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Physics 2113 Jonathan Dowling Lecture 33: WED 12 NOV Electrical Oscillations, LC Circuits, Alternating Current I Nikolai Tesla

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**What are we going to learn? A road map**

Electric charge Electric force on other electric charges Electric field, and electric potential Moving electric charges : current Electronic circuit components: batteries, resistors, capacitors Electric currents Magnetic field Magnetic force on moving charges Time-varying magnetic field Electric Field More circuit components: inductors. Electromagnetic waves light waves Geometrical Optics (light rays). Physical optics (light waves)

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**Energy Density in E and B Fields**

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**Oscillators in Physics**

Oscillators are very useful in practical applications, for instance, to keep time, or to focus energy in a system. All oscillators can store energy in more than one way and exchange it back and forth between the different storage possibilities. For instance, in pendulums (and swings) one exchanges energy between kinetic and potential form. We have studied that inductors and capacitors are devices that can store electromagnetic energy. In the inductor it is stored in a B field, in the capacitor in an E field.

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**PHYS2110: A Mechanical Oscillator**

Newton’s law F=ma!

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**PHYS2113 An Electromagnetic LC Oscillator**

Capacitor initially charged. Initially, current is zero, energy is all stored in the E-field of the capacitor. Capacitor discharges completely, yet current keeps going. Energy is all in the B-field of the inductor all fluxed up. A current gets going, energy gets split between the capacitor and the inductor. The magnetic field on the coil starts to deflux, which will start to recharge the capacitor. Finally, we reach the same state we started with (with opposite polarity) and the cycle restarts.

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**Electric Oscillators: the Math**

Energy Cons. Or loop rule! Both give Diffy-Q: Solution to Diffy-Q: LC Frequency In Radians/Sec

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**Electric Oscillators: the Math**

Voltage as Function of Time Energy as Function of Time

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LC Circuit: At t=0 1/3 Of Energy Utotal is on Capacitor C and Two Thirds On Inductor L. Find Everything! (Phase φ0=?)

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**Analogy Between Electrical And Mechanical Oscillations**

Charqe q -> Position x Current i=q’ -> Velocity v=x’ Dt-Current i’=q’’-> Acceleration a=v’=x’’

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**LC Circuit: Conservation of Energy**

The energy is constant and equal to what we started with.

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**LC Circuit: Phase Relations**

The current runs 90° out of phase with respect to the charge.

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**Example 1 : Tuning a Radio Receiver**

FM radio stations: frequency is in MHz. The inductor and capacitor in my car radio have one program at L = 1 mH & C = 3.18 pF. Which is the FM station? (a) KLSU 91.1 (b) WRKF 89.3 (c) Eagle 98.1 WDGL

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**Example 2 ω = 2500 rad/s T = period of one complete cycle**

In an LC circuit, L = 40 mH; C = 4 μF At t = 0, the current is a maximum; When will the capacitor be fully charged for the first time? ω = 2500 rad/s T = period of one complete cycle T = 2π/ω = 2.5 ms Capacitor will be charged after T=1/4 cycle i.e at t = T/4 = 0.6 ms

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Example 3 In the circuit shown, the switch is in position “a” for a long time. It is then thrown to position “b.” Calculate the amplitude ωq0 of the resulting oscillating current. b a E=10 V 1 mH 1 mF Switch in position “a”: q=CV = (1 mF)(10 V) = 10 mC Switch in position “b”: maximum charge on C = q0 = 10 mC So, amplitude of oscillating current = 0.316 A

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**Example 4 In an LC circuit, the maximum current is 1.0 A.**

If L = 1mH, C = 10 mF what is the maximum charge q0 on the capacitor during a cycle of oscillation? Maximum current is i0=ωq0 Maximum charge: q0=i0/ω Angular frequency w=1/√LC=(1mH 10 mF)–1/2 = (10-8)–1/2 = 104 rad/s Maximum charge is q0=i0/ω = 1A/104 rad/s = 10–4 C

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