PHYS 1444 – Section 004 Lecture #21Example 32 – 1 Example 32 – 1 Maxwell’s Amazing Leap of FaithProduction of EM WavesProduction of EM WavesProperties of Radiation FieldsProperties of Radiation FieldsEM WavesEM Waves and Their SpeedsMaxwell’s Equations in free spaceEM Waves from Maxwell’s EquationsLight as EM WaveLight as EM WaveElectromagnetic SpectrumExample 32 – 2Monday, April 16, 2007 1PHYS 1444-004, Spring 2007Dr. Andrew BrandtPHYS 1444 – Section 004Lecture #21Monday, April 30, 2007Dr. Andrew Brandt• Maxwell’s Equations• Production of Electromagnetic Waves• EM Waves from Maxwell’s Equations• Speed of EM WavesMonday, April 16, 2007 PHYS 1444-004, Spring 2007Dr. Andrew Brandt2Example 32 – 1 Charging capacitor. A 30-pF air-gap capacitor has circular plates of area A=100cm2. It is charged by a 70-V battery through a 2.0-Ω resistor. At the instant the battery is connected, the electric field between the plates is changing most rapidly. At this instance, calculate (a) the current into the plates, and (b) the rate of change of electric field between the plates. (c) Determine the magnetic field induced between the plates. Assume E is uniform between the plates at any instant and is zero at all points beyond the edges of the plates. Since this is an RC circuit, the charge on the plates is: For the initial current (t=0), we differentiate the charge with respect to time.QThe electric field is =E=0I=dEdt=0CVChange of the electric field is ()1tRCe−−0tdQdt==00tRCtCVeRC−==0VR=70352.0VA=Ω⋅0σε=0QAε0dQ dtAε=()()1412 2 2 2 2354.0 108.85 10 1.0 10AVmsCNm m−−=×⋅×⋅⋅×Monday, April 16, 2007 PHYS 1444-004, Spring 2007Dr. Andrew Brandt3Example 32 – 1 (c) Determine the magnetic field induced between the plates. Assume E is uniform between the plates at any instant and is zero at all points beyond the edges of the plates. The magnetic field lines generated by changing electric field isperpendicular to E and is circular due to symmetry Whose law can we use to determine B?We choose a circular path of radius r, centered at the center of the plane, following the B. EΦ=Bdl⋅=∫GGvExtended Ampere’s Law w/ Ιencl=0!since E is uniform throughout the plate For r<rplate, the electric flux is So from Ampere’s law, we obtain ()2Brπ⋅=Since we assume E=0 for r>rplate, the electric flux beyond the plate is fully contained inside the surface.EΦ=So from Ampere’s law, we obtain ()2Brπ⋅=002rdEBdtμε=Solving for BFor r<rplate2002platerdEBrdtμε=For r>rplateSolving for B00EddtμεΦEA=2Erπ()200dErdtπμε=200dErdtμεπEA=2plateErπ()200platedErdtπμε=200 platedErdtμεπMonday, April 16, 2007 PHYS 1444-004, Spring 2007Dr. Andrew Brandt4Maxwell’s Amazing Leap of Faith• According to Maxwell, a magnetic field will be produced even in an empty space if there is a changing electric field– He then took this concept one step further and concluded that• If a changing magnetic field produces an electric field, the electric field is also changing in time.• This changing electric field in turn produces a magnetic field that also changes• This changing magnetic field then in turn produces the electric field that changes• This process continues– With the manipulation of the equations, Maxwell found that the net result of this interacting changing fields is a wave of electric and magnetic fields that can actually propagate (travel) through thespaceMonday, April 16, 2007 PHYS 1444-004, Spring 2007Dr. Andrew Brandt5Production of EM Waves• Consider two conducting rods connected to a DC power source– What do you think will happen when the switch is closed?• The rod connected to the positive terminal acquires a positive charge and the other a negative one• Then an electric field will be generated between the two rods• Since there is current that flows through the rods, a magnetic field around them will be generated• How far would the electric and magnetic fields extend?– In the static case, the field extends indefinitely– When the switch is closed, the fields are formed near the rods quickly but – The stored energy in the fields won’t propagate w/ infinite speedMonday, April 16, 2007 PHYS 1444-004, Spring 2007Dr. Andrew Brandt6Production of EM Waves• What happens if the antenna is connected to an ac power source?– When the connection is initially made, the rods are charging up quickly w/ the current flowing in one direction as shown in the figure• The field lines form as in the dc case• The field lines propagate away from the antenna– Then the direction of the voltage reverses• New field lines in the opposite direction forms • While the original field lines still propagate farther away from the rod – Since the original field propagates through empty space, the field lines must form a closed loop (no charge exist)• Since changing electric and magnetic fields produce changing magnetic and electric fields, the fields moving outward are self-supporting and do not need antenna with flowing charge– The field far from the antenna is called the radiation field– Both electric and magnetic fields form closed loops perpendicular to each otherMonday, April 16, 2007 PHYS 1444-004, Spring 2007Dr. Andrew Brandt7Properties of Radiation Fields• The fields are propogated throughout all space on both sides of the antenna • The field strengths are greatest in the direction perpendicular to the oscillating charge while along the parallel direction the fields are zero• The magnitudes of E and B in the radiation field decrease with distance ~1/r• The energy carried by the EM wave is proportional to the square of the amplitude, E2or B2– So the intensity of wave decreases as 1/r2Monday, April 16, 2007 PHYS 1444-004, Spring 2007Dr. Andrew Brandt8Properties of Radiation Fields• The electric and magnetic fields at any point are perpendicular to each other and to the direction of motion • The fields alternate in direction– The field strengths vary from maximum in one direction, to 0 and to maximum in the opposite direction• The electric and magnetic fields are in phase• Very far from the antenna, the field lines are pretty flat over a reasonably large area– Called plane wavesMonday, April 16, 2007 PHYS 1444-004, Spring 2007Dr. Andrew Brandt9EM Waves• If the voltage of the source varies sinusoidally, the field strengths of the radiation field vary sinusoidally• We call these waves EM waves•
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