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Current in an LC Circuit t Q Q0 cos LC I dQ dt Current in an LC circuit Q0 t I sin LC LC Period: T 2 LC Frequency: f 1 / 2 LC Question (Chap. 23) A capacitor C was charged and contains charge +Q0 and –Q0 on 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? Q0 A. There will be no current in the circuit at any time because of the opposing emf in the inductor. B. The current in the circuit will maximize at time t when the capacitor will have charge Q(t)=0. C. The current in the circuit will maximize at time t when capacitor will have full charge Q(t)=Q0. D. The current will decay exponentially. 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 A) Clockwise B) Counterclockwise when viewed from above 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 Faraday’s Law: Applications Faraday’s Law: Applications Induction microphone Chapter 24 Classical Theory of Electromagnetic Radiation Maxwell’s Equations E nˆdA q B nˆA 0 inside 0 Gauss’s law for electricity Gauss’s law for magnetism 𝑑 𝐸 ∙ 𝑑𝑙 = − 𝑑𝑡 𝐵 ∙ 𝑛𝑑𝐴 Complete Faraday’s law Ampere’s law B d l I 0 inside_ path (Incomplete Ampere-Maxwell law) Ampere’s Law B dl 0 Iinside_ path Current pierces surface No current inside B dl 0 0 2 I B 4 r B dl 0 I 0 2 I B dl 4 r 2r 0 I Maxwell’s Approach Time varying magnetic field leads to curly electric field. Time varying electric field leads to curly magnetic field? elec E nˆdA Q Q A cos 0 elec 0 A 0 d elec 1 dQ 1 I dt 0 dt 0 I d elec ‘equivalent’ current I 0 dt combine with current in Ampere’s law B dl 0 Iinside_ path The Ampere-Maxwell Law d elec B dl 0 I inside_ path 0 dt Works! Maxwell’s Equations Four equations (integral form) : q Gauss’s law E nˆdA Gauss’s law for magnetism B nˆA 0 Faraday’s law Ampere-Maxwell law + Lorentz force inside 0 d E dl dt B nˆdA d elec B dl 0 I inside_ path 0 dt F qE qv B 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 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) A Pulse and Gauss’s Laws E nˆdA q inside 0 E nˆdA 0 Pulse is consistent with Gauss’s law B nˆA 0 Pulse is consistent with Gauss’s law for magnetism A Pulse and Faraday’s Law emf d mag dt Since pulse is ‘moving’, B depends on time and thus causes E mag Bhv t mag d mag Bhv t dt emf emf E dl Eh Is direction right? Area does not move E=Bv A Pulse and Ampere-Maxwell Law =0 d elec B dl 0 I inside_ path 0 dt elec Ehvt elec d elec Ehv t dt B dl Bh Bh 0 0 Evh B 0 0vE A Pulse: Speed of Propagation B 0 0vE E=Bv B 0 0vBv 1 0 0v 2 v 1 0 0 3 108 m/s E=cB Based on Maxwell’s equations, pulse must propagate at speed of light Question At this instant, the magnetic flux mag through the entire rectangle is: A) B; B) Bx; C) Bwh; D) Bxh; E) Bvh Question In a time t, what is mag? A) 0; B) Bvt; C) Bhvt; D) Bxh; E) B(x+vt)h Question emf = mag/t = ? A) 0; B) Bvh; C) Bv; D) Bxh; E) B(x+v)h Question What is 𝐸 ∙ 𝑑 𝑙 around the full rectangular path? A) Eh; B) Ew+Eh; C) 2Ew+2Eh; D) Eh+2Ex+2Evt; E)2Evt Question d mag emf Bvh dt r r 𝐸 ∙dl𝑑 𝑙 = Eh𝐸ℎ — Eg What is E? A) Bvh; B) Bv; C) Bvh/(2h+2x); D) B; E) Bvh/x 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? E cB E 3x108 m / s 1x105 T 3000V / m Force on charge q moving with velocity v perpendicular to B: 𝐹𝑚𝑎𝑔 𝑣𝐵 𝑣𝐵 𝑣 = = = 𝐹𝑒𝑙 𝐸 𝑐𝐵 𝑐 Direction of Propagation Direction of speed is given by vector product EB Accelerated Charges Electromagnetic pulse can propagate in space How can we initiate such a pulse? Short pulse of transverse electric field Accelerated Charges 1. Transverse pulse propagates at speed of light 2. Since E(t) there must be B 3. Direction of v is given by: E B E v B Magnitude of the Transverse Electric Field We can qualitatively predict the direction. What is the magnitude? Magnitude can be derived from Gauss’s law Field ~ -qa Eradiative 1 qa 40 c 2 r 1. The direction of the field is opposite to qa 2. The electric field falls off at a rate 1/r Exercise a An electron is briefly accelerated in the direction shown. Draw the electric and magnetic vectors of radiative field. E B 1. The direction of the field is opposite to qa 2. The direction of propagation is given by E B Stability of Atoms v 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 a Sinusoidal Electromagnetic Radiation Acceleration: d2y a 2 ymax 2 sin t dt 1 qa Eradiative 40 c 2 r f 2 T 1/ f 1 qymax 2 Eradiative sin t ĵ 2 40 c r Sinusoidal E/M field Sinusoidal E/M Radiation: Wavelength Instead of period can use wavelength: cT c f f 2 T 1/ f Example of sinusoidal E/M radiation: atoms radio stations E/M noise from AC wires Freeze picture in time: Electromagnetic Spectrum 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 Polarized E/M Radiation AC voltage (~300 MHz) E/M radiation can be polarized along one axis… no light …and it can be unpolarized: 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