Tài liệu Chapter XIII Electromagnetic Oscilation, Eletromagnetic Field and Wave doc

4/8/2008 1 GENERAL PHYSICS II Electromagnetism & Thermal Physics 4/8/2008 2 Chapter XIII Electromagnetic Oscilation, Eletromagnetic Field and Wave §1. Oscillating circuits §2. System of Maxwell’s equations §3. Maxwell’s equations and electromagnetic waves 4/8/2008 3  We have known the close connection between changing eletric fields and magnetic fields. They can create each other and form a system of electromagnetic fields.  Electromagnetic fields can propagate in the space (vacuum or material environment). We call them electromagnetic waves. They play a very important role in science and technology. In this chapter we will study how can describe electromagnetic fields, what are their properties (in comparison with mechanical waves).  First we consider the oscillating circuits in which there exist oscillating currents and voltages. They are sources for electromagnetic fields 4/8/2008 4 §1. Oscillating circuits: 1.1 L-C circuits and electrical oscillations: • Consider the RC and LC series circuits shown: • Suppose that the circuits are formed at t=0 with the capacitor charged to value Q. There is a qualitative difference in the time development of the currents produced in these two cases. Why?? L C C R ++++ - - - - ++++ - - - - 1.1.1 Consider from point of view of energy (qualitatively): • In the RC circuit, any current developed will cause energy to be dissipated in the resistor. • In the LC circuit, there is NO mechanism for energy dissipation; energy can be stored both in the capacitor and the inductor! 4/8/2008 5 RC: current decays exponentially C R 0 t I 0 I Q +++ - - - L C LC: current oscillates I 0 0 t I Q +++ - - - 4/8/2008 6 Recall: Energy in the Electric and Magnetic Fields 2 1 2 U LI 2 magnetic 0 1 2 B u   … energy density . Energy stored in an inductor …. B Energy stored in a capacitor . 2 1 2 U C V 2 electric 0 1 2 u E   … energy density . +++ +++ - - - - - - E 4/8/2008 7    L C + + - - 0I 0 QQ  L C + + - - 0I 0 QQ  L C 0 II  0Q  L C 0 II  0Q Energy is stored in the capacitor Energy is stored in the inductor Energy is stored in the capacitorEnergy is stored in the inductor 4/8/2008 8 where  and Q 0 determined from initial conditions • Differentiate above form for Q(t) and substitute into the differential equation we can find   L C I 1.1.2 Equations for Q(t) and I(t): 2 2 dt Qd L dt dI L C Q VV LC  )cos( 00   tQQ remember: 0 2 2 d x m kx dt   1 2 At t = 0 the switch is transfered from the position 1 to the position 2: 0 2 2  C Q dt Qd L The solution Q(t) has the form analogue to SHM (simple hamonic motion): 4/8/2008 9 )sin( 000   tQ dt dQ )cos( 00 2 0 2 2   tQ dt Qd      0)cos( 1 )cos( 0000 2 0   tQ C tQL 0 1 2 0  C L  Therefore, LC 1 0   The oscillation of the current in LC circuit is determined by the following equation: )sin( 000   tQ dt dQ I 00 QI m  ),sin( 0   tII m 4/8/2008 10 I 1 2 L C R 1.2 LCR circuit and damped oscillation: dt dI LRI C Q VVV LRC  0 2 2  C Q dt dQ R dt Qd L When the swicth is transferred to 2: The solution Q(t) has the form of a damped oscillation: where L R 2   )'cos( 0     t βt eQQ o          2 2 4 1 L R LC ' o  and The frequency of oscilation (In an LRC circuit, depends also on R) Damping constant [...]... electromagnetism These equations predict the existence of electromagnetic disturbances consisting of time-varying electric and magnetic fields that can propagate from one region of space to another 4/8/2008 electromagnetic waves 31 §3 Maxwell’s equations and electromagnetic waves: First we review of what we have known about “waves”:  The one-dimensional wave equation: 2 2 h 1 h 2 2 x v t 2 has a general... 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  h x , t A cos kx  t  A     2 2 k   f  2  T 4/8/2008  v   f k x A = amplitude  = wavelength f = frequency v = speed k = wave number 32 3.1 Plane Harmonic Wave: An important specific case of electromagnetic waves... only x-component and the magnetic field B has only y-component These components vary in t and z sinusoidally • The fields E and B must be perpendicular to each other and to the direction of propagation Later we will show that this specific solution is consistent with Maxwell’s equations Why do we call this solution “plane wave ? For any given value of z, the magnitude of the electric field is uniform... x-y plane with that z value In other words, the wave front is plane x The similar situation is true for the magnitude of z magnetic field 4/8/2008 y 34 3.2 Derivation of the electromagnetic wave equation: Now we show that • The components of electric and magnetic fields obey wave equations which are derived from the Maxwell’s equations • Harmonic plane waves described before are really consistent with... equations:  In the previous chapters we have known the basis laws concerning to electric and magnetic phenomena:  Gaussian law  Amperian law  Faraday’s law Combining these results and adding some complementary ideas, Maxwell derived a system of equations which allows describe electromagnetic phenomena in a general way 4/8/2008 26 2.1 Maxwell’s statement on displacement current and Maxwel-Amper’s equation:... magnetically induced part which is non-conservative 30 Remark on the symmetry between electric and magnetic fields in the Maxwell’s equations: • In empty space (where there is no charge) the Gauss’s laws for E and B have the same form • In empty space (the charge Q=0 and the induction current I = 0 we can write Amper’s and Faraday’s laws in the symmetrical form: d B E 0 0 .d l  dt .d A d E B .d l dt... But what is the “displacement current” and where does come from?!  We know that current is a flow of moving charges  Although there is no actual charge moving between the plates, something is changing – the electric field between them → a changing electric field is equivalent to a fictitious current  We can calculate quantitively this current:  The electric field E between the plates of the capacitor... through a capacitor (DC does not pass): the displacement current connects induction currents which come into one plate of capacitor and come out the other plate 4/8/2008 29 2.2 Maxwell’s equations: Resume what we have known about the relationships between electric, magnetic fields and their sources: Gauss’s law for E: Gauss’s law for B : Qencl E d A   0 B d A 0 Amper’s law including displacement current:... harmonic wave in which E and B have the following form: E x E 0 sin( kz  t )  B y B0 sin( kz  t )  where  kc x z y  By is in phase with Ex  B0 E0 / c ˆ  The direction of propagation s is given by the cross product  ˆ ˆb s e ˆ ˆˆ where e, b are the unit vectors in the (E,B) directions Note cyclical relation: 4/8/2008 ˆˆ ˆb s e  ˆs e ˆˆ b  ˆˆb s e ˆ 33 Some remarks: • The electric field. .. frequencies and passes high frequencies 4/8/2008 16 1.3.3  Circuit: L IL dI VL L L  m s i n   d I  s i n t d t  t m L dt   L  L   m   I L  dI L   cos  m sin t  / 2  t    L L In this case the voltage across L leads the current through L by one-quarter cycle (90 )  m VL 4/8/2008 1 0 IL 0 0  m  m L t   m 1 L 0 t 17 Phasor diagram: Two vectors representing voltage and current . Electromagnetism & Thermal Physics 4/8/2008 2 Chapter XIII Electromagnetic Oscilation, Eletromagnetic Field and Wave §1. Oscillating circuits §2. System of. Maxwell’s equations and electromagnetic waves 4/8/2008 3  We have known the close connection between changing eletric fields and magnetic fields. They can