HR31_post

# HR31_post - Chapter 31 Electromagnetic Oscillations and...

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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 AC-circuits • 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 τ , 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 capacitor’s electric field and the inductor’s 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 (cont’d) 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( φ ϖ + = 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 ϖ is the angular frequency of the electromagnetic oscillations. It is given by (31-1) LC Oscillations (cont’d) ) cos( φ ϖ + = t Q q 5 • The amplitude I of this current is I = w Q . So we can rewrite the above equation as: h φ is the phase constant • Taking the first derivative of (31-1) with respect to time gives current i : ) sin( φ ϖ +- = = t ω Q dt dq i ) sin( φ ϖ +- = t I i LC ω 1 = • Thus, in an...
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HR31_post - Chapter 31 Electromagnetic Oscillations and...

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