Physics-272Physics-272 Lecture Lecture 21 21
Description of Wave Motion Electromagnetic Waves Maxwell’s Equations Again ! Maxwell’s Equations
James Clerk Maxwell (1831-1879) • Generalized Ampere’s Law • And made equations symmetric: – a changing magnetic field produces an electric field – a changing electric field produces a magnetic field
• Showed that Maxwell’s equations predicted electromagnetic waves and c =1/ √ε0µ0
• Unified electricity and magnetism and light .
Maxwell’s Equations (integral form) Name Equation Description Gauss’ Law for r r Q Charge and electric Electricity ∫ E ⋅ dA = fields ε0 Gauss’ Law for r r Magnetic fields Magnetism ∫ B ⋅ dA = 0 r r Electrical effects dΦB from changing B Faraday’s Law ∫ E ⋅ ld = − dt field Ampere’s Law r Magnetic effects dΦE from current and (modified by ∫ B ⋅ dl = µ0ic + ε0 Maxwell) dt Changing E field
All of electricity and magnetism is summarized by Maxwell’s Equations. On to Waves!! • Note the symmetry now of Maxwell’s Equations in free space, meaning when no charges or currents are present r r r r ∫ E ⋅ dA = 0 ∫ B ⋅ dA = 0 r r dΦ r dΦ ∫ E ⋅ ld = − B B ⋅ dl = µ ε E dt ∫ 0 0 dt • Combining these equations leads to wave equations for E and B, e.g., ∂ ∂ 2E 2 E x= µ ε x ∂ ∂ z20 0 t 2
• Do you remember the wave equation??? 2 2 ∂∂ h1 h h is the variable that is changing 2= 2 2 in space (x) and time ( t). v is the ∂∂ x v t velocity of the wave. Review of Waves from Physics 170 2 2 • The one-dimensional wave equation: ∂∂ h1 h 2= 2 2 has a general solution of the form: ∂∂ x v t
h(x,t) = h1(x − vt ) + h2 (x + vt )
where h1 represents a wave traveling in the +x direction and h2 represents a wave traveling in the -x direction. • A specific solution for harmonic waves traveling in the +x direction is: h λ h(x,t )= A cos (kx − ωt ) A
2π 2π x k = ω π=2 f = λ T A = amplitude ω λλλ = wavelength v = λ f = f = frequency k v = speed k = wave number Clicker problem y • Snapshots of a wave with angular frequency ωωωare shown at 3 times: 0 t = 0 2A – Which of the following expressions describes this wave? (a) y = sin( kx -ωωωt) (b) y = sin( kx +ωωωt) π 0 t = (c) y = cos( kx +ωωωt) 2ω
π 0 t = ω 2B • In what direction is this wave traveling? (a) +x direction (b) -x direction 0 x Clicker problem y • Snapshots of a wave with angular frequency ωωωare shown at 3 times: 0 t = 0 2A – Which of the following expressions describes this wave? (a) y = sin( kx -ωωωt) (b) y = sin( kx +ωωωt) π 0 t = (c) y = cos( kx +ωωωt) 2ω • The t =0 snapshot ⇒ at t =0 , y = sin kx • At t =πππ/2ωωωand x=0 , (a) ⇒ y = sin (-πππ/2) = -1 • At t =πππ/2ωωωand x =0 , (b) ⇒ y = sin (+ πππ/2) = +1 π 0 t = ω
0 x Clicker problem y • Snapshots of a wave with angular frequency ωωωare shown at 3 times: 0 t = 0 2A – Which of the following expressions describes this wave? (a) y = sin( kx -ωωωt) (b) y = sin( kx +ωωωt) π 0 t = (c) y = cos( kx +ωωωt) 2ω
π 0 t = ω 2B • In what direction is this wave traveling? (a) + x direction (b) -x direction 0 x • We claim this wave moves in the -x direction. • The orange dot marks a point of constant phase. • It clearly moves in the -x direction as time increases!! Velocity of Electromagnetic Waves
E 2 2 • The wave equation for x ∂ ∂ E E x= µ ε x (Maxwell derived it in ~1865!): ∂ ∂ z20 0 t 2
• Comparing to the general wave equation: ∂ ∂ 2h1 2 h = ∂∂ x2 v 2 t 2 we have the velocity of electromagnetic waves in free space: 1 v= =×3.00 108 m/s ≡ c µ ε0 0 • This value is essentially identical to the speed of light measured by Foucault in 1860! – Maxwell identified light as an electromagnetic wave. E & B in Electromagnetic Wave • Plane Harmonic Wave: E = E sin( kz − ωt) x 0 where ω = kc B y = B0 sin( kz − ωt)
x
z y
By is in phase with Ex B0= E 0 / c The direction of propagation s ˆis given by the cross product sˆ = eˆ × bˆ where ( eˆ , b ˆ ) are the unit vectors in the (E,B) directions.
Nothing special about (Ex, By); e.g., could have (Ey, -Bx) Clicker Problem • Suppose the electric fieldr in an e-m wave is given by:
E=− yEˆ 0 cos( −+ kzω t ) 3A In what direction is this wave traveling ? (a) + z direction (b) - z direction
3B Which of the following expressions describes the magnetic field associated with this wave?
(a) Bx = -(Eo/c) cos( kz +ω t ) ω (b) Bx = +( Eo/c) cos( kz -ω t ) ω (c) Bx = +( Eo/c) sin( kz -ω t ) ω Clicker Problem
• Suppose the electric rfield in an e-m wave is given by:
E=− yEˆ 0 cos( −+ kzω t ) 3A – In what direction is this wave traveling ? (a) + z direction (b) - z direction ω • To determine the direction, set phase = 0 : − kz +ωt = 0 z = + t k • Therefore wave moves in + z direction! • Another way: Relative signs opposite means + direction Clicker Problem
• Suppose the electric rfield in an e-m wave is given by:
E=− yEˆ 0 cos( −+ kzω t ) 3A – In what direction is this wave traveling ? (a) + z direction (b) - z direction
3B • Which of the following expressions describes the magnetic field associated with this wave?
(a) Bx = -(Eo/c) cos( kz +ω t ) ω (b) Bx = +( Eo/c) cos( kz -ω t ) ω) (c) Bx = +( Eo/c) sin( kz -ω t ) ω • B is in phase with E and has direction determined from: bˆ=sˆ×eˆ
• At t=0 , z=0 , Ey = -Eo ˆ ˆ ˆ ˆ • Therefore at t=0 , z=0 , b=sˆ×eˆ =k×(−j) =i r E B= + iˆ 0 cos( kz − ω t ) c 10) An electromagnetic wave is travelling along the x-axis, with its electric field oscillating along the y-axis. In what direction does the magnetic field oscillate?
a) along the x-axis b) along the z-axis c) along the y-axis Note: the direction of propagation sˆ is given by the cross product sˆ = eˆ × bˆ where ( eˆ , b ˆ ) are the unit vectors in the (E,B) directions. In this case, direction of s ˆ is x and direction of e ˆ is y xˆ= y ˆ × zˆ
The fields must be perpendicular to each other and to the direction of propagation.
Leads to the possibility of “polarization” of light to be discussed in next chapter. Electromagnetic (EM) Waves in free space
Maxwell’s eqns predict electric & magnetic fields can propagate in vacuum. Examples of EM waves include; radio/TV waves, light, x-rays, and microwaves. ∂ ∂ 2E 2 E x= µ ε x ∂ ∂ z20 0 t 2 • A specific solution for harmonic waves traveling in the +x direction is: h λ h(x,t )= A cos (kx − ωt ) A 2π 2π x k = ω π=2 f = λ T A = amplitude ω λλλ = wavelength v = λ f = f = frequency k v = speed k = wave number No – moving in the minus z direction No – has Ey rather than Ex
Most of the EM spectrum is invisible to our eyes
X-rays are deeply penetrating because of their short wavelengths
The earth’s atmosphere blocks out much of the EM spectrum (e.g. much of UV and X-ray band) Students must become familiar with the wavelengths of common types of EM radiation Wavelength is equal to the speed of light divided by the frequency.
c 300,000,000 1 λ = = = f 900,000,000 3 y r x ˆ An electromagnetic wave is described by: E= jE0 cos( kz − ω t ) where ˆ j is the unit vector in the +y direction. z
Which of the following graphs represents the z-dependence of Bx at t = 0?
X X
(A) (B) (C) (D)
E and B are “in phase” (or 180 o out of phase) r ˆ E= jE0 cos( kz − ω t ) Wave moves in +z direction y r r E× B points in direction of propagation E x r ˆ B= − iB0 cos( kz − ω t ) B z y x r iˆ+ ˆj An electromagnetic wave is described by: E= Ecos( kz + ω t ) 2 0 z What is the form of B for this wave? r iˆ+ ˆj r −iˆ − ˆj B=( E / c )cos( kz + ω t ) A) B=( E0 / c )cos( kz + ω t ) B) 0 2 2
r iˆ− ˆj r −iˆ − ˆj B=( E / c )cos( kz + ω t ) B=( E0 / c )cos( kz + ω t ) D) 0 C) 2 2
r iˆ+ ˆj E= Ecos( kz + ω t ) Wave moves in –z direction 2 0 y E +z points out of screen -z points into screen x
r r B E× B points in negative z-direction B must be perp to E Plane Waves
• For any given value of z, the magnitude of the electric field is uniform everywhere in the x-y plane with that z value.
x
z
y Shown is an EM wave at an instant in time. Points A, B, and C lie in the same x-y plane.
3) Compare the magnitudes of the electric field at points A and B.
a) Ea < E b
b) Ea = E b
c) Ea > E b
4) Compare the magnitudes of the electric field at points A and C.
a) Ea < Ec
b) Ea = Ec
c) Ea > Ec Students said … x
z
Magnitude of field is determined only by value of z (and t) !!! y
Right: • for any given value of z, the magnitude of the electric field is uniform everywhere in the x-y plane with that z value
Wrong: • The point A is where the electric field and magnetic field are zero. So the points c and b have a greater field. (a common misconception) EM wave; energy momentum, angular momentum Previously, we demonstrated the energy density existed in E fields in capacitors and in B fields in inductors. We can sum these energies,
1 2 1 2 u = ε 0 E + B 2 2µ0
Since, E=cB and c = /1 µ 0 ε 0 then u in terms of E,
1 1 E 2 1 1 u = ε E2 + µ = ε E2 + ε E2 = ε E2 2 0 2 0 c 2 0 2 0 0
=> EM waves have energy and can transport energy at speed c
Almost all energy resources on the earth come directly or indirectly via sunlight. What doesn’t? EM wave; energy flow & Poynting Vector ct Suppose we have transverse E and B fields moving A at velocity c, to the right. If the energy density is u = ε E2 and the field is moving through area 0 A at velocity c covers volume Act , that has energy, uAct . Hence the amount of energy flow per unit time per unit surface area is, 1 dU 1 = ()uAct = uc = ε cE 2 A dt At 0 This quantity is called, S. In terms of E&B field magnitudes, 2 S = ε0cE = ε0cEcE / c = ε0cEcB = EB / µ0 We also define the energy flow with direction, called the Poynting vector, r 1 r r , which is in the direction of propagation S = E × B µ0 EM wave; Intensity The intensity, I, is the time average of the magnitude of the Poynting vector and represents the average energy flow per unit area . r 1 r r I = S()x, y, z,t = E()()x, y, z,t × B x, y, z,t µ0 In Y&F, section 32.4, the average S for a sinusoidal plane wave (squared) is worked out, r E B E B I = S()x, y, z,t = max max 1+ cos ()2kx − 2ωt = max max 2µ0 2µ0 The can be rewritten as, r Emax Bmax ε0c 2 ε0 2 I = S()x, y, z,t = = Emax = Bmax 2µ0 2 2c Cell Phone Problem 32.22) A sinusoidal electromagnetic wave emitted by a cell phone has a wavelength of 35.4 cm and an electric field amplitude of 5.4 x 10 -2 V/m at a distance of 250 m from the antenna. (a) What is the magnetic field amplitude? (b) The intensity? (C) The total average power?
a.) E = c B; B= E/c = 5.4 x 10 -2 (V/m)/3 x 10 8 m/s
B = 1.8 x 10 -10 T 1 b.) I = ε cE 2 ; 2 0 1 I = 85.8( ×10 −12 C 2 / Nm 2 )( 3×10 8 m / s)( 4.5 ×10 −2V / m) 2 I = 87.3 ×10 −6W / m 2
c.) Assume isotropic: I=P/A; P=4 πr2 I = 4 π (250m) 2 I P = 3 W Waves Carry Energy Intensity
Intensity = energy delivered per unit time, per unit area 1 dU I = (U = energy) A dt
dU= u ⋅ volume = u Acdt Area = A Length = c dt
I= c u
Sunlight on Earth: I ~ 1000 J/s/m 2 ~ 1 kW/m 2 Comment on the Poynting Vector Just another way to keep track of all this - It’s the same thing as I but it has a direction Light has Momentum! If it has energy and it’s moving, then it also has momentum:
Analogy from mechanics: p2 E = 2m dE2 pdp mv dp = = = vF dt2 mdt m dt
dE dU For E-M waves: → = IA v→ c dt dt IA= cF
I I F P = = pressure c c A Radiation pressure 3 G cell phone networks in the US use 1710-1756 MHz frequencies.
What is the corresponding wavelength range ?
3× 108m / s /(1710 × 10 6 Hz ) = 0.175 m
3× 108m / s /(1756 × 10 6 Hz ) = 0.170 m