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Unformatted text preview: Chapter 31 Electromagnetic Oscillations and Alternating Current Will cover the following topics Electromagnetic oscillations in an LC circuit Alternating current (AC) circuits with apacitors 1 capacitors Resonance in an RLC circuits Power in ACcircuits Transformers , AC power transmission We have learned three circuit elements: resistance R , capacitance C , and inductance L Two series combinations of the above three elements have been discussed: RC circuit and RL circuit In both cases, charge, current and voltage grow and decay exponentially . How fast this happens is defined by time onstant which is either apacitive C ircuit) or ductive LC circuit 2 constant , which is either capacitive ( RC circuit) or inductive ( RL circuit) time constant In this chapter we will examine LC circuit We will see that in this case the charge, current and voltage do not decay exponentially with time; they rather vary sinusoidally . The resulting oscillations of the capacitors electric field and the inductors magnetic field are said to be electromagnetic oscillations L C i + + + + The circuit shown in the figure consists of capacitor C and inductor L . There is no resistance R in the circuit We charge the capacitor with an initial charge Q and observe what happens. We will see that the capacitor will discharge through the inductor resulting in time dependent current i Let us examine quantitatively the LC circuit: LC Oscillations 3 At any given time, the total energy of the circuit is the sum of the energies stored in an electric field of the capacitor C and in a magnetic field of the inductor L : 2 2 2 2 Li U C q U U U U B E B E = = + = , where , The last two equations correspond to the electric energy of the capacitor ( q is the charge of the capacitor) and the magnetic energy of the inductor ( i is the current in the circuit) at a given instant of time t , respectively L C i + + + + The total energy of the circuit: 2 2 2 2 Li C q U + = Since we have assumed that resistance of the circuit is zero, no energy is transferred to thermal energy, and therefore U remains constant over time. That is: = dt dU 2 2 2 q i LC Oscillations (contd) 4 2 2 2 = + = + = + = dt q d Li i C q dt di Li dt dq C q Li C q dt d dt dU 1 2 2 = + q C dt q d L The above equation is a homogeneous, second order differential equation. Its solution is: ) cos( + = t Q q L C i + + + + Thus, in LC circuit, charge q of the capacitor oscillates with time: Here h Q is the amplitude of the charge variations h is the angular frequency of the electromagnetic oscillations. It is given by (311) LC Oscillations (contd) ) cos( + = t Q q 5 The amplitude I of this current is I = w Q . So we can rewrite the above equation as: h is the phase constant Taking the first derivative of (311) with respect to time gives current i : ) sin( + = = t Q dt dq i ) sin( + = t I i LC 1 = Thus, in an...
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 Fall '09

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