Maxwell s Equations & Electromagnetic Waves. The Equations So Far...

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Magnus Long
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1 Maxwell s Equations & Electromagnetic Waves Maxwell s equations contain the wave equation Velocity of electromagnetic waves c = x 1 8 m/s Relationship between E and B in an EM wave Energy in EM waves: the Poynting vector x y z The Equations So Far... Gauss Law for E Fields Gauss Law for B Fields E n da = 1 Q S ε inside B n da = S Faraday s Law " E d l = d B n da S Ampere s Law B d l " = I 4 1

2 A Problem with Ampere s Law onsider a wire and a capacitor is a loop Time dependent case urrent flows in the wire as the capacitor charges up or down Apply Ampère's law to S 1 Intersects the wire B d l " = I S1 Static case apacitor acts like a break No current flows & no magnetic field is generated Ampère's law works Apply Ampère's law to S 2 Passes between the plates of the capacitor Flux = B d l " = I S2 = => Ampère's law is ambiguous Maxwell s Displacement urrent, I d Maxwell replaced I by (I + I d ) dφ I d = ε e d = ε " B d l = I +I d ( ) E d A Generalized Ampère s law B d l d " = I + E d A A changing electric flux dφ e generates a magnetic field A changing magnetic flux dφ m generates an electric field 2

Maxwell s Displacement urrent, I d Maxwell replaced I by (I + I d ) dφ I d = ε e d = ε " B d l = I +I d E d A Generalized Ampère s law B d l d " = I + Faraday s law and the generalized Ampère s law

3 Maxwell s Displacement urrent, I d Maxwell replaced I by (I + I d ) dφ I d = ε e d = ε " B d l = I +I d E d A Generalized Ampère s law B d l d " = I + Faraday s law and the generalized Ampère s law imply existence of electromagnetic waves ( ) E d A alculating Displacement urrent onsider a parallel plate capacitor with circular plates of radius R. If charge is flowing onto one plate and off the other plate at a rate I = dq/ what is I d? The displacement current is not an actual current Represents magnetic fields generated by timevarying electric fields φ e = EA = σ A = Q / A A = Q dφ I d = ε e d Q = ε = dq 8 3

alculating the B Field - Example " B d l = I +I d ( ) B 2πr = dφ E φ E = EA B = A de de V / m V Assume : A = 1m 2 =14 = 1 5.1s m s B = A de N 2 V B = 4π1 7 A 2 8.

4 alculating the B Field - Example " B d l = I +I d ( ) B 2πr = dφ E φ E = EA B = A de de V / m V Assume : A = 1m 2 =14 = 1 5.1s m s B = A de N 2 V B = 4π1 7 A N m 2 1m2 1 5 m s B 1 12 T => Ampere s Law works well because B is very 9 small Gauss law Maxwell s Equations (1865) Q inside E n da = 1 S Gauss law for magnetism B n da = S Faraday s law # E d l = d B n da = B n da S S Ampère-Maxwell law # B d l = I +I d ( ) = I + d E n da S In SI units S E = I + ε n da 4

Maxwell s Equations Note the symmetry of Maxwell s Equations in free space If no charges or currents are present E n da = S # E d l = dφ B B n da = S B d dφ # l = ε E an predict the existence of

5 Maxwell s Equations Note the symmetry of Maxwell s Equations in free space If no charges or currents are present E n da = S # E d l = dφ B B n da = S B d dφ # l = ε E an predict the existence of electromagnetic waves Wave equation contained in these equations 2 h x = 1 2 h 2 v 2 2 h is changing in space (x) and time (t) v is the wave velocity Review of Waves from Mechanics 1-D wave equation 2 h x = 1 2 h 2 v 2 2 General solution of the form: h(x,t) = h 1 (x vt) + h 2 (x +vt) h 1 = wave traveling in the +x direction h 2 = wave traveling in the -x direction A solution for waves traveling in the +x direction is: 5

6 Wave on a String Wave Examples F d 2 y dx 2 = µ d 2 y 2 v = F µ Wave speed of a guitar string Proportional to frequency of fundamental Electromagnetic wave What is being displaced? E and M fields Rewrite Maxwell s equations as Wave velocity, v, related to and μ 2 E x z = 1 2 E x 2 v Four Step Plane Wave Derivation Step 1 Assume a plane wave propagating in z (i.e. E, B not functions of x or y) Example: E x = E sin(kz ωt) Step 2 Apply Faraday s Law to infinitesimal loop in x-z plane " E i dl = dφ B x E x E x Δx ( E x ( z 2 ) E x ( z 1 ))Δx = ΔxΔz B y E x z ΔzΔx = ΔxΔz B y y B y z 1 z 2 ΔZ E x z = B y z 6

7 Four Step Plane Wave Derivation Step 3 Apply Ampère s law to infinitesimal loop in y-z plane ( B y ( z 1 ) B y ( z 2 ))Δy = µ o ΔyΔz E x dφ " B i dl = ε E B y z = µ ε E x B y z ΔzΔy = µ ε ΔyΔz E x Step 4 Use results from steps 2 and 3 to eliminate B 2 B y z = µ ε 2 E x 2 E x z = B y 2 E x z = µ ε 2 E x B y z = 2 E x z 2 y B y x E x Z z 1 z 2 B y y z Velocity of Electromagnetic Waves We derived the wave equation for E x : 2 E x z = µ ε 2 E x 2 2 The velocity of electromagnetic waves in free space is: v = 1 Putting in the measured values for μ &, we get: v = m / s c Identical to the measured speed of light => Light is an electromagnetic wave 7

Question If the magnetic field of a light wave oscillates parallel to a y axis and is given by B y = B m sin(kz-ωt) in what direction does the wave travel? A. -y B. -z. y D.

8 Question If the magnetic field of a light wave oscillates parallel to a y axis and is given by B y = B m sin(kz-ωt) in what direction does the wave travel? A. -y B. -z. y D. z E. -x 19 Electromagnetic Spectrum ~185: Only known forms of electromagnetic waves Infrared, visible, and ultraviolet light Visible light (human eye) Frequency f ~ 1 15 Hz 8

9 Electromagnetic Spectrum 22 Sunscreen Absorbs UV px-UV_and_Vis_Sunscreen.jpg 9

What a Bee Sees Evening primrose in two different wavelenghts Upper = visible Lower = UV Honey guides visible Guide bees to nectar 24 How is B Related to E?

10 What a Bee Sees Evening primrose in two different wavelenghts Upper = visible Lower = UV Honey guides visible Guide bees to nectar 24 How is B Related to E? Derived wave eqn for E x 2 E x z = 1 2 E x 2 c 2 2 ould have derived for B y 2 B y z = 1 2 B y 2 c 2 2 How are Ex and B y related in phase and magnitude? onsider the harmonic solution E x = E sin(kz ωt) ω = kc Result from step 2 (Faraday s law) B y = E x z = ke cos(kz ωt) B y = ke cos(kz ωt) = k E ω sin(kz ωt) B y is in phase with E x B = E / c 1

11 E & B in Electromagnetic Waves Plane Wave: E x = E sin(kz ωt) B y = B sin(kz ωt) where: ω = kc E = cb x z y Direction of propagation given by the cross product ŝ = ê ˆb ( ) ê, ˆb are the unit vectors in the (E,B) directions Nothing special about (E x,b y ) (could also have E y,-b x ) Note cyclical relation ê ˆb = ŝ ˆb ŝ = ê ŝ ê = ˆb Energy in Electromagnetic Waves Electromagnetic waves contain energy Energy density stored in E and B fields u E = 1 2 E 2 u B = 1 2 In an EM wave, B = E/c Total energy density in an EM wave = u u = u E +u B = E 2 = B2 B 2 ub = 1 E 2 = 1 2 c 2 2 ε E 2 = u E 11

12 Energy in Electromagnetic Waves Intensity of a wave Average power (P av =U av /Δt) transmitted per unit area = Average energy density x wave velocity I = c E 2 = E 2 c For ease in calculation define Z as Z = c = 377Ω I = E 2 377Ω The Poynting Vector The direction of the propagation of an electromagnetic wave is given by: ŝ = ê ˆb The energy transported by the wave defined by the Poynting vector S as: E B S S has the direction of propagation of the wave The magnitude of S is directly related to the energy being transported by the wave: S = EB = E 2 c = E 2 Z = E 2 377Ω Intensity for harmonic waves is given by S = E 2 = E 2 sin 2 (kz ωt) = 1 Z Z 2 E 2 Z S = E 2 377Ω = I E rms = E 2 B rms = B 2 12

99792458 1 8 m/s (exact) z x S 3 Summary of Electromagnetic Radiation ombined Faraday s Law and Ampere s Law Time-varying B-field

13 1. E B haracteristics 2. If E = E sin(kx ωt), B = B sin(kx ωt) 3. The Poynting vector S E B = is in the direction of propagation. 4. E and B are both S 5. c = 1 6. E = cb m/s (exact) z x S 3 Summary of Electromagnetic Radiation ombined Faraday s Law and Ampere s Law Time-varying B-field induces E-field x Time-varying E-field induces B-field 2 E x z 2 = µ o ε o 2 E x 2 y z E-field and B-field are perpendicular Energy density u = ε o E 2 = B 2 / µ o Poynting Vector describes power flow E B S = units: watts/m 2 ê ˆb = ŝ B E S 13

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Maxwell Equations: Electromagnetic Waves

Maxwell Equations: Electromagnetic Waves Maxwell s Equations contain the wave equation The velocity of electromagnetic waves: c = 2.99792458 x 10 8 m/s The relationship between E and B in an EM wave Energy