Electromagnetic fields and waves - Tran Thi Ngoc Dung

Nội dung text: Electromagnetic fields and waves - Tran Thi Ngoc Dung

1. ELECTROMAGNETIC FIELDS AND WAVES Tran Thi Ngoc Dung – Huynh Quang Linh – Physics A2 HCMUT 2016
2. CONTENTS - Maxwell’s Equations - Displacement current - Plane Electromagnetic Waves – Wave equation - Energy Carried by Electromagnetic Waves, Poynting vector
3. Maxwell’s Equations and Electromagnetic Waves • Maxwell discovered that the basic principles of ectromagnetism can be expressed in terms of the four equations that we now call Maxwell’s equations. • These four equations are (1) Faraday’s law, (2) Ampere’s law, including displacement current; (3) Gauss’s law for electric fields; (4) Gauss’s law for magnetic fields, showing the absence of magnetic monopoles
4. Review : Gauss’s Law for electric field The flux of the electric field (the area integral of the electric field) over any closed surface (S) is equal to the net charge inside the surface (S) divided by the permittivity o. qin E E dS closed 0 surfaceS
5. Review : Gauss’s law of magnetism Gauss’s law of magnetism states that the net magnetic flux through any closed surface is zero: B.dS 0 dS B S (S) The magnetic field lines are closed lines. The number of magnetic field lines that exit equal The number of magnetic field lines that enter the closed surface.
6.  I B(r,,z) o e Review: AMPÈRE’S LAW 2 r  Ampere'slawsaysthat thelineintegralof B.d d drer rde dzez  I  I aroundanyclosedpath equalsμ I, B.d o e .(dre rde dze ) o d o 2 r  r  z 2 whereIis the totalcurrent through anysurfaced  I  I B.d o d o d bounded by theclosedpath (C) (C) 2 2 (C) B.d o Ii (C) going through _ surface bounded by theclosed path I it dependson thedirection of thecurrent relativeto thedirection of thelineintegral r that Ii 0or Ii 0 d H.d Ii B   H + B  o r (C) i  I B o Magnetic filed B is sometimes called magnetic induction, 2 r (vector cảm ứng từ) Magnetic field H , magnetic field strength , (cường độ từ trường)
7. Review: Faraday’s law of induction when the magnetic flux through the loop changes with time, there is an emf induced in a loop d d  B B.dS dt dt S where B B.dS is the magnetic flux through the loop (S)
8. Maxwell-Faraday’s equation d B E.d B.dS rotE (C) dt S t Maxwell - Faraday’s equation states that a time varying magnetic field induces an electric field. Maxwell -Faraday’s equation states that the emf, which is the line integral of the electric field around any closed path, equals the rate of change of magnetic flux through any surface bounded by that path. B(t) (C)
9. Maxwell- Ampere’s equation 1. Time-changing electric fields induces magnetic fields 2. Displacement current Conduction currents cause Magnetic field ( motion of charged particles) Time changing electric fields also cause Magnetic field => Time changing electric fields is equivalent to a current. We call it dispalcement current. 3. Displacement current density D E jd or Electric field E, vector cường độ điện trường t t Electric Displacement field D : vector cảm ứng điện D orE
10. Maxwell Ampère’s equation D D H.d j .dS rotH j (C) S t t “Line integral of vector H over a closed path (C) is equal to the total current going through any surface bounded by the closed path. The total current is equal to the sum of conduction current and displacement current Itotal Iconduction Idisplcement ( j jd ).dS S
11. Maxwell’s equations d B Time changing magnetic fields Maxwell- E.d B.dS rotE dt t induces electric fields Faraday (C) S D D Maxwell – H.d ( j ).dS rotH j Time changing electric fields Ampère (C) S t t induces magnetic fields This law relates an electric Maxwell – D.dS qin dV; divD field to the charge (S) (The distribution that creates it. Gauss for tich V) E-field  the number of magnetic field lines Maxwell – B.dS 0 divB 0 that enter a closed surface must equal Gauss for (S) to the number that leave that surface. B-field There is no magnetic charge. -1 D orE B orH j E : electrical conductivity (.m) ; volume charge density C/m3
12. ELECTROMAGNETIC WAVE IN FREE SPACE
13. We have learned equation of the wave on a string y Acos(t kx) y is vertical position of an element of the string dy d2y Asin(t kx); A2 cos(t kx) dt dt 2 dy d2y Aksin(t kx); Ak2 cos(t kx) dx dx 2 2 2  k  v.T v d2y 2 2 dt v2 d2y k 2 dx 2 Wave equation of the Wave on a string 2y 1 2y 0 x2 v2 t2
14. The Wave Equation For any function, f: f(x – vt) describes a wave moving in the positive x direction. f(x + vt) describes a wave moving in the negative x direction. What is the origin of these functional forms? They are solutions to a wave equation: 2f 1 2f 0 2x v2 2t The harmonic wave, f = cos(kx ± t), satisfies the wave equation. 2p 1 2p Sound waves: 0 p is pressure 2x v2 2t Electromagnetic waves: 2E 1 2E 2B 2B x x 0 y 1 y 2 2 2 0  z v  t 2z v2 2t
15. Maxwell’s Maxwell ‘s equation in free Equations space B rotE t I n free space 0, 0, j 0, D B  H,D  E rotH j o o t B divB 0 M - Fadaday's law : rotE t divD E M - Ampère's law : rotB oo j E t D orE M - Gauss's law for B field : divB 0 B orH M - Gauss's law for E field : divE 0
16. 2. Wave equation. We use the formula rotrot graddiv rotrotE graddivE E 1 oo rotrotB graddivB B c2 B E rotE rotB   t o o t   rotrotE rotB rotrotB oo rotE t t  E  B , graddivE E (  ) graddiv B B oo ( ) o o t t 0 t t 0 2 2 1  B 1  E B 0 E 0 2 2 c2 t 2 c t
17. E  B  c E B c
18. Electromagnetic Energy 2 1 2 1 B 1 EM energy density w em oE (E.D B.H) 2 2 o 2 2 1 2 1 B EM energy in a W emdV ( oE )dV volume V volumeV Volume V 2 2 o 1 B2 1 E2 1  E2 2 o 2 o 2 oc 2 2 w em oE 1 Average EM energy w  E2  E2 ; density em o 2 o m 1 (E Em cos(t kx);oo ;B E / c) c2
19. POYNTING VECTOR  E  B S (W / m2 ) Poynting vector o  E2 S u oc 2 Eo Saverage 2oc u : unit vectoralong thedirection of propagation The magnitude of the Poynting vector represents the rate at which energy passes through a unit surface area perpendicular to the direction of wave propagation The direction of the Poynting vector is along the direction of wave propagation
20. Wave Intensity  E  B S (W / m2 ) Poynting vector o  E2 S u oc 2 2 Eo cBo I Saverage 2oc 2o u : unit vectoralong thedirection of propagation
21. Relationship between Poynting Vector and Electromagnetic Energy Density : Poynting vector is a vector that has the magnitude equal to the energy going through a unit area perpendicular to the wave propagation direction per unit time Energy that goes across area A during time interval dt is contained in the volume with cross section A, length cdt dW wem (A cdt) Energy that goes across a unit area during per a unit time wo A Energy Density dW 1 2 3 cdt S wem.c w  E (J/m ) Adt em 2 o m 2 S wem.c (W / m )
22. Quick Quiz 34.2 What is the phase difference between the sinusoidal oscillations of the electric and magnetic fields in Figure 34.8? (a) 180° (b) 90° (c) 0 (d) impossible to determine ANS: c
23. Quick Quiz 34.3 An electromagnetic wave propagates in the negative y direction. The electric field at a point in space is momentarily oriented in the positive x direction. In which direction is the magnetic field at that point momentarily oriented? (a) the negative x direction (b) the positive y direction (c) the positive z direction (d) the negative z direction ANS: c
24. Example 34.2 An Electromagnetic Wave A sinusoidal electromagnetic wave of frequency 40.0 MHz travels in free space in the x direction as in Figure 34.9. (A) Determine the wavelength and period of the wave. (B) At some point and at some instant, the electric field has its maximum value of 750 N/C and is directed along the y axis. Calculate the magnitude and direction of the magnetic field at this position and time. the magnitude of the magnetic field:
25. Example 34.3 Fields on the Page • Estimate the maximum magnitudes of the electric and magnetic fields of the light that is incident on this page because of the visible light coming from your desk lamp. Treat the lightbulb as a point source of electromagnetic radiation that is 5% efficient at transforming energy coming in by electrical transmission to energy leaving by visible light. The visible light output of a 60-W lightbulb operating at 5% P E2 I ave max efficiency is approximately 3.0 W by visible light. 2 (The remaining energy transfers out of the lightbulb by 4 r 2oc thermal conduction and invisible radiation.) 2ocPave Emax 2 A reasonable distance from the lightbulb to the page might 4 r be 0.30 m.
26. Review Statement Write the equation • Maxwell -Faraday’s law states that a changing magnetic field with time (or a changing magnetic flux) induces an electric field. • Maxwell – Ampere ‘s law states that both conduction current and displacement current (the changing electric filed) are the sources of magnetic filed. • Maxwell – Gauss ‘s law for electric fields states that the surface integral of electric field over any closed surface is equal to the total charges enclosed within the closed surface divided by o • Maxwell – Gauss ‘s law for magnetic fields states that the surface integral of magnetic field over any closed surface is 0