Current in an LC Circuit

1. Current in an LC Circuit Current in an LC circuit Period: Frequency:

2. Question (Chap. 23) Q0 A capacitor C was charged and contains charge +Q0and –Q0on each of its plates, respectively. It is then connected to an inductor (coil) L. Assuming the ideal case (wires have no resistance) which is true? • There will be no current in the circuit at any time because of the opposing emf in the inductor. • The current in the circuit will maximize at time t when the capacitor will have charge Q(t)=0. • The current in the circuit will maximize at time t when capacitor will have full charge Q(t)=Q0. • The current will decay exponentially.

3. Question Two metal rings lie side-by-side on a table. Current in the left ring runs clockwise and is increasing with time. This induces a current in the right ring. This current runs Clockwise Counterclockwise when viewed from above

4. Faraday’s Law: Applications Single home current: 100 A service Vwires=IRwires Transformer:emfHV IHV = emfhomeIhome Single home current in HV: <0.1 A Power loss in wires ~ I2

5. Faraday’s Law: Applications

6. Faraday’s Law: Applications Induction microphone

7. Chapter 24 Classical Theory of Electromagnetic Radiation

8. Maxwell’s Equations Gauss’s law for electricity Gauss’s law for magnetism Complete Faraday’s law Ampere’s law (Incomplete Ampere-Maxwell law)

9. Ampere’s Law Current pierces surface No current inside

10. Maxwell’s Approach I ‘equivalent’ current Time varying magnetic field leads to curly electric field. Time varying electric field leads to curly magnetic field? combine with current in Ampere’s law

11. The Ampere-Maxwell Law Works!

12. Maxwell’s Equations Four equations (integral form) : Gauss’s law Gauss’s law for magnetism Faraday’s law Ampere-Maxwell law + Lorentz force

13. Fields Without Charges Time varying magnetic field makes electric field Time varying electric field makes magnetic field Do we need any charges around to sustain the fields? Is it possible to create such a time varying field configuration which is consistent with Maxwell’s equation? Solution plan: • Propose particular configuration • Check if it is consistent with Maxwell’s eqs • Show the way to produce such field • Identify the effects such field will have on matter • Analyze phenomena involving such fields

14. A Simple Configuration of Traveling Fields Key idea: Fields travel in space at certain speed Disturbance moving in space – a wave? 1. Simplest case: a pulse (moving slab)

15. A Pulse and Gauss’s Laws Pulse is consistent with Gauss’s law Pulse is consistent with Gauss’s law for magnetism

16. A Pulse and Faraday’s Law Area does not move emf Since pulse is ‘moving’, B depends on time and thus causes E E=Bv Is direction right?

17. A Pulse and Ampere-Maxwell Law =0

18. A Pulse: Speed of Propagation E=Bv E=cB Based on Maxwell’s equations, pulse must propagate at speed of light

19. Question At this instant, the magnetic flux Fmag through the entire rectangle is: • B; B) Bx; C) Bwh; D) Bxh; E) Bvh

20. Question In a time Dt, what is DFmag? A) 0; B) BvDt; C) BhvDt; D) Bxh; E) B(x+vDt)h

21. Question emf = DFmag/Dt = ? A) 0; B) Bvh; C) Bv; D) Bxh; E) B(x+v)h

22. Question What is around the full rectangular path? • Eh; B) Ew+Eh; C) 2Ew+2Eh; D) Eh+2Ex+2EvDt; E)2EvDt

23. Question What is E? A) Bvh; B) Bv; C) Bvh/(2h+2x); D) B; E) Bvh/x

24. Exercise If the magnetic field in a particular pulse has a magnitude of 1x10-5 tesla (comparable to the Earth’s magnetic field), what is the magnitude of the associated electric field? Force on charge q moving with velocity v perpendicular to B:

25. Direction of Propagation Direction of speed is given by vector product

26. Accelerated Charges Electromagnetic pulse can propagate in space How can we initiate such a pulse? Short pulse of transverse electric field

27. Accelerated Charges E v B • Transverse pulse propagates at speed of light • Since E(t) there must be B • Direction of v is given by:

28. Magnitude of the Transverse Electric Field Field ~ -qa 1.The direction of the field is opposite toqa We can qualitatively predict the direction. What is the magnitude? Magnitude can be derived from Gauss’s law 2. The electric field falls off at a rate 1/r

29. Exercise a E B 2. The direction of propagation is given by 1. The direction of the field is opposite to qa An electron is briefly accelerated in the direction shown. Draw the electric and magnetic vectors of radiative field.

30. Stability of Atoms v a Circular motion: Is there radiation emitted? • Classical physics says “YES” • orbiting particle must lose energy! • speed decreases • particle comes closer to center Classical model of atom: Electrons should fall on nucleus! To explain the facts - introduction of quantum mechanics: Electrons can move around certain orbits only and emit E/M radiation only when jumping from one orbit to another

31. Sinusoidal Electromagnetic Radiation Acceleration: Sinusoidal E/M field

32. Sinusoidal E/M Radiation: Wavelength Instead of period can use wavelength: Freeze picture in time: Example of sinusoidal E/M radiation: atoms radio stations E/M noise from AC wires

33. Electromagnetic Spectrum

34. E/M Radiation Transmitters How can we produce electromagnetic radiation of a desired frequency? Need to create oscillating motion of electrons Radio frequency LC circuit: can produce oscillating motion of charges To increase effect: connect to antenna Visible light Heat up atoms, atomic vibration can reach visible frequency range Transitions of electrons between different quantized levels

35. Polarized E/M Radiation AC voltage (~300 MHz) E/M radiation can be polarizedalong one axis… no light …and it can be unpolarized:

36. Polarized Light Making polarized light Turning polarization Polaroid sunglasses and camera filters: reflected light is highly polarized: can block it Considered: using polarized car lights and polarizers-windshields