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Chapter 34 The Laws of Electromagnetism Maxwell’s Equations Displacement Current Electromagnetic Radiation

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The Electromagnetic Spectrum Radio waves -wave infra -red -rays x-rays ultra -violet

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The Equations of Electromagnetism (at this point …) Gauss’ Law for Electrostatics Gauss’ Law for Magnetism Faraday’s Law of Induction Ampere’s Law

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1 2 The Equations of Electromagnetism..monopole.. ?...there’s no magnetic monopole....!! Gauss’s Laws

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4 The Equations of Electromagnetism 3.. if you change a magnetic field you induce an electric field........... if you change a magnetic field you induce an electric field................is the reverse true..?.......is the reverse true..? Faraday’s Law Ampere’s Law

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...lets take a look at charge flowing into a capacitor......when we derived Ampere’s Law we assumed constant current... E B

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...lets take a look at charge flowing into a capacitor... E...when we derived Ampere’s Law we assumed constant current..... if the loop encloses one plate of the capacitor..there is a problem … I = 0 B Side view: (Surface is now like a bag:) E B

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Maxwell solved this problem by realizing that.... B E Inside the capacitor there must be an induced magnetic field... How?.

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Maxwell solved this problem by realizing that.... B E x x x x x x x x x A changing electric field induces a magnetic field Inside the capacitor there must be an induced magnetic field... How?. Inside the capacitor there is a changing E E B

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Maxwell solved this problem by realizing that.... B E x x x x x x x x x A changing electric field induces a magnetic field Inside the capacitor there must be an induced magnetic field... How?. Inside the capacitor there is a changing E where I d is called the displacement current E B

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Maxwell solved this problem by realizing that.... B E x x x x x x x x x A changing electric field induces a magnetic field Inside the capacitor there must be an induced magnetic field... How?. Inside the capacitor there is a changing E where I d is called the displacement current Therefore, Maxwell’s revision of Ampere’s Law becomes.... E B

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Derivation of Displacement Current For a capacitor, and. Now, the electric flux is given by EA, so:, where this current, not being associated with charges, is called the “Displacement current”, I d. Hence: and:

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Derivation of Displacement Current For a capacitor, and. Now, the electric flux is given by EA, so:, where this current, not being associated with charges, is called the “Displacement Current”, I d. Hence: and:

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Maxwell’s Equations of Electromagnetism Gauss’ Law for Electrostatics Gauss’ Law for Magnetism Faraday’s Law of Induction Ampere’s Law

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Maxwell’s Equations of Electromagnetism in Vacuum (no charges, no masses) Consider these equations in a vacuum...........no mass, no charges. no currents.....

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Maxwell’s Equations of Electromagnetism in Vacuum (no charges, no masses)

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Electromagnetic Waves Faraday’s law: dB/dt electric field Maxwell’s modification of Ampere’s law dE/dt magnetic field These two equations can be solved simultaneously. The result is: E(x, t) = E P sin (kx- t) B(x, t) = B P sin (kx- t) ˆ z ˆ j

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B E Electromagnetic Waves

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B E Special case..PLANE WAVES... satisfy the wave equation Maxwell’s Solution Electromagnetic Waves

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Plane Electromagnetic Waves x EyEy BzBz E(x, t) = E P sin (kx- t) B(x, t) = B P sin (kx- t) ˆ z ˆ j c

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Static wave F(x) = F P sin (kx + ) k = 2 k = wavenumber = wavelength F(x) x Moving wave F(x, t) = F P sin (kx - t ) = 2 f = angular frequency f = frequency v = / k F(x) x v

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x v Moving wave F(x, t) = F P sin (kx - t ) What happens at x = 0 as a function of time? F(0, t) = F P sin (- t) F For x = 0 and t = 0 F(0, 0) = F P sin (0) For x = 0 and t = t F (0, t) = F P sin (0 – t) = F P sin (– t) This is equivalent to: kx = - t x = - ( /k) t F(x=0) at time t is the same as F[x=-( /k)t] at time 0 The wave moves to the right with speed /k

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Plane Electromagnetic Waves x EyEy BzBz Notes: Waves are in Phase, but fields oriented at 90 0. k= Speed of wave is c= /k (= f At all times E=cB. E(x, t) = E P sin (kx- t) B(x, t) = B P sin (kx- t) ˆ z ˆ j c

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MAXWELL’S EQUATIONS 1. 2 Maxwell’s Equations in differential form.

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