Physics 2113 Jonathan Dowling Lecture 34 FRI 13

Physics 2113 Jonathan Dowling Lecture 34: FRI 13 NOV Electrical Oscillations, LC Circuits, Alternating Current I Nikolai Tesla

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)

Energy Density in E and B Fields

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.

PHYS 2110: A Mechanical Oscillator Newton’s law F=ma!

PHYS 2113 An Electromagnetic LC Oscillator Capacitor initially charged. Initially, current is zero, energy is all stored in the E-field of the capacitor. A current gets going, energy gets split between the capacitor and the inductor. Capacitor discharges completely, yet current keeps going. Energy is all in the B-field of the inductor all fluxed up. 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.

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

Electric Oscillators: the Math Energy as Function of Time Voltage as Function of Time

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

Analogy Between Electrical And Mechanical Oscillations Charqe q -> Position x Current i=q’ -> Velocity v=x’ Dt-Current i’=q’’-> Acceleration a=v’=x’’

LC Circuit: Conservation of Energy The energy is constant and equal to what we started with.

LC Circuit: Phase Relations The current runs 90° out of phase with respect to the charge.

Example 1 : Tuning a Radio Receiver The inductor and capacitor in my car radio have one program at L = 1 m. H & C = 3. 18 p. F. Which is the FM station? (b) WRKF 89. 3 What is wavelength of radio wave? How about for WJBO 1150 AM? FM radio stations: frequency is in MHz.

Example 2 • In an LC circuit, L = 40 m. H; 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

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 ωq 0 of the resulting oscillating current. 1 m. H 1 m. F b E=10 V a • Switch in position “a”: q=CV = (1 m. F)(10 V) = 10 m. C • Switch in position “b”: maximum charge on C = q 0 = 10 m. C • So, amplitude of oscillating current = 0. 316 A

Example 4 In an LC circuit, the maximum current is 1. 0 A. If L = 1 m. H, C = 10 m. F what is the maximum charge q 0 on the capacitor during a cycle of oscillation? Maximum current is i 0=ωq 0 Maximum charge: q 0=i 0/ω Angular frequency w=1/√LC=(1 m. H 10 m. F)– 1/2 = (10 -8)– 1/2 = 104 rad/s Maximum charge is q 0=i 0/ω = 1 A/104 rad/s = 10– 4 C