Lecture 5 Phys. 07: Waves and Light Physics Department Yarmouk University 63 Irbid Jordan &KDSWHUElectromagnetic Oscillations and Alternating urrent L ircuit In this chapter you will see how the electric charge varies with time in a circuit made up of an inductor L, and a capacitor. From another point of view, we shall discuss how energy shuttles back and forth between the magnetic field of the inductor and the electric field of the capacitor. Dr. Nidal Ershaidat L Fig. http://ctaps.yu.edu.jo/physics/ourses/phys07/lec4- Dr. N. Ershaidat Phys. 07 hapter 4: Electromagnetic Oscillations and Alternating urrent Lecture 5 L Oscillation - Qualitative The energy stored in the capacitor is called the electric energy because it is associated with the energy stored between the capacitor plates as electric field. Which is eual to: The energy stored in the inductors is call the magnetic energy because it is associated with the energy stored in the inductor as magnetic field Which is eual to: U B Li U E Fig. -a Dr. N. Ershaidat Phys. 07 hapter 4: Electromagnetic Oscillations and Alternating urrent Lecture 5 3 L Oscillator onsider the L circuit. According to Kirchhoff s second law: the (algebraic) sum of potential differences euals zero, i.e. V L V 0 + di L + 0 di V L L d di d I d d L + 0 + 0 L Thus we get a homogeneous linear nd order DE: &+ & ω 0 where ω L V Dr. N. Ershaidat Phys. 07 hapter 4: Electromagnetic Oscillations and Alternating urrent Lecture 5 4
L Oscillator We know that the solution is of the form: Q cos ( ωt which is similar to x x cos( ωt The current in this circuit is given by: ( ωt I sin( ω i Q ωsin t ( ω which is similar to v v0 sin t 0 5 Electric and Magnetic Energy Oscillations Q U E cos ( ω t U B L i Lω Q sin ω t + φ ω Lω L U Q sin ω t + φ B Q U U E + U B Animation of L ircuit ( ω x x cos t 0 6 Dr. N. Ershaidat Phys. 07 hapter 4: Electromagnetic Oscillations and Alternating urrent Lecture 5 Dr. N. Ershaidat Phys. 07 hapter 4: Electromagnetic Oscillations and Alternating urrent Lecture 5 L Oscillation - Energy The inductor and capacitor transfer energy from one to the other as shown below: Fig. 3 Dr. N. Ershaidat Phys. 07 hapter 4: Electromagnetic Oscillations and Alternating urrent Lecture 5 7 4-3 The Electrical Mechanical Analogy L Oscillation - Energy The Energy in Two Oscillating Systems ompared Block Spring System Element Spring Block dx v Energy K x m v L Oscillator Element apacitor Inductor Energy L i d i Dr. N. Ershaidat Phys. 07 hapter 4: Electromagnetic Oscillations and Alternating urrent Lecture 5 8
Lecture 6 Phys. 07: Waves and Light Physics Department Yarmouk University 63 Irbid Jordan 3UREOHPVRQElectromagnetic Oscillations and Alternating urrent Problem 33-5 The freuency of oscillation of a certain L circuit is 00 khz. At time t 0, plate A of the capacitor has maximum positive charge. At what times t > 0 will (a) plate A again have maximum positive charge, (b) the other plate of the capacitor have maximum positive charge, and (c) the inductor have maximum magnetic field? 0 Dr. Nidal Ershaidat L http://ctaps.yu.edu.jo/physics/ourses/phys07/lec4- Dr. N. Ershaidat Phys. 07 hapter 4: Electromagnetic Oscillations and Alternating urrent Lecture 5 Problem 33-5 - Solution a) harge on the capacitor () t Q cos( ωt at t 0 Q Q Q cos φ φ 0 Q cosω t b) Q at t T / f / 00000 5µ s at t T /. 5 µ s c) The Magnetic field is maximum when the current is maximum and that occurs at t T/4 when 0, at t.5 µs not that: i I sin ω t i I at t T / 4 Dr. N. Ershaidat Phys. 07 hapter 4: Electromagnetic Oscillations and Alternating urrent Lecture 5 Problem 33-3 In the circuit shown in Fig. 33-3 the switch is kept in position a for a long time. It is then thrown to position b. (a) alculate the freuency of the resulting oscillating current. (b) What is the amplitude of the current oscillations? Solution a) f π L f 75 Hz π 54 0 6. 0 b) I Q ω π f Q Q V 6.µ F 34.0V 0. 8µ I π 75 0.8 0 6 0. 364 A Dr. N. Ershaidat Phys. 07 hapter 4: Electromagnetic Oscillations and Alternating urrent Lecture 5 3
Problem 33-7 The energy in an oscillating L circuit containing a.5 H inductor is 5.70 µj. The maximum charge on the capacitor is 75.0 µ. Find (a) the mass: The mass m corresponds to the inductance, i.e. m.5 kg. b) the spring constant: The spring constant k corresponds to the reciprocal of the capacitance. Since the total energy is given by U Q /, where Q is the maximum charge on the capacitor and is the capacitance then: Q 75 0.69 0 F k 37 N m U 5.70 0 J.69 0 Dr. N. Ershaidat Phys. 07 hapter 4: Electromagnetic Oscillations and Alternating urrent Lecture 5 3 Problem 33-7 c) the maximum displacement, The maximum displacement x m corresponds to the maximum charge, thus x m 75 0-6 m 75.0 µm and (d) the maximum speed for a mechanical system with the same period. The maximum speed v m corresponds to the maximum current. The maximum current is: I Q ω Q L Thus v 3.0 0-3 m/s 75 0 3.0 0 A (.5 H )(.69 0 F ) Dr. N. Ershaidat Phys. 07 hapter 4: Electromagnetic Oscillations and Alternating urrent Lecture 5 4 Problem 33-7 In an oscillating L circuit, L 5.0 mh and 7.80 mf. At time t 0 the current is 9.0 ma, the charge on the capacitor is 3.80 m, and the capacitor is charging. (a) What is the total energy in the circuit? The total energy U is the sum of the energies in the inductor and capacitor. If is the charge on the capacitor, is the capacitance, i is the current, and L is the inductance, then: i L U U E + U B + ( 3.80 0 ) ( 9. 0 A) ( 5.0 0 H ) + 7.80 0 F.98 0 J.98 µ J Dr. N. Ershaidat Phys. 07 hapter 4: Electromagnetic Oscillations and Alternating urrent Lecture 5 5 Problem 33-7 (b) What is the maximum charge on the capacitor? Solve U Q / for the maximum charge Q: Q U (7.80 0 F )(.98 0 J ) 5.56 0 (c) What is the maximum current? U (7.80 0 F ) I. 6 ma L 5.0 0 H (d) If the charge on the capacitor is given by Q cos(ωt, what is the phase angle φ? If 0 is the charge on the capacitor at time t 0, then : 0 Q cos φ 0 3.80µ φ cos cos ± 49.6 Q 5.56µ Dr. N. Ershaidat Phys. 07 hapter 4: Electromagnetic Oscillations and Alternating urrent Lecture 5 6 4
Problem 33-7 (e) Suppose the data are the same, except that the capacitor is discharging at t 0. What then is φ? Now you want the derivative to be negative and sinφ to be positive. Take φ +46.9. For φ +46.9, the charge on the capacitor is decreasing; for φ - 46.9, it is increasing. To check this, calculate the derivative of with respect to time, evaluated for t 0. You should get - ω Q sinφ. You want this to be positive. Since sin(+46.9 ) is positive and sin(-46.9 ) is negative, the correct value for increasing charge is φ -46.9 7 Next Lecture hapter 5 Electromagnetic Waves Dr. N. Ershaidat Phys. 07 hapter 4: Electromagnetic Oscillations and Alternating urrent Lecture 5 (QGRI/HFWXUH 5