changed the world! Maxwell is the first one who recognized it because of the term he put in. It was a wave equation with speed equal to the speed of light: v p = c = 1 µ0 ɛ m/s What about the B field? We can do the same exercise: B = µ 0 ɛ 0 B It is very important that the associated magnetic field also satisfies the wave equation. From the Maxwell equation E creates B and B creates E, therefore they can not exist without each other Galileo: speed of light is large 1676 Romer: m/s 179 James Bradley: m/s This means that in vacuum you can excite EM waves! What is oscillating? The field! Before we tackle EM waves, let s review divergence and curl briefly. *Field: Scalar field: every positing in the space gets a number. Temperature is an example. Vector field: Instead of a number or scalar, every point gets a vector. A(x, y, z) = A xˆx + A y ŷ + A z ẑ The electric and magnetic fields are vector fields, e.g.: F = q( E + v B) To understand the structure of vector fields: Divergence (using our definition of from above): A = A x x + A y y + A z z 3

4 The divergence is a measure of how much the vector v spreads out (diverges) from a point: The divergence of this vector field is positive. The divergence of this vector field is zero. Curl: A ˆx ŷ ẑ = x y z = A x A y A z ( Az y A ) ( y Ax ˆx + z z A ) ( z Ay ŷ + x x A ) x ẑ y What exactly does curl mean? It is a measure of how much the vector A curls around a point. This vector field has a large curl. This vector field has no curl. Gauss Theorem (or the Divergence Theorem): V ( ) A dτ = S A da Which allows us to relate the integral of the divergence over the whole volume (RHS) to a -D surface integral (LHS). Stokes Theorem: S ( ) A da = P A dl 4

5 Which allows us to related the surface integral over the curl (LHS) to a line integral integral over a closed path (RHS). Gauss Theorem Stokes theorem *Consider a plane wave solution: [ E = Re E 0 e i(kz ωt)ˆx ] = {E 0 cos(kz ωt), 0, 0} Only in the ˆx direction. Check if it satisfies E = µ 0 ɛ 0 E E x z ˆx = µ Ex 0ɛ 0 ˆx In ˆx direction: E 0 k cos(kz ωt) = µ 0 ɛ 0 ω E 0 cos(kz ωt) *What about B? ω k = 1 = c Condition needed to satisfy the wave equation. µ0 ɛ 0 E = B ˆx ŷ ẑ 0 = x y z = E x E x 0 0 z ŷ E x y ẑ = ke 0 sin(kz ωt)ŷ B = k ω E 0 cos(kz ωt)ŷ = E 0 c cos(kz ωt)ŷ What did we learn from this exercise? 1. E must come with B. In vacuum: the two fields are perpendicular and they are in phase. If k is the direction of propagation then B = 1 c ˆk E The amplitude of the magnetic field is equal to the amplitude of the electric field divided by the speed of light.. The EM wave is non-dispersive, meaning that the speed of the wave c is independent of the wave number k: ω k = c = 1 µ0 ɛ 0 5

6 3. The direction of the propagating EM wave is E B In general a propagating EM wave can be written as: [ E(r, ] t) = Re E0 e i( k r ωt+φ) Where E 0 E 0x ˆx + E 0y ŷ + E 0z ẑ, r xˆx + yŷ + zẑ and ω ck Given this electric field, we can get the magnetic field: B(r, t) = 1 c ˆk E Example: k = π λ { ˆx + ŷ E 0 = E 0 ˆx + E 0 ŷ } k r = π λ (x + y) ( E(x, y, z) = E 0 ˆx + ŷ ) ( ) π cos (x + y) ωt λ 6

7 B = 1 c ˆk E B(x, y, z) = E ( ) 0 π c ẑ cos (x + y) ωt λ If there is no other material, this EM wave will travel forever... Now let s put something into the game: A perfect conductor A busy world inside this system! All the little charges are moving around without cost of energy (there is no dissipation). Incident wave: E I = E 0 cos(kz ωt)ˆx B I = E 0 c cos(kz ωt)ŷ To satisfy the boundary conditions E = 0 at z = 0 we need a reflected wave! E R = E 0 cos( kz ωt)ˆx B R = E 0 cos( kz ωt)ŷ c 7

8 E = E I + E R = E 0 (cos(kz ωt) cos( kz ωt)) ˆx = E 0 sin(ωt) sin(kz)ˆx B = B I + B R = E 0 (cos(kz ωt) + cos( kz ωt))ŷ c = E 0 c cos(ωt) cos(kz)ŷ Energy density? U E = 1 ɛ be = ɛ 0 E 0 sin ωt sin kz U B = 1 µ 0 B = ɛ 0 E 0 cos ωt cos kz Poynting vector: directional energy flux, or the rate of energy transfer per unit area: S = E B µ 0 = 1 µ 0 E x B y ẑ = E 0 sin ωt cos ωt sin kz cos kzẑ µ 0 c This is how a microwave oven works! = E 0 sin(ωt) sin(kz)ẑ 4µ 0 c *The EM waves are bounced around inside the oven *EM waves increase the vibration of the molecules in the oven and increase the temperature of the food. 8

9 MIT OpenCourseWare SC Physics III: Vibrations and Waves Fall 016 For information about citing these materials or our Terms of Use, visit: