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UIUC Physics 436 EM Fields & Sources II Fall Semester, 2015 Lect. Notes 5 Prof. Steven Errede LECTURE NOTES 5 ELECTROMAGNETIC WAVES IN VACUUM THE WAVE EQUATION(S) FOR E AND B In regions of free space (i.e. the vacuum), where no electric charges, no electric currents and no matter of any kind are present, Maxwell’s equations (in differential form) are: 1) E r , t 0 B r ,t 3) E r , t t 2) B r , t 0 Set of coupled first-order partial differential equations E r , t 1 E r , t 2 4) B r , t o o t c t 2 c 1 o o We can de-couple Maxwell’s equations e.g. by applying the curl operator to equations 3) and 4): 1 E B B 2 E c t t 0 0 1 E 2 E B B 2 B 2 E t c t 1 B 1 E 2 2 E 2 B 2 t c t c t t 1 2 E 1 2 B 2 2 B 2 2 E 2 2 c t c t These are three-dimensional de-coupled wave equations for E and B - note that they have exactly the same structure – both are linear, homogeneous, 2nd order differential equations. Remember that each of the above equations is explicitly dependent on space and time, i.e. E E r , t and B B r , t : or: 2 E r,t 1 2 E r , t 2 c t 2 2 B r,t 1 2 B r , t 2 c t 2 2 1 E r ,t E r ,t 2 0 c t 2 2 1 B r ,t B r ,t 2 0 c t 2 2 2 Thus, Maxwell’s equations implies that empty space – the vacuum {which is not empty, at the microscopic scale} – supports the propagation of {macroscopic} electromagnetic waves, which propagate at the speed of light {in vacuum}: c 1 o o 3 108 m s . © Professor Steven Errede, Department of Physics, University of Illinois at Urbana-Champaign, Illinois 2005-2015. All Rights Reserved. 1 UIUC Physics 436 EM Fields & Sources II Fall Semester, 2015 Lect. Notes 5 Prof. Steven Errede EM waves have associated with them a frequency f and wavelength , related to each other via c f . At the microscopic level, EM waves consist of large numbers of {massless} real photons, each carrying energy E hf hc , linear momentum p h hf c E c and angular momentum z 1 where h = Planck’s constant = 6.626 1034 Joule-sec and h 2 . EM waves can have any frequency/any wavelength – the continuum of EM waves over the frequency region 0 f (c.p.s. or Hertz {aka Hz}), or equivalently, over the wavelength region 0 (m) is known as the electromagnetic spectrum, which has been divided up (for convenience) into eight bands as shown in the figure below (kindly provided by Prof. Louis E. Keiner, of Coastal Carolina University, Conway, SC): 2 © Professor Steven Errede, Department of Physics, University of Illinois at Urbana-Champaign, Illinois 2005-2015. All Rights Reserved. UIUC Physics 436 EM Fields & Sources II Fall Semester, 2015 Lect. Notes 5 Prof. Steven Errede Monochromatic EM Plane Waves: Monochromatic EM plane waves propagating in free space/the vacuum are sinusoidal EM plane waves consisting of a single frequency f , wavelength c f , angular frequency 2 f and wavenumber k 2 . They propagate with speed c f k . In the visible region of the EM spectrum {~380 nm (violet) ≤ λ ≤ ~ 780 nm (red)}, EM light waves (consisting of real photons) of a given frequency / wavelength are perceived by the human eye as having a specific, single color. Hence we call such single-frequency, sinusoidal EM waves mono-chromatic. EM waves that propagate e.g. in the ẑ direction but which additionally have no explicit x- or y-dependence are known as plane waves, because for a given time, t the wave front(s) of the EM wave lie in a plane which is to the ẑ -axis, as shown in the figure below: x̂ ŷ ẑ The planar wavefront associated with a plane EM wave propagating in the kˆ zˆ direction lies in the x-y plane. constant everywhere in (x,y) on this plane. Note that there also exist spherical EM waves – e.g. emitted from a point source, such as an atom, a small antenna or a pinhole aperture – the wavefronts associated with these EM waves are spherical, and thus do not lie in a plane to the direction of propagation of the EM wave: Portion of a spherical wavefront associated with a spherical wave n.b. If the point source is infinitely far away from observer, then a spherical wave → plane wave. In this limit, the radius of curvature RC → ∞). i.e. a spherical surface becomes planar as RC → ∞. A criterion for a {good} approximation of spherical wave as a plane wave is: RC © Professor Steven Errede, Department of Physics, University of Illinois at Urbana-Champaign, Illinois 2005-2015. All Rights Reserved. 3 UIUC Physics 436 EM Fields & Sources II Fall Semester, 2015 Lect. Notes 5 Prof. Steven Errede Monochromatic traveling EM plane waves can be represented by complex E and B fields: E z , t E o ei kz t B z , t Bo ei kz t Propagating in the Propagating in the kˆ zˆ direction kˆ zˆ direction n.b. complex vectors: E o E o xˆ E o ei xˆ Eo ei xˆ e.g. n.b. complex vectors: Bo Bo yˆ Bo ei yˆ Bo ei yˆ e.g. n.b. The real, physical instantaneous time-domain EM fields are related to their corresponding complex time-domain fields via: E r , t Re E r , t B r , t Re B r , t Note that Maxwell’s equations for free space impose additional constraints on E o and Bo . → Not just any E o and/or Bo is acceptable / allowed !!! Since: E 0 Re E 0 B 0 Re B 0 These two relations can only be satisfied r , t if E 0 r , t and B 0 r , t . and: xˆ yˆ zˆ x y z In Cartesian coordinates: Thus: E 0 and B 0 become: i kz t i kz t ˆ ˆ ˆ ˆ ˆ 0 0 x y z E e and x y zˆ Bo e o y z y z x x Now suppose we do allow: E o Eox xˆ Eoy yˆ Eoz zˆ ei Eo ei polarization in xˆ yˆ zˆ 3 D Bo Box xˆ Boy yˆ Boz zˆ ei Bo ei polarization in xˆ yˆ zˆ 3 D Then: 4 yˆ zˆ Eox xˆ Eoy yˆ Eoz zˆ ei ei kz t 0 xˆ y z x yˆ zˆ Box xˆ Boy yˆ Boz zˆ ei ei kz t 0 xˆ y z x © Professor Steven Errede, Department of Physics, University of Illinois at Urbana-Champaign, Illinois 2005-2015. All Rights Reserved. UIUC Physics 436 EM Fields & Sources II Fall Semester, 2015 Lect. Notes 5 Prof. Steven Errede yˆ zˆ Eox xˆ Eoy yˆ Eoz zˆ ei kz t ei 0 xˆ y z x yˆ zˆ Box xˆ Boy yˆ Boz zˆ ei kz t ei 0 xˆ y z x Or: Now: Eox, Eoy, Eoz = Amplitudes (constants) of the electric field components in x, y, z directions respectively. Box, Boy, Boz = Amplitudes (constants) of the magnetic field components in x, y, z directions respectively. We see that: And: ˆ i kz t ei 0 ← has no explicit x-dependence xˆ Eox xe x ˆ i kz t ei 0 ← has no explicit y-dependence yˆ Eoy ye y ˆ i kz t ei 0 ← has no explicit x-dependence xˆ Box xe x ˆ i kz t ei 0 ← has no explicit y-dependence yˆ Boy ye y And: However: Thus: az e ae az z ˆ i kz t ei ikEoz ei kz t ei 0 true iff Eoz 0 !!! zˆ Eoz ze z ˆ i kz t ei ik oz ei kz t ei 0 true iff Boz 0 !!! zˆ Boz ze z Thus, Maxwell’s equations additionally tell us/impose the restriction that an electromagnetic plane wave cannot have any component of E or B to (or anti- to) the propagation direction (in this case here, the kˆ zˆ -direction) Another way of stating this is that an EM plane wave cannot have any longitudinal components of E and B (i.e. components of E and B lying along the propagation direction). Thus, Maxwell’s equations additionally tell us that an EM plane wave is a purely transverse wave (at least while it is propagating in free space) – i.e. the components of E and B must be to propagation direction. The plane of polarization of an EM plane wave is defined (by convention) to be parallel to E . © Professor Steven Errede, Department of Physics, University of Illinois at Urbana-Champaign, Illinois 2005-2015. All Rights Reserved. 5 UIUC Physics 436 EM Fields & Sources II Fall Semester, 2015 Lect. Notes 5 Prof. Steven Errede Furthermore: Maxwell’s equations impose yet another restriction on the allowed form of E and B for an EM wave: 1 E B E B 2 and/or: c t t B 1 E Re E Re Re B Re 2 t c t B E t Can only be satisfied r , t iff: and/or: 1 E B 2 c t Thus: 0 0 E z E y Ex E y ˆ E x z x z y B 0 0 0 B y E y E x B x B z yˆ x y zˆ t xˆ t yˆ t ẑ 0 0 0 0 0 B x B y B y B x B z B y 1 E x 1 E y 1 E z xˆ yˆ x y zˆ c 2 t xˆ c 2 t yˆ c 2 t ẑ y z x z 0 0 i kz t i e With: E E x xˆ E y yˆ E z zˆ Eox xˆ Eoy yˆ Eoz zˆ e 0 0 i kz t i B B x xˆ B y yˆ B z zˆ Box xˆ Boy yˆ Boz zˆ e e i kz t i E e Thus: E x xˆ E y yˆ Eox xˆ Eoy yˆ e B B x xˆ B y yˆ Box xˆ Boy yˆ ei kz t ei 6 E B E B E y xˆ x yˆ x xˆ y yˆ z z t t B B 1 E 1 E y yˆ B y xˆ x yˆ 2 x xˆ 2 c t c t z z Can only be satisfied / can only be true iff the xˆ and yˆ relations are separately / independently satisfied r , t ! © Professor Steven Errede, Department of Physics, University of Illinois at Urbana-Champaign, Illinois 2005-2015. All Rights Reserved. UIUC Physics 436 EM Fields & Sources II i.e. E y B xˆ x xˆ E : z t B y E x yˆ yˆ z t B y 1 xˆ 2 B : c z B 1 x yˆ 2 c z From (1): ikE oy i Box From (2): ikE ox i Boy From (3): ikBoy From (4): ikBox E : (1) (2) B : (3) (4) E y z E x z B x t B y t Lect. Notes 5 Prof. Steven Errede ikEoy i Box (1) ikEox i Boy (2) B E x 1 E 1 xˆ y 2 x ikBoy 2 i Eox (3) c t z c t E y B x 1 E y 1 2 yˆ ikBox 2 i Eoy (4) c t z c t 1 i Eox c2 1 i Eoy c2 Now: c f 2 f 2 Fall Semester, 2015 Eoy Box or: k Eox Boy or: k 1 Boy 2 Eox c k 1 Box 2 Eoy c k k and: 1 Box Eoy c 1 Boy Eox c 1 Boy Eox c 1 Box Eoy c k Box Eoy k Boy Eox k 2 1 k c Maxwell’s equations also have some redundancy encrypted into them! 1 So we really / actually only have two independent relations: Box Eoy c But: xˆ yˆ zˆ Very Useful Table: yˆ zˆ xˆ ẑ xˆ yˆ ẑ yˆ xˆ 1 and Boy Eox c ẑ xˆ yˆ yˆ xˆ zˆ ẑ yˆ xˆ xˆ zˆ yˆ 1 ˆ We can write the above two relations succinctly/compactly with one relation: Bo c k E o © Professor Steven Errede, Department of Physics, University of Illinois at Urbana-Champaign, Illinois 2005-2015. All Rights Reserved. 7 UIUC Physics 436 EM Fields & Sources II Fall Semester, 2015 Lect. Notes 5 Prof. Steven Errede Physically, the mathematical relation Bo 1c kˆ E o tells us that for a monochromatic EM plane wave propagating in free space, E and B are: a.) in phase with each other. b.) mutually perpendicular to each other .and. each is perpendicular to the propagation direction: B kˆ ( kˆ zˆ = propagation direction) The E and B fields associated with this monochromatic plane EM wave are purely transverse { n.b. this is as also required by relativity at the microscopic level – for the extreme relativistic particles – the (massless) real photons traveling at the speed of light c that make up the macroscopic monochromatic plane EM wave.} The purely real/physical amplitudes of E and B are {also} related to each other by: Bo 1c Eo 2 2 with Bo Box Boy 2 2 and Eo Eox Eoy Griffiths Example 9.2: A monochromatic (single-frequency) plane EM wave that is plane polarized/linearly polarized in the x̂ direction, propagating in the kˆ zˆ direction in free space, has: E E xˆ definition of linearly polarized EM wave, polarized in the x̂ direction. B 1c kˆ E 1c zˆ Exˆ 1c E zˆ xˆ 1c Eyˆ yˆ 1 1 ˆ BcE Bo 1c Eo With: B c k E , and: i kz t i kz t i i kz t xˆ Eo e e xˆ Eo e xˆ Then: E z , t E o e i kz t i kz t i i kz t B z , t B e yˆ B e e yˆ B e yˆ o o o ei cos i sin The physical instantaneous electric and magnetic fields are given by the following expressions: imaginary real E z, t Re E z, t Re Eo cos kz t xˆ i Eo sin kz t xˆ E z, t Eo cos kz t xˆ imaginary real B z, t Re B z , t Re Bo cos kz t yˆ i Bo sin kz t yˆ B z , t Bo cos kz t yˆ 1c Eo cos kz t yˆ 8 The physical instantaneous E and B fields are in-phase with each other for a linearly polarized EM plane wave © Professor Steven Errede, Department of Physics, University of Illinois at Urbana-Champaign, Illinois 2005-2015. All Rights Reserved. UIUC Physics 436 EM Fields & Sources II Note that: E B zˆ E zˆ, B zˆ Fall Semester, 2015 Lect. Notes 5 Prof. Steven Errede ( ẑ = direction of propagation of EM wave) Instantaneous Poynting’s vector for a linearly polarized EM plane wave propagating in free space: S z, t S z, t S z, t 1 o 1 o E z, t B z, t 1 o Re E z, t Re B z , t Eo Bo cos 2 kz t xˆ yˆ zˆ 1 o Eo Bo cos 2 kz t zˆ Watts 2 m EM power flows in the direction of propagation of the EM plane wave (here, kˆ zˆ direction) Generalization for Propagation of Monochromatic Plane EM Waves in an Arbitrary Direction Obviously, there is nothing special / profound with regard to plane EM waves propagating in a specific direction in free space / the vacuum. They can propagate in any direction. We can easily generalize the mathematical description for monochromatic plane EM waves traveling in an arbitrary direction as follows: Introduce the notion / concept of a wave vector (or propagation vector) k which points in the direction of propagation, whose magnitude k k . Then the scalar product k r is the appropriate 3-D generalization of kz: k kzˆ with k k and: r xxˆ yyˆ zzˆ with r r x 2 y 2 z 2 1-D: If: Then: k r kzˆ xxˆ yyˆ zzˆ kz 3-D: If: Then: k k x xˆ k y yˆ k z zˆ with k k x2 k y2 k z2 and: k r k x x k y y k z z with r xxˆ yyˆ zzˆ r x2 y2 z 2 © Professor Steven Errede, Department of Physics, University of Illinois at Urbana-Champaign, Illinois 2005-2015. All Rights Reserved. 9 UIUC Physics 436 EM Fields & Sources II Now: Fall Semester, 2015 Prof. Steven Errede k x k cos x k y k cos y where cos x , cos y , cos z = direction cosines w.r.t. k z k cos z (with respect to) the xˆ , yˆ , zˆ -axes respectively cos x kˆ xˆ sin cos cos kˆ yˆ sin sin Direction Cosines: Lect. Notes 5 in spherical-polar coordinates y cos z kˆ zˆ cos cos 2 x cos 2 y cos 2 z Note: sin 2 cos 2 sin 2 sin 2 cos 2 sin 2 cos 2 1 If e.g. k r then: k r kr . We explicitly demonstrate this in spherical polar coordinates: k x k cos x k sin cos For k r : k y k cos y k sin sin x r cos x r sin cos y r cos y r sin sin and: k z k cos z k cos Then: z r cos z r cos k r k x k y k z kx cos x y z x ky cos y kz cos z kr cos 2 x kr cos 2 y kr cos 2 z kr sin 2 cos 2 kr sin 2 sin 2 kr cos 2 kr sin 2 cos 2 sin 2 sin 2 cos 2 kr sin 2 cos 2 sin 2 cos 2 kr sin 2 cos 2 kr Thus, most generally, we can write the E r , t and B r , t -fields as: i k r t E r , t E o e nˆ where: n̂ polarization vector n̂ kˆ i k r t ˆ ˆ ˆ ˆ B r , t Bo e kn i.e. nk 0 because E is transverse i k r t i k r t B r , t 1c kˆ E r , t 1c E o e kˆ nˆ Bo e kˆ nˆ We must have: B r , t E r , t kˆ i.e. E B 0 10 and E kˆ 0 and B kˆ 0 © Professor Steven Errede, Department of Physics, University of Illinois at Urbana-Champaign, Illinois 2005-2015. All Rights Reserved. UIUC Physics 436 EM Fields & Sources II Fall Semester, 2015 Lect. Notes 5 Prof. Steven Errede The Direction of Propagation of a Monochromatic Plane EM Wave: k̂ The Real/Physical (Instantaneous) EM Fields are: E r , t Re E r , t Eo cos k r t nˆ where: n̂ polarization vector E B r , t Re B r , t Bo cos k r t Bo 1c Eo kˆ nˆ in free space Instantaneous Energy, Linear & Angular Momentum in EM Plane Waves (Free Space) Instantaneous Energy Density Associated with an EM Plane Wave (Free Space): 1 1 2 u EM r , t o E 2 r , t B r , t uelect r , t umag r , t o 2 1 1 2 1 2 B r ,t oE2 r ,t where: uelect r , t o E r , t and umag r , t 2 2 o 2 1 2 1 E and 2 o o for EM waves propagating in vacuum/free space 2 c c But: B2 Thus: o o 2 1 1 uEM r , t o E 2 r , t E r , t o E 2 r , t o E 2 r , t o E 2 r , t 2 2 o Or: uEM r , t o E 2 r , t o Eo2 cos 2 k r t Joules 3 m n.b. for EM plane waves propagating in the vacuum: umag r , t uelect r , t and/or: umag r , t uelect r , t 1 © Professor Steven Errede, Department of Physics, University of Illinois at Urbana-Champaign, Illinois 2005-2015. All Rights Reserved. 11 UIUC Physics 436 EM Fields & Sources II Fall Semester, 2015 Lect. Notes 5 Prof. Steven Errede Instantaneous Poynting’s Vector Associated with an EM Plane Wave (Free Space): S r,t E r,t H r ,t 1 o E r ,t B r ,t Re E z , t Re B z, t o 1 Watts 2 m For a linearly polarized monochromatic plane EM plane wave propagating in the vacuum, e.g.: E r , t Eo cos kz t xˆ and: B r , t Bo cos kz t yˆ Then: S r , t o Thus: S r , t o c 1 1 Eo Bo cos 2 kz t zˆ but: Bo 1c Eo for EM plane waves in vacuum. c Eo2 cos 2 kz t zˆ ← multiply RHS by 1 c 1 2 1 E cos 2 kz t zˆ but: 2 o o Hence: S r , t c 2 o c o c o o Thus: S r , t c o But: 2 Eo cos 2 kz t zˆ c o Eo2 cos 2 kz t zˆ u EM r , t o E 2 r , t o Eo2 cos 2 kz t S r , t cu EM r , t zˆ Here, the propagation velocity of EM field energy: vE czˆ Poynting’s Vector = Energy Density * (Energy) Propagation Velocity: S r , t uEM r , t vE Instantaneous Linear Momentum Density Associated with an EM Plane Wave (Free Space): 1 kg EM r , t o o S r , t 2 S r , t 2 c m -sec For linearly polarized monochromatic plane EM waves propagating in the vacuum: 1 1 EM 2 c o Eo2 cos 2 kz t zˆ o Eo2 cos 2 kz t zˆ c c uEM But: u EM r , t o E 2 r , t o Eo2 cos 2 kz t 1 1 kg EM r , t o o S r , t 2 S r , t uEM r , t zˆ 2 c c m -sec 12 © Professor Steven Errede, Department of Physics, University of Illinois at Urbana-Champaign, Illinois 2005-2015. All Rights Reserved. UIUC Physics 436 EM Fields & Sources II Fall Semester, 2015 Lect. Notes 5 Prof. Steven Errede Instantaneous Angular Momentum Density Associated with an EM Plane Wave (Free Space): But: kg EM r , t r EM r , t m-sec 1 1 kg EM r , t o o S r , t 2 S r , t uEM r , t zˆ 2 c c m -sec for an EM plane wave propagating in the ẑ direction: 1 1 kg EM r , t 2 r S r , t uEM r , t r zˆ c c m-sec n.b. depends on the choice of origin The instantaneous EM power flowing into/out of volume v with bounding surface S enclosing volume v (containing EM fields in the volume v) is: U EM t uEM r , t PEM t d S r , t da v S t t (Watts) n.b. closed surface S enclosing volume v. The instantaneous EM power crossing an (imaginary) surface (e.g. a 2-D plane – a window!) is: PEM t S r , t da S The instantaneous total EM energy contained in volume v is: U EM t v uEM r , t d (Joules) The instantaneous total EM linear momentum contained in the volume v is: pEM t EM r , t d v kg-m sec The instantaneous total EM angular momentum contained in the volume v is: LEM t EM r , t d v kg-m 2 sec © Professor Steven Errede, Department of Physics, University of Illinois at Urbana-Champaign, Illinois 2005-2015. All Rights Reserved. 13 UIUC Physics 436 EM Fields & Sources II Fall Semester, 2015 Lect. Notes 5 Prof. Steven Errede 3-D Vector Impedance Associated with an EM Plane Wave (Free Space): Z r ,t E r,t 1 H r ,t (Ohms) = Ohm’s law for EM fields! (n.b. a vector quantity) Analog of: Ohm’s law for AC circuits: Z t V t I t (n.b. a scalar quantity) 2 * Complex form of Ohm’s Law: Z t V t I t V t I t I t What {precisely} is the mathematical meaning of a “generic” reciprocal vector 1 A r , t ??? The magnitude of the reciprocal vector 1 A r , t 1 A r , t is invariant (i.e. cannot change) for arbitrary rotations & translations of the coordinate system. The direction that the reciprocal vector 1 A r , t points in space {at time t} is also invariant for arbitrary rotations & translations of the coordinate system. Note further that inverse unit vectors {such as 1 xˆ , 1 yˆ , 1 zˆ } are meaningless! For a purely real “generic” vector A r , t Ax r , t xˆ Ay r , t yˆ Az r , t zˆ {e.g. expressed in rectangular/Cartesian coordinates}, the mathematical definition of a purely real reciprocal vector 1 A r , t , satisfying all of the above requirements is: 1 1 Aˆ r , t Ar ,t Ar ,t where: ˆ ˆ ˆ ˆA r , t A r , t Ax r , t x Ay r , t y Az r , t z Ar ,t Ar ,t Hence, we also see that: Ax r , t xˆ Ay r , t yˆ Az r , t zˆ Aˆ r , t Ar ,t 1 2 2 Ar ,t Ar ,t Ar ,t Ar ,t Using rectangular / Cartesian coordinates Using rectangular / Cartesian coordinates Note: For the more general case of complex reciprocal vectors 1 A r , t these relations become: 1 1 Aˆ * r , t where: Aˆ * r , t Ar ,t Ar ,t A* r , t A x* r , t xˆ A *y r , t yˆ A z* r , t zˆ Ar ,t Ar ,t Hence, we also see that: Aˆ * r , t A* r , t A x* r , t xˆ A *y r , t yˆ A z* r , t zˆ 1 2 2 A r , t A r , t Ar ,t Ar ,t 14 Using rectangular / Cartesian coordinates © Professor Steven Errede, Department of Physics, University of Illinois at Urbana-Champaign, Illinois 2005-2015. All Rights Reserved. Using rectangular / Cartesian coordinates UIUC Physics 436 EM Fields & Sources II Fall Semester, 2015 Lect. Notes 5 Prof. Steven Errede Thus, e.g. for a linearly polarized monochromatic EM plane wave propagating in the vacuum in the kˆ zˆ direction, with instantaneous/physical purely real time-domain EM fields of: E r , t Eo cos kz t xˆ and: B r , t Bo cos kz t yˆ and: H r , t 1 o B r,t 1 o Bo cos kz t yˆ 1 o c 1 with: Bo Eo c Eo cos kz t yˆ The vector impedance Z r , t associated with a monochromatic plane EM plane wave propagating in the kˆ zˆ direction in free space is: Z r ,t E r ,t 1 H r,t S r ,t ˆ E r ,t H r ,t E r ,t H r ,t S r,t 2 2 H r ,t H r,t H r,t c o Eo2 cos 2 kz t zˆ 1 o c 2 Eo2 cos 2 kz t o2 c3 o zˆ o c c 2 o o zˆ 1 1 zˆ o c o o zˆ o c zˆ o o o o o Thus, in free space: Z r , t Z o zˆ oo zˆ (Ohms) where: Z o o o o zˆ Z o zˆ o Note the cancellations!!! Here, Z has no spatial and/or temporal dependence – for a monochromatic EM plane wave propagating in free space! is known as the {scalar!} characteristic impedance of free space. The vector impedance Z r , t associated with an EM field is a physical property of the medium that the EM field is propagating – which in this case {here} – is the vacuum. Microscopically, the quantum numbers of the {QED} vacuum – free space {which, at the microscopic level is not empty!} – must all be associated with scalar-type quantities – spin = 0, even parity (+) for both space inversion operation P and charge conjugation C, i.e. the quantum numbers of the {QED} vacuum are J PC 0 . Note further that all of the physical macroscopic (mean-field) parameters of the vacuum must be invariant {i.e. unchanged} under arbitrary rotations, translations and Lorentz boosts - from one reference frame to any other. This means that all macroscopic physical parameters of the vacuum intrinsically must have no spatial and/or temporal dependence – they are constants: o 8.85 1012 Farads m = electric permittivity of free space o 4 107 Henrys m = magnetic permeablity of free space c 1 o o 3 108 m s = speed of EM waves propagating in in free space Zo o o 376.82 = characteristic impedance of EM waves propagating in free space © Professor Steven Errede, Department of Physics, University of Illinois at Urbana-Champaign, Illinois 2005-2015. All Rights Reserved. 15 UIUC Physics 436 EM Fields & Sources II Fall Semester, 2015 Lect. Notes 5 Prof. Steven Errede The vectorial nature of Z r , t is simply associated with the direction of propagation of the EM wave – here, in this case {i.e. the vacuum} the kˆ zˆ direction, so: Z r , t Z o zˆ oo zˆ . For EM waves propagating in the vacuum, it can’t physically matter which direction they are propagating in – any direction k̂ will give Z r , t Z kˆ o kˆ ! o o Note also from the above derivations, that we also have a relation between the vector impedance Z r , t and Poynting’s vector S r , t associated with a propagating EM wave: Z r ,t E r,t 1 H r ,t S r ,t E r , t Hˆ r , t E r , t H r , t 2 H r,t H r ,t S r,t 2 H r,t For the complex time-domain representation of EM fields – at least those associated with monochromatic (i.e. single-frequency) EM waves, then in general we have: E r , t ; E r ; e it and: H r , t ; H r ; e it , and thus the complex vector impedance is: E r , t ; Hˆ * r , t ; E r , t ; H * r , t ; Z r , t; E r , t; 1 H r , t; Ohms 2 H r , t; H r , t; E r ; e it H * r ; e it H r ; e it H * r ; e it S r ; * * E r ; H r ; E r ; H r ; 2 H r ; H * r ; H r ; S r ; 2 Z r ; H ; where S r , and Z r ; are the complex frequency-domain Poynting’s vector and vector impedance, respectively. Note that – at least for monochromatic/single-frequency EM waves – that: Z r , t ; Z r ; , i.e. the complex vector impedance associated with monochromatic EM waves has no time dependence! It is a manifestly frequency-domain quantity! For monochromatic EM plane waves propagating in free space/the vacuum in the kˆ zˆ direction, the complex vector impedance is a purely real quantity: Z r , t ; Z r ; Z o zˆ oo zˆ 376.82 zˆ Physically, the real part of the complex vector impedance e Z r ; is associated with propagating EM waves/propagating EM wave energy, whereas the imaginary part of the complex vector impedance m Z r ; is associated with non-propagating EM wave energy – i.e. EM wave energy that simply sloshes back and forth locally, 2× per cycle of oscillation! 16 © Professor Steven Errede, Department of Physics, University of Illinois at Urbana-Champaign, Illinois 2005-2015. All Rights Reserved. UIUC Physics 436 EM Fields & Sources II Fall Semester, 2015 Lect. Notes 5 Prof. Steven Errede Time-Averaged Quantities Associated with EM Waves: Frequently, we are not interested in knowing the instantaneous power P(t), energy / energy density, Poynting’s vector, linear and angular momentum, etc.- e.g. simply because experimental measurements of these quantities are very often only averages over many extremely fast cycles of oscillation… 1 (e.g. period of oscillation of a light wave: light 1 flight 15 1015 sec cycle 1 femto-sec ) 10 cps We want/need time-averaged expressions for each of these quantities (e.g. in order to compare directly with experimental data) e.g. for monochromatic plane EM light waves: 2 If we have e.g. a “generic” instantaneous physical quantity of the form: Q t Qo cos t The time-average of Q t is defined as: Q t Q 1 t t 0 Q t dt Qo t t 0 cos 2 t dt Q(t) = Qocos (t) 2 Qo 1 Q Q t Qo 2 t The time average of the cos 2 t function: 1 0 But: cos 2 t dt 2 f 1 0 1 t sin 2t 2 4 and: cos 2 t dt t t 0 1 sin 2 1 sin 2 0 0 2 2 2 2 f 1 2 2 sin sin 2 0 1 1 1 Q t Q Qo 2 2 2 The time-averaged quantities associated with an EM plane wave propagating in free space are: uEM r , t S r,t Poynting’s Vector: Linear Momentum Density: EM r , t Angular Momentum Density: EM r , t EM Energy Density: uEM r , t Total EM Energy: U EM t U EM t S EM r , t EM Power: PEM t PEM t EM r , t Linear Momentum: pEM t pEM t EM r , t Angular Momentum: LEM t LEM t © Professor Steven Errede, Department of Physics, University of Illinois at Urbana-Champaign, Illinois 2005-2015. All Rights Reserved. 17 UIUC Physics 436 EM Fields & Sources II Fall Semester, 2015 Lect. Notes 5 Prof. Steven Errede For a monochromatic EM plane wave propagating in free space / vacuum in the kˆ zˆ direction: 1 Joules uEM r , t o Eo2 3 2 m 1 Watts S r , t c o Eo2 zˆ c uEM r , t zˆ 2 2 m 1 1 1 kg EM r , t o Eo2 zˆ 2 S r , t uEM r , t zˆ 2 c c 2c m -sec 1 1 kg EM r , t r EM r , t 2 r S r , t u EM r , t rˆ zˆ c c m-sec Time – averaged quantities for EM plane wave propagating in the ẑ direction We define the intensity I associated with an EM wave as the time average of the magnitude of Poynting’s vector: 1 2 Watts Intensity of an EM wave: I r S r , t S r , t c uEM r , t c o Eo 2 2 m The intensity of an EM wave is also known as the irradiance of the EM wave – it is the so-called radiant power incident per unit area on a surface. When working with time-averaged quantities such as uEM r , t , S r , t , EM r , t , EM r , t , etc. it is convenient/useful to define the so-called root-mean-square ( RMS) values of the E and B electric and magnetic field amplitudes (using the mathematical definition of RMS from probability and statistics): For a monochromatic (i.e. single frequency, sinusoidally-varying) EM wave (only): 1 Erms E 2 1 Brms B 2 Where: Eorms 1 Eo 0.707 Eo 2 1 Bo 0.707 Bo 2 Eo = peak (i.e. max) value of the E -field = amplitude of the Bo = peak (i.e. max) value of the B -field = amplitude of the Borms E -field. B -field. E z, t Eorms 18 Eo 1 Eo 2 Eo Eorms 1 Eo 0.707 Eo 2 z or t © Professor Steven Errede, Department of Physics, University of Illinois at Urbana-Champaign, Illinois 2005-2015. All Rights Reserved. UIUC Physics 436 EM Fields & Sources II Fall Semester, 2015 Lect. Notes 5 Prof. Steven Errede Thus we see that: 1 1 1 1 1 1 Erms Erms E E E E and Brms Brms B B B B 2 2 2 2 2 2 i.e. that: Then: For monochromatic EM plane waves (only): 2 Erms 1 2 1 2 1 1 1 2 1 2 2 2 E E peak Eo2rms Eo2 and Brms B 2 B peak Borms Bo 2 2 2 2 2 2 1 1 1 1 1 RMS Joules uEM t o Eo2 o Eo2 o Eo2rms 2 2 2 2 4 m3 1 1 rms Srms t S t c uEM t zˆ c uEM t zˆ RMS Watts 2 2 2 m 1 1 1 1 rms kg rms S t uEM t zˆ 2 Srms t uEM t zˆ RMS EM t 2 2 c c 2c 2c m -sec rms RMS kg 1 1 1 rms EM t r EM t r rms uEM t r zˆ m-sec EM t 2 r S rms t c c 2 1 1 1 rms I rms S rms t S rms t I S t c uEM t c o Eo2rms RMS Watts 2 2 2 2 m rms uEM t Real world example: Here in the U.S., 120 Vac/60 Hz “wall power” refers to the RMS AC voltage! The peak voltage (i.e. the voltage amplitude) is: V peak 2Vrms 2 120 169.7 170.0 Volts. n.b. For EM waves ≠ sinusoidal waves, the root-mean-square (RMS) must be defined properly / or triangle wave mathematically – e.g. the RMS value of square amplitudes (from Fourier analysis these consist of linear combinations of infinite # of harmonics) 1 1 rms rms (See/refer to probability & statistics reference books!!) 2 2 The Relationship(s) Between the Complex Time-Domain Poynting’s Vector and the Complex Vector Impedance/Admittance of an EM Plane Wave: Complex Time-Domain Poynting’s Vector: S r , t ; E r , t ; H r , t ; Watts m 2 Complex Vector Impedance of an EM Plane Wave: Z r , t ; E r , t ; H 1 r , t ; E r , t ; H * r , t ; 2 H r , t ; Ohms Complex Vector Admittance = Reciprocal of Complex Vector Impedance: Y r , t ; Z 1 r , t ; E 1 r , t ; H r , t ; E * r , t ; H r , t ; 2 E r , t ; Siemens © Professor Steven Errede, Department of Physics, University of Illinois at Urbana-Champaign, Illinois 2005-2015. All Rights Reserved. 19 UIUC Physics 436 EM Fields & Sources II Fall Semester, 2015 Lect. Notes 5 Prof. Steven Errede If we start with Poynting’s vector, we show that it is linearly related to vector admittance and/or reciprocal vector impedance {we suppress (here) the argument r , t ; for notation clarity}: ( E E ) 1 S E H H (EE) E H ( E E )Y ( E E ) Z 1 Watts m 2 E Y 2 Note that the units of E Volts m , hence: ( E E ) Z 1 Volts m Ohms Watts m 2 !!! We can also obtain the alternate relations: ( H H ) 1 1 S E H E (H H ) E H ( H H ) Z ( H H )Y Watts m 2 H Z 2 Note that the units of H Amps m , hence: ( H H ) Z Amp m Ohms Watts m 2 !!! Note that the above complex relations are the vector analogs of the complex scalar power and Ohm’s law relations associated with AC circuits {suppressing the arguments t ; for notational clarity}: Complex time-domain AC power: P V I Volts Amps Watts Complex Ohm’s Law: 2 Z V I (V I * ) I Volts Amps Ohms Complex Scalar Admittance = Reciprocal of Complex Scalar Impedance: 2 Y 1 Z I V I V * V Amps Volts Siemens Ohms 1 Starting with complex time-domain AC power, we show that it is linearly related to scalar admittance and/or reciprocal scalar impedance: (V V ) I P V I I (V V ) (V V )Y (V V ) Z Watts V V Note that the units of V Volts , hence: (V V ) Z Volts 2 Ohms Watts !!! We can also obtain the alternate relations: ( I I ) V P V I V ( I I ) ( I I ) Z ( I I ) Y Watts I I Note that the units of I Amps , hence: ( I I ) Z 20 Amp 2 Ohms Watts !!! © Professor Steven Errede, Department of Physics, University of Illinois at Urbana-Champaign, Illinois 2005-2015. All Rights Reserved. UIUC Physics 436 EM Fields & Sources II Fall Semester, 2015 Lect. Notes 5 Prof. Steven Errede RMS Newtons Radiation Pressure: rad m2 When an EM wave impinges (i.e. is incident) on a perfect absorber (e.g. a totally black object with absorbance {aka absorption coefficient} A = 1, as “seen” at the frequency of the EM wave), all of the EM energy (by definition) is absorbed {ultimately winding up as heat…}. By conservation of energy, linear momentum & angular momentum the object being irradiated by the incident EM wave acquires energy, linear momentum & angular momentum from the incident EM wave. The EM Radiation Pressure acting on a perfect absorber for a normally incident EM wave is defined as: net FEM t RMS Newtons Time-Averaged Force perfect Rad EM absorber A 1 A Unit Area m2 However, the time-averaged EM force is defined as: net d pEM t pEM t FEM t = dt t the EM Radiation Pressure at normal incidence is: time rate of change of the timeaveraged linear momentum Rad EM perfect absorber A1 pEM t t 1 RMS Newtons A m2 In a time interval t 1 f , the time-averaged magnitude of the EM linear momentum transfer pEM t at normal incidence to a perfect absorber of EM radiation is: pEM t EM t V EM Linear momentum density Volume of EM wave associated with time interval t The volume associated with an EM wave propagating in free space over a time interval t is: V A ct where ct = distance traveled by the EM wave in the time interval t . Rad EM perfect absorber A1 pEM t t EM t V 1 EM t A c t 1 c EM t A t A t A Thus, we see that for a monochromatic EM plane wave propagating in free space normally incident on a perfect absorber (A = 1): 1 perfect Rad o Eo2 uEM I c EM absorber A 1 c EM t 2 RMS Newtons m2 © Professor Steven Errede, Department of Physics, University of Illinois at Urbana-Champaign, Illinois 2005-2015. All Rights Reserved. 21 UIUC Physics 436 EM Fields & Sources II Fall Semester, 2015 Lect. Notes 5 Prof. Steven Errede For a perfect reflector (e.g. a perfect mirror, with reflection coefficient R = 1{A = 0}), note that: pEM t perfect reflector 2 pEM t perfect absorber initial final final initial Since pEM pEM pEM and pEM pEM for an EM wave reflecting off of a perfect initial final initial initial initial reflector, then pEM pEM pEM pEM pEM 2 pEM i.e. an EM wave that reflects off of (i.e. “bounces” off of) a perfect reflector delivers twice (2×) the momentum kick (i.e. impulse) to the perfect reflector than the same EM wave that is absorbed by a perfect absorber! Thus at normal incidence: Rad perfect Rad perfect EM reflector R 1 2 EM absorber A1 2 I c RMS Newtons m2 Note that for a partially reflecting surface, with reflection coefficient R < 1, since R + A = 1, the radiation pressure associated with an EM wave propagating in free space and reflecting off of a partially reflecting surface at normal incidence is given by: c partial Rad perfect Rad perfect I Rad EM reflector R A 1 A EM absorber A1 2 R EM absorber R 1 A 2 R RMS Newtons m2 Since A = 1 – R, we can equivalently re-write this relation as: c 1 R 2R I c 1 R I c partial I Rad EM reflector R A 1 A 2 R RMS Newtons m2 If the EM wave is not at normal incidence on the absorbing/reflecting surface, but instead makes a finite angle with respect to the unit normal of the surface, these relations need to be modified, due to the cosine factor S nˆ S cos I cos associated with the flux of EM energy/momentum EM t nˆ EM t cos o o S t cos c12 S t cos c12 I cos crossing the surface area A at a finite angle : c cos RMS Newtons m2 c cos RMS Newtons m2 perfect I Rad EM absorber A 1 perfect I Rad EM reflector R 1 2 c cos 1 R I c cos partial I Rad EM reflector R A 1 A 2 R 22 RMS Newtons m2 © Professor Steven Errede, Department of Physics, University of Illinois at Urbana-Champaign, Illinois 2005-2015. All Rights Reserved. UIUC Physics 436 EM Fields & Sources II Fall Semester, 2015 Lect. Notes 5 Prof. Steven Errede Maxwell’s equations (and relativity) for the macroscopic E and B fields associated with an EM plane wave propagating in free space mandate / require that E B propagation direction (here, kˆ zˆ ) v prop czˆ , as shown in the figure below: Compare this microscopic picture to that of a classical / macroscopic EM plane wave, polarized in the x-hat direction: Macroscopic EM plane waves propagating in free space are purely transverse waves, i.e. E B , and both of the E and B fields are also to the propagation direction of the EM plane wave, e.g. v prop czˆ . Thus: E v prop czˆ and: B v prop czˆ . The behavior of the macroscopic E and B fields associated with e.g. a monochromatic EM plane wave propagating in free space, at the microscopic scale is simply the sum over (i.e. linear superposition of) the E and B -field contributions from {large numbers of} individual real photons making up the EM field. Each real photon has associated with it, its own E and B field – e.g. a linearly polarized real photon, polarized in x̂ direction: x̂ E Eo cos kz t xˆ ( x̂ = polarization direction) Photon Real Photon Momentum: ẑ p h zˆ Photon Poynting’s vector: S 1o E B zˆ ŷ B Bo cos kz t yˆ 1 1 B kˆ E where the unit wavevector kˆ zˆ {here} and Bo Eo in vacuum. c c 0 Real photon energy: E hf p c p c (Total Relativistic Energy2 = E2 p2 c 2 m2 c 4 ) Real photon momentum (deBroglie relation): p h and c f m c 2 0 for real photon c = speed of light (in vacuum) = 3 × 108 m/sec © Professor Steven Errede, Department of Physics, University of Illinois at Urbana-Champaign, Illinois 2005-2015. All Rights Reserved. 23 UIUC Physics 436 EM Fields & Sources II Fall Semester, 2015 Lect. Notes 5 Prof. Steven Errede Question: How many real visible-light photons per second are emitted e.g. from a EM power = 10 mW laser? (mW = milli-Watt = 103 Watt) Answer: The rate at which visible-light photons from a 10 mW laser depends on the color (i.e. the wavelength λ, frequency f, and/or photon energy Eγ) of the laser beam! Eγ = hf = hc/. When we say a 10 mW power laser, what precisely does this mean/refer to? It refers to the time-averaged EM power: Plaser t 10 mW RMS 10 103 Watts RMS 0.010 Watts RMS Let’s assume that the laser beam points in the ẑ direction. Also assume that the diameter of the laser beam is D = 1 mm = 0.001 m (typical). Further assume (for simplicity’s sake): Power flux density = intensity profile I(x,y) is uniform in x and y over the diameter of the laser beam (not true in real life – laser beams have ~ Gaussian 2 2 intensity profiles in x and y (i.e. I I o e 2 ); note that there also exist e.g. diffraction {beam-spreading} effects that should/need to be taken into account…) I x, y S x, y , t In t = 1 second, the time-averaged energy associated with the 10 (RMS) mW laser beam is: Elaser t Plaser t t Elaser t 0.010 RMS Watts 1 sec Elaser t 0.010 Elaser t 24 RMS Joules 1 sec sec 0.010 RMS Joules = Time-averaged energy of laser beam © Professor Steven Errede, Department of Physics, University of Illinois at Urbana-Champaign, Illinois 2005-2015. All Rights Reserved. UIUC Physics 436 EM Fields & Sources II Fall Semester, 2015 Lect. Notes 5 Prof. Steven Errede The {instantaneous} energy of the laser beam crosses an imaginary planar surface that is to the laser beam. If the laser has red light, e.g. λred = 750 nm (n.b. 1 nm = 1 nano-meter = 109 meters) or if the laser has blue light, e.g. λblue = 400 nm Since f = c/λ the corresponding photon frequencies associated with red and blue laser light are: c 3 108 m / s f red 4.0 1014 cycles/sec (= Hertz, or Hz) 9 red 750 10 m fblue c blue 3 108 m / s 7.5 1014 cycles/sec (= Hertz, or Hz) 9 400 10 m The energy associated with a single, real photon is: E hf hc , where h = Planck’s constant: h = 6.626 x 1034 Joule-sec and c = 3 x 108 m/sec (speed of light in vacuum). Thus, the corresponding photon energies associated with red and blue laser light are: Ered hf red hc red hc blue and: Eblue hf blue since f = c/λ Ered hf red 6.626 1034 Joule / sec 4.0 1014 / sec 2.6504 1019 Joules (red light) Eblue hfblue 6.626 1034 Joule / sec 7.5 1014 / sec 4.9695 1019 Joules (blue light) In a time interval of t 1 sec, the time-averaged energy Elaser t N t E where N t is the {time-averaged} number of photons crossing a area in the time interval t . Thus, the number of red (blue) photons emitted from a red (blue) laser in a t 1 sec time interval is: # red photons: Nred t # blue photons: Nblue t Elaser t red E Elaser t blue 0.010 Joules 3.7730 1016 19 2.6504 10 Joules/photon 0.010 Joules 2.0123 1016 19 4.9695 10 Joules/photon Thus, the {time-averaged} rate of emission of red (blue) photons from a red (blue) laser is: red R blue R t t Nred t t Nblue t t 3.7730 1016 red photons/sec 2.0123 1016 blue photons/sec Note: In a time interval of t 1 sec, photons (of any color / / f / E ) will travel a distance of d ct 3 108 m/s 1 s 3 108 meters © Professor Steven Errede, Department of Physics, University of Illinois at Urbana-Champaign, Illinois 2005-2015. All Rights Reserved. 25 UIUC Physics 436 EM Fields & Sources II Fall Semester, 2015 Lect. Notes 5 Prof. Steven Errede If the flux of photons is assumed (for simplicity) to be uniform across the D = 1 mm diameter laser beam, then the time-averaged flux of photons (#/m2/sec) is: F F red blue t t Rred t laser A Rblue t laser A sec 4.8039 10 red m / sec m 3.7730 1016 103 2 22 2 sec 2.562 10 blue m / sec m 2.0123 1016 103 2 2 22 2 2 If each photon has E Joules of energy, then power associated with red (blue) laser beam: Pred t Alaser Sred t Ered F red t m 2 /sec m 2 /sec 2.6504 1019 Joules 4.8039 1022 red 1.2732 104 Watts m 2 Pblue t laser A Sblue t Eblue F blue t 4.9695 1019 Joules 2.56211022 blue 1.2732 104 Watts m 2 Thus we see that: Pred t Alaser Pblue t Alaser Sred t Sblue t 1.2732 104 Watts/m 2 ←10 mW laser n.b. This is precisely why you shouldn’t look into a laser beam {with your one remaining eye}!!! Time-averaged linear momentum density: 1 1 red t o o Sred t 2 Sred t ured t zˆ 1.4147 1013 kg/m 2 -sec c c blue blue 1 blue 1 blue t o o S t 2 S t u t zˆ 1.4147 1013 kg/m 2 -se c c c Thus: red blue 1.4147 1013 kg/m 2 -sec Momentum density, Poyntings vector, energy density are independent of frequency / wavelength / photon energy The time-averaged linear momentum contained in t 1 second’s worth of laser beam: Time averaged linear momentum: p t = momentum density t x volume V Volume V Alaser ct m 3 Distance light travels in t sec. 26 © Professor Steven Errede, Department of Physics, University of Illinois at Urbana-Champaign, Illinois 2005-2015. All Rights Reserved. UIUC Physics 436 EM Fields & Sources II Fall Semester, 2015 Lect. Notes 5 Prof. Steven Errede Red light momentum: 2 0.001 kg-m pred t red t ctA 1.4147 1013 3 108 1 2 sec 3.3333 1011 kg-m sec Blue light momentum: 2 0.001 kg-m pblue t blue t ctA 1.4147 1013 3 108 1 2 sec 3.3333 1011 kg-m sec pred t pblue t 3.3333 1011 kg-m sec Thus: {“TRICK”}: For an EM plane wave, the time-averaged energy density uEM t = time-averaged momentum density EM t c (Since photon energy, E p c ). Thus: kg 8 5 3 ured t red t c 1.4147 1013 2 3 10 m/s 4.2441 10 Joules/m m /sec kg 8 5 3 ublue t blue t c 1.4147 1013 2 3 10 m/s 4.244110 Joules/m m /sec 2 kg-m Joule kg Joule 2 2 s m m/s 2 The time-averaged energy contained in t = 1 second’s worth of laser beam is: The time-averaged energy U t = time-averaged energy density u t volume V V Alaser ct red U t 2 u red Joules 0.001 8 3 t A ct 4.244110 3 3 10 1 m m 2 0.010 Joules 10 mJ 5 2 Joules 0.001 8 3 U blue t ublue t A ct 4.2441105 3 10 1 m 3 m 2 0.010 Joules 10 mJ The time-averaged power in the laser beam: Time-averaged Power (Watts) = d U t dt red Plaser t Joules sec U laser t t blue 10mW laser t t = 1 sec © Professor Steven Errede, Department of Physics, University of Illinois at Urbana-Champaign, Illinois 2005-2015. All Rights Reserved. 27 UIUC Physics 436 EM Fields & Sources II Fall Semester, 2015 Lect. Notes 5 Prof. Steven Errede Note: Plaser (laser power) is measured by the total time-averaged energy U t deposited in (a very accurately) known time interval t using an absolutely calibrated photodiode (e.g. by NIST). A typical time interval t = 10 secs → t (oscillation period) = 1 f !! red 1 f 2.500 1015 sec 2.500 femto-sec 2.500 fs red blue 1 f 1.333 1015 sec 1.333 femto-sec 1.333 fs blue 1 peak Plaser t 2 Consider (the time-averaged) energy density associated with this 10 mW laser: Joules uEM t 4.2441 105 3 m 1 1 peak Now: uEM t uelect t umag t o Eo2 uEM t 2 2 1 1 And because: B t E t for EM plane waves propagating in free space / vacuum ( 2 o o ) c c → The laser power measured is time-averaged power, i.e. Plaser t We showed that: uelect t umag t uelect t 1 1 2 o Eo 2 2 Bo2 1 2 Eo o o Eo2 2 c 1 1 2 1 o o 2 1 1 Bo Eo o Eo2 2 2o 2 2 o 2 2 E z , t Eo cos kz t xˆ Now: Eo = amplitude of the macroscopic electric field: Bo = amplitude of the macroscopic magnetic field: B z , t Bo cos kz t yˆ umag t Define the RMS (Root-Mean-Square) amplitudes of the E and B fields: 1 1 Eo Eo2rms Eo2 2 2 1 1 1 Bo Bo2rms Bo2 2 Eo2 in free space / vacuum 2 2c 2 Eorms Borms Then: 28 uelect t 1 1 1 2 2 3 o Eo o Eorms (Joules/m ) 2 2 2 umag t 1 1 2 1 2 1 Bo Borms o Eo2rms in free space / vacuum 2 2 o 2 o 2 © Professor Steven Errede, Department of Physics, University of Illinois at Urbana-Champaign, Illinois 2005-2015. All Rights Reserved. UIUC Physics 436 EM Fields & Sources II So if: Fall Semester, 2015 Lect. Notes 5 Prof. Steven Errede uEM t uelect t umag t 2 uelect t in free space / vacuum 2 o Eo2rms 4.2441105 Joules/m3 Then: Eo2rms 1 uEM t where o 8.85 1012 Farads/m = electric permittivity of free space o 1 4.2441105 Joules/m 2 U EM t Thus: E (Volts/m) 12 o 8.85 10 Farads/m 3 Eolaser rms 3.0970 10 3097 RMS Volts/m (n.b. same for red vs. blue laser light!) laser orms 1 laser 2 Eolaser Then: Eolaser E peak rms 4380 Volts/m 1 Eolaser 10.3232 106 RMS Tesla 10.3232 102 RMS Gauss Then: Bolaser rms c rms Note: 1 Tesla {SI/MKS units} = 104 Gauss {CGS units} Thus: 1 Bolaser Eolaser 2 Bolaser 14.5970 106 Tesla 14.5970 102 Gauss rms c Now earlier (above) we calculated the (time-averaged) number of photons present in the {red and blue} laser beams that were emitted in a time interval of t = 1 sec. # red photons emitted in t = 1 sec: Nred t 3.7730 1016 red photons # blue photons emitted in t = 1 sec: Nblue t 2.0123 1016 blue photons The volume associated with a D = 1 mm diameter laser beam turned on for t = 1 sec is: 2 2 D 0.001 8 3 V A ct ct 3 10 1 235.6194 m 2 2 The (time-averaged) number density n t red n t nblue t Nred t V Nblue t V N t V of {red and blue} photons in the laser beam is: 3.7730 1016 1.6009 1014 red photons/m3 2 2.3562 10 2.0123 1016 8.5405 1013 blue photons/m3 2 2.3562 10 Then the (time-averaged) energy density uEM t of the {red and blue} laser beam is: Red photon energy: Ered hfred 2.6504 1019 Joules Blue photon energy: Eblue hfblue 4.9695 1019 Joules © Professor Steven Errede, Department of Physics, University of Illinois at Urbana-Champaign, Illinois 2005-2015. All Rights Reserved. 29 UIUC Physics 436 EM Fields & Sources II Fall Semester, 2015 Lect. Notes 5 Prof. Steven Errede red uEM t nred t Ered 1.6009 1014 2.6504 1019 4.2442 105 Joules/m3 blue uEM t nblue t Eblue 8.5405 1013 4.9695 1019 4.2442 105 Joules/m3 The (time-averaged) energy U EM t uEM t V of the {red and blue} laser beams is: red red U EM t uEM t V 4.2442 105 2.3562 102 0.010 Joules 10 mJoules blue blue U EM t uEM t V 4.2442 105 2.3562 102 0.010 Joules 10 mJoules Now here is something quite interesting: Given that Eorms Eo 2 for a monochromatic EM plane wave propagating in free space/the vacuum, with time-averaged EM energy density: Joules uEM t o Eo2rms 12 o Eo2 3 m Joules n t = photon number density (#/m3) in laser beam But: uEM t n t E 3 m E hf hc = energy/photon (Joules) o Eo2rms n t E 2 Or: Eorms n t o E This formula explicitly connects the amplitudes of the macroscopic E and B fields (since Bo Eo c ) with the microscopic constituents of the and B fields (i.e. the photons)!!! n.b. This formula physically says that the number of {real} photons in the EM wave (each of photon energy E ) is proportional to Eo2 = the square of the macroscopic electric field amplitude! We can write this as: n t o Eo2rms E and note also that: Eorms n t o E !!! Thus, we can now see that the {time-averaged} EM energy density: uEM r , t o Eo2rms n r , t E with: v u EM r , t d U EM plays a role analogous to that of the probability density in quantum mechanics: 2 P r , t r , t | r , t r , t with: Since: n r , t uEM r , t E 2 o Eo2rms r E and: Then: P r , t n r , t P r , t d 1 v v n r , t d N , 2 N r , t | r , t r , t !!! Thus, we also see that the macroscopic electric field E r , t plays a role analogous to that of the probability density amplitude r , t in quantum mechanics!!! 30 © Professor Steven Errede, Department of Physics, University of Illinois at Urbana-Champaign, Illinois 2005-2015. All Rights Reserved. UIUC Physics 436 EM Fields & Sources II Fall Semester, 2015 Lect. Notes 5 The (real) photon number density in the laser beam is: n t Then: o Eo2rms A ct N t E R t But: Eo2rms 1 2 o F t N t t R t A c or: Eo2rms N t o A ct N t V Prof. Steven Errede N t A ct # 3 m E = the time averaged rate of photons in laser beam (#/sec) E and N t A t R t # A m 2 -s = flux of photons in the laser beam 1 Joules F t E n r , t E 3 c m # Thus, we see that the {real} photon flux: F t c n r , t 2 m -s Thus, the intensity {aka irradiance} of the laser beam is: Eo2rms 1 F t E and oc uEM t o Eo2rms I S t kˆ c uEM t o Eo2rms F t E c n t E Watts 2 m The {time-averaged} <longitudinal separation distance> between photons is defined as: d t ct N t For t = 1 sec: dred t dblue t (m) 3 108 m 7.85 109 m ~ 8 109 m 8 nm (1 nm = 109 m) 16 3.773 10 ' s 3 108 m 1.49 108 m ~ 15 109 m 15 nm 16 2.012 10 ' s Recall that: red 750 nm and blue 400 nm Thus: d t for either red or blue laser light. The {time-averaged} <transverse separation distance> between photons is defined as: d t A N t © Professor Steven Errede, Department of Physics, University of Illinois at Urbana-Champaign, Illinois 2005-2015. All Rights Reserved. 31 UIUC Physics 436 EM Fields & Sources II Fall Semester, 2015 Lect. Notes 5 Prof. Steven Errede Thus: 2 d red t 0.001 2 2.35 1020 m 16 3.773 10 2 d blue t 0.001 2 4.40 1020 m 16 2.0123 10 We showed above that the time-averaged/mean number density of photons in the laser beam is: n r , t uEM r , t E 12 o Eo2 E photons m3 . The instantaneous number density of photons in the laser beam is: n r , t uEM r , t E o Eo2 cos 2 kz t E photons m3 We can normalize the instantaneous photon number density to obtain an instantaneous 3-D photon probability density, P3-D z , t 1 m3 . Recall that the laser beam intensity is uniform/constant in the cross-sectional area A of the laser beam. The 3-D photon probability density, P3-D r , t is: P3-D r , t n r , t N uEM r , t N E o Eo2 cos 2 kz t N E 1 m3 E U EM Joules in time interval of t secs . Then also note that: U EM uEM r , t Vol 12 o Eo2 A n r , t E A . But note that: 3-D Thus: P N n r , t u EM r , t r,t U EM N o Eo2 1 2 o Eo2 A cos 2 kz t 2 cos 2 kz t 1 m3 A with: v P3-D r , t d 2 A z ct z ct 2 2 kz t dz da cos 2 kz t dz cos A z 0 0 z A A 2 1 z c t cos 2 kz t dz 1 0 z 1 2 We thus see that the instantaneous normalized photon number density P3-D r , t n r , t N plays a role analogous to that of the probability density in quantum mechanics 2 P r , t r , t | r , t r , t , with: P r , t d 1 . Hence, we also see that the v 32 © Professor Steven Errede, Department of Physics, University of Illinois at Urbana-Champaign, Illinois 2005-2015. All Rights Reserved. UIUC Physics 436 EM Fields & Sources II macroscopic electric field amplitude Eo Fall Semester, 2015 1 o Lect. Notes 5 Prof. Steven Errede n r , t E plays a role analogous to the probability density amplitude in quantum mechanics! We can calculate the time rate of change of the normalized 3-D photon probability density, P3-D r , t : P3-D r , t t 2 4 cos 2 kz t sin kz t cos kz t A t A We can define a normalized 3-D photon probability current density as: J 3-D r , t c P3-D r , t zˆ . We calculate the divergence of the normalized 3-D photon probability current density: 2 J3-D r , t c P3-D r , t zˆ c cos 2 kz t zˆ A c 2 4k 2 sin kz t cos kz t cos kz t c A z A We thus show that the photons in this laser beam obey the continuity equation for photons: P3-D r , t 3-D J r , t 0 t From above: P3-D r , t 4k 4 sin kz t cos kz t sin kz t cos kz t and: J3-D r , t c t A A However, in free space, we have: ck . Hence: Thus: P3-D r , t t P3-D r , t t 4ck sin kz t cos kz t J3-D r , t A J3-D r , t 0 i.e. microscopically, photons neither disappear, nor are they created in propagating as this laser beam! © Professor Steven Errede, Department of Physics, University of Illinois at Urbana-Champaign, Illinois 2005-2015. All Rights Reserved. 33