Lecture 19 Ch 31 - PH 222-3A Spring 2007 Electromagnetic Oscillations and Alternating Current Lecture 19 Chapter 31(Halliday/Resnick/Walker

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PH 222-3A Spring 2007 lectromagnetic Oscillations and Electromagnetic Oscillations and Alternating Current Lecture 19 Chapter 31 (Halliday/Resnick/Walker, Fundamentals of Physics 8 th edition) 1
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Chapter 31 Electromagnetic Oscillations and Alternating Current this chapter we will cover the following topics: In this chapter we will cover the following topics: -Electromagnetic oscillations in an LC circuit lternating current (AC) circuits with capacitors -Alternating current (AC) circuits with capacitors -Resonance in RCL circuits - ower in AC circuits Power in AC circuits -Transformers, AC power transmission 2
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L C The circuit shown in the figure consists of a capacitor C LC Oscillations and an inductor . We give the capacitor an initial charge and then observe what happens. The capacitor ill discharge throu L Q h the inductor resulting in a time- will discharge through the inductor, resulting in a time dependent current . i e will show that the charge on the ca acitor plates as well as the current i We will show that the charge on the capacitor plates as well as the current 1 in the inductor oscillate with constant amplitude at an angular frequency . qi LC ω= The total energy in the circuit is U 22 the sum of the energy stored in the electric field of the capacitor and the magnetic field of the inductor: . 2 EB qL i UU U =+= + The total energy of the circuit does not change with time. Thus C d 2 2 0. U dt U q dq di dq di d q q = 2 2 0. 1 0 dU Li i dt C dt dt dt dt dq Lq dt C dt =+ = = + =→ = 3
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L C 2 2 2 2 1 0( ) 1 0 dq Lq dt C q dt LC ⎛⎞ += =→ + eq. 1 This is a homogeneous, second order, linear differential equation that we have encountered previously. We used it to describe the simple harmonic oscillator (SHO). 2 2 2 With sol () tion: ( ) cos( ) dx x dt t X t ω + eq. 2 co s qt Q t φ =+ With solution: () cos ( xt ωφ 1 LC ω= If we compare eq. 1 with eq. 2 we find that the solution to the differential equation that describes the circuit (eq. 1) is: LC 1 ( ) cos where , and is the phase angle. Q t LC = The current d i = ( ) sin . q Qt dt =− + 4
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L C 22 The energy stored in the electric field of the capacitor: Q ( ) 2 2 2 2 cos The energy stored in the magnetic field of the inductor: E qQ Ut CC ω φ = =+ () sin sin 2 The total energy : B EB Li L Q Q t C UU U ωφ == + = + Q U = ()() cos sin Q tt ⎛⎞ ⎡⎤ ++ + = ⎜⎟ ⎣⎦ ⎝⎠ The total energy is constant; energy is conserved. 2 3 he energy of the has a value of at QT T T ectric field maximum 2 The energy of the value of at 0 ,, , , . . . . 2 35 The energy of the has a value of at , , ,.... 4 4 4 tT C T T t = = electric field magnetic field maximum 24 When is maximum, is C Note : zero, and vice versa. 5
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/8 tT = /4 = 3/ 8 = 4 3 2 5 T 1 0 t = /2 = 6 / 8 T 8 1 3 5 5/ 8 = 7/ = 7 2 4 6 4 = 7 8 6
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If we add a resistor in an circuit (see figure) we must RL Damped Oscillations in an Circuit RCL 2 modify the energy equation, because now energy is being dissipated on the resistor: .
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This note was uploaded on 09/26/2008 for the course PH 222 taught by Professor Mirov during the Spring '08 term at University of Alabama at Birmingham.

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Lecture 19 Ch 31 - PH 222-3A Spring 2007 Electromagnetic Oscillations and Alternating Current Lecture 19 Chapter 31(Halliday/Resnick/Walker

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