Chapter XIII Electromagnetic Oscilation, Eletromagnetic Field and Wave

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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.

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Nội dung Text: Chapter XIII Electromagnetic Oscilation, Eletromagnetic Field and Wave

GENERAL PHYSICS II Electromagnetism & Thermal Physics 4/8/2008 1
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 2
 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 3
§1. Oscillating circuits: 1.1 L-C circuits and electrical oscillations: • Consider the RC and LC series circuits shown: ++++ ++++ • Suppose that the circuits are C R C formed at t=0 with the capacitor ---- ---- L charged to value Q. There is a qualitative difference in the time development of the currents produced in these two cases. Why?? 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 4
I I Q +++ Q +++ C L C R --- --- RC: LC: current decays exponentially current oscillates I I 0 0 0 0 t t 4/8/2008 5
Recall: Energy in the Electric and Magnetic Fields Energy stored in a capacitor ... +++ +++ E 1 --- -- - U  CV 2 2 1 2 … energy density ... uelectric  0E 2 Energy stored in an inductor …. B 1 2 U  LI 2 1 B2 … energy density ... umagnetic  2 0 4/8/2008 6
Energy is stored in the capacitor Energy is stored in the inductor I0 I  0 I + + C L  C L - - Q  Q0  Q 0   I 0 I  0 I - - C L C L  + + Q0 Q  Q0  Energy is stored in the inductor Energy is stored in the capacitor 4/8/2008 7
1.1.2 Equations for Q(t) and I(t): 1 2 I At t = 0 the switch is transfered from the position 1 to the position 2: L 2 C Q dI d Q V C V L  L L C dt dt 2 d 2Q Q L 2  0 remember: d 2x dt C m 2  x 0 k dt The solution Q(t) has the form analogue to SHM (simple hamonic motion): Q  0 cos(t  ) Q 0  where  and Q0 determined from initial conditions • Differentiate above form for Q(t) and substitute into the differential equation we can find  4/8/2008 8
dQ d 2Q  0Q 0 sin( 0 t  )   2  02 Q 0 cos( 0 t  )   dt dt L Q0 cos( t  )  0 cos( t  )  0   0 L  0 1 1 02 0  Q 0    2 C C 1 Therefore,  0 LC The oscillation of the current in LC circuit is determined by the following equation: dQ I  0 Q0 sin(  t  )  0  dt I I m sin(  t  ), 0   I m  0Q0 4/8/2008 9
1.2 LCR circuit and damped oscillation: R 1 2 When the swicth is transferred to 2: I Q dI V C  R V L  V RI L C dt C L d 2Q dQ Q L 2 R  0 dt dt C The solution Q(t) has the form of a damped oscillation: t β Q Q 0 e cos( 'o t  )   R 1 R2  where and ' o    2   2L LC 4L   The frequency of oscilation Damping constant 4/8/2008 (In an LRC circuit, depends also on R) 10
Some remarks: R R 0 Q • The amplitude of oscillation is damping, 0 since energy is dissipated in the resistor. The role of resistance in oscillations of the current in LCR circuit is analogue to friction in mechanical oscillations ! 1 t • The resistance R increases → the amplitude of oscillation decreases faster. Oscillations Q R0 R  will occure as long as 0 is real, so there ’ 4 exists a critical resistance 0 1 R c2 L  2  Rc  4 1 LC 4L C If R R c the circuit is called underdamped; t R R c : critically damped; R  R c : overdamped. 4/8/2008 11
1.3 LRC circuit with alternating current (AC) source:  This is the case analogue to the mechanical R driven oscillations with a periodic force. Suppose that the emf of the source has C L the following form:    sint = m The equation for Q(t) must be modified  by adding AC emf in the right hand side: d 2Q dQ Q L 2 R   m sin t  dt dt C First we consider the particular cases: the circuit  or R,  C, or  only. L 4/8/2008 12
1.3.1  Circuit: R R The above equation leads to dQ R  sin t I dt m  The formulas for the voltage and current  across R are as follows  V R  R I R  s i n t  IR  m s i n t m R   m R m VR IR 0 0 - m  m 1 1 0 R 0 t t 4/8/2008 13
• Remark: Both the current and the voltage vary with time as sin . t We say they are “in phase” . Phasor diagram: • Vectors represent the maximum voltage and the maximum current • Both vectors make an angle t  with the positive x-axis • With time both vectors rotate counter-clockwise • The vertical component of each vector represents the instantaneous value of voltage or current. Impedance: The ratio of the maximum voltage to the maximum current For a resistor Impedance of a resistor 4/8/2008 don’t depend on frequency. 14
1.3.2  Circuit: C Q VC   m sin   t  Q C  sin  m t C IC C   IC  dQ  C c o s   m t  dt  In this case the voltage on C and the current through C are not in phase, we say that they are ”out of phase”.  The current has peaks at an earlier time than the voltage. The current leads the voltage by one-quarter cycle or 90 . m   C m VC IC 0 0   m1  C 1  m 0 t 0 t 4/8/2008 15
Phasor diagram: The vectors which represent the current and the voltage are perpendicular each to other, as shown in the picture. Impedance: We can calculate the impedance for capacitor Note that the impedance of a capacitor depends on, beside C, also the frequency. The impedance will be large at low frequencies. The capacitor can play a role as a filter which stops low frequencies and passes high frequencies. 4/8/2008 16
1.3.3  Circuit: L IL L VL L L  m s i n   d I  s i n t d t dI  t m  dt L L       I L  dI L  cos  m sin t  / 2  m t   L L In this case the voltage across L leads the current through L by one-quarter cycle (90 ).  m m VL L IL 0 0 m 1   m 1 4/8/2008 0 t L 0 17 t
Phasor diagram: Two vectors representing voltage and current are perpendicular each to other. (Comparison: For  circuits, the current C leads the voltage, but for  circuits, the voltage L leads the current). Calculate the impedance for inductor: The impedance of an inductor depends on L and frequency . The impedance will be large at high frequencies. The inductor can play a role as a filter which stops high frequencies and passes low frequencies. 4/8/2008 18
1.3.4 Driven LCR circuit: R Using the obtained results, we can consider circuits LCR with AC source. C L The current at any time is equal in all the circuit  element:  Phasor diagram for the circuit is shown in the picture. According to the phasor diagram, we have Using the definition of impedance: 4/8/2008 19
Factoring out imax we can calculate the total impedance of the circuit: Usually, the symbol Z is used for the total impedance of a circuit: The phasor diagram gives also the formula for phasor angle: 4/8/2008 20