Chapter XIII Electromagnetic Oscilation, Eletromagnetic Field and Wave

Chia sẻ: Thanh An | Ngày: | Loại File: PDF | Số trang:49

Thêm vào BST

Báo xấu

108
lượt xem 19
download

Chapter XIII Electromagnetic Oscilation, Eletromagnetic Field and Wave

Mô tả tài liệu

Download Vui lòng tải xuống để xem tài liệu đầy đủ

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.

Chủ đề:

Bình luận(0) Đăng nhập để gửi bình luận!

Lưu

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

CÓ THỂ BẠN MUỐN DOWNLOAD