Announcements WebAssign HW Set 7 due this Friday Problems cover material from Chapters 20 and 21 We re skipping Sections 21.1-21.7 (alternating current circuits) Review: Self-inductance induced ε induced QUESTIONS? PLEASE ASK! Review: Self-inductance 20.7 RL Circuits Self-inductance occurs when the changing flux through a circuit arises from the circuit itself The self-induced EMF must be proportional to the time rate of change of the current ΔΦ ε = N Δ t ΔI ε = L Δt L is inductance of the device; the negative sign indicates that a changing current induces an EMF in opposition to that change The inductance of a coil depends on geometric factors The SI unit of self-inductance is the Henry: 1 H = 1 (V s) / A N The expression for L is L N ΔΦ Φ = = Δ I I Consists of voltage source, a resistor, and an inductor: When the current reaches its maximum, the rate of change and the back emf are zero The time constant, τ, for an RL circuit is the time required for the current in the circuit to reach 63.2% of its final value: L τ = R The current at any time can be found by ε I = R t / τ ( 1 e ) 1
20.8 Energy Stored in a Magnetic Field The battery has to do work to produce a current against the back EMF This work can be thought of as energy stored by the inductor in its magnetic field > potential energy! The energy stored in an inductor is: PE L = ½ L I 2 Analogous to energy stored in a capacitor: E = ½ Q 2 /C Problem: 20.50, p694 In the circuit to the right, ε = 6.00V, L = 8.00 mh, and R = 4.00 Ω. (a) (b) (c) (d) What is the inductive time constant of the circuit? Calculate the current in the circuit 250 μs after the switch is closed. What is the final value of the steady state current? How long does it take the current to reach 80% of its maximum value? 20.8: Maxwell s Theory of Electromagnetism Chapter 21 Electromagnetic (We re skipping Sections 21.1-21.7) Electricity and magnetism were originally thought to be unrelated in 1865, James Clerk Maxwell developed a unified theory of electromagnetism Started from the following observations (thanks to Gauss and Faraday!) Electric field lines originate on positive charges and terminate on negative charges Magnetic field lines always form closed loops they do not begin or end anywhere A varying magnetic field induces an EMF and hence an electric field (Faraday s Law) Magnetic fields are generated by moving charges or currents (Ampère s Law) 2
Maxwell Equations All fundamental information about E&M is contained in these equations! Source: Wikipedia Don t worry - you are not expected to know this for PHY 2054 (If you want to, you should major in physics!) Hertz s Confirmation: Generation of Radio Hertz used an LC circuit It oscillates! It radiates! How does it work? Switch closes; current flows to capacitor An EMF source is connected to provide current and then taken out of the circuit The capacitor charges fully total energy of the circuit is stored in capacitor (electric field); the current is zero and no energy is stored in the inductor The capacitor then discharges energy stored in the electric field decreases and the current increases energy stored in the inductor (magnetic field) increases When the capacitor is fully discharged, there is no energy stored in its electric field When the current is a maximum, all the energy is stored in the magnetic field in the inductor The process repeats in the opposite direction The energy sloshes around the circuit; there is a continuous transfer of energy between the inductor and the capacitor E = ½ Q 2 /C E = ½ L I 2 Hertz Measures the Speed of the Hertz measured the speed of the waves from the transmitter He used the waves to form an interference pattern and calculated the wavelength From v = f λ, v was found v was very close to 3 x 10 8 m/s, the known speed of light Why this works - Electromagnetic Produced by an Antenna When a charged particle accelerates, it radiates energy When currents in an AC circuit change rapidly, energy is radiate in the form of EM waves EM waves are radiated by any circuit carrying alternating current An alternating voltage applied to the wires of an antenna forces the electric charges in the antenna to oscillate 3
Electromagnetic are Transverse The E and fields are perpendicular to each other oth fields are perpendicular to the direction of motion E&M waves are transverse waves Electromagnetic waves travel at the speed of light c = 1 μ ε ecause EM waves travel at a speed that is precisely the speed of light, light is an electromagnetic wave o o Properties of EM (stated with out proof) For an E&M wave: E c = Energy is carried by EM waves; it is shared equally by the electric and magnetic fields Average power per unit area = 2 2 Emaxmax Emax cmax I = = = 2μ 2μ c 2μ o o o Intensity (I) is average power per unit area Electromagnetic waves transport momentum as well as energy For complete absorption of energy U, p=u/c For complete reflection of energy U, p=(2u)/c The Spectrum of EM EM waves are distinguished by their frequencies and wavelengths c = ƒλ Wavelengths for visible light range from 400 nm to 700 nm There is no sharp division between one kind of EM wave and the next Doppler Equation for EM The Doppler effect for EM waves u fo fs 1 ± c f o is the observed frequency f s is the frequency emitted by the source u is the relative speed between the source and the observer The equation is valid only when u is much smaller than c 4