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Physics 1304: Lecture 17, Pg 1 f()x x f(x x z y

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Physics 1304: Lecture 17, Pg 2 Lecture Outline l Electromagnetic Waves: Experimental l Ampere’s Law Is Incomplete: Displacement Current l Review of Wave Properties l Electromagnetic Waves: Theory çMaxwell’s Equations contain the wave equation! çThe velocity of electromagnetic waves = c çThe relationship between E and B in an e-m wave çEnergy in e-m waves: the Poynting vector Text Reference: Chapter 34

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Physics 1304: Lecture 17, Pg 3 Fields from Circuits? l We have been focusing on what happens within the circuits we have been studying (eg currents, voltages, etc.) l What’s happening outside the circuits?? çWe know that: »charges create electric fields and »moving charges (currents) create magnetic fields. çCan we detect these fields? çDemo: Tesla coil & fluorescence bulbs… what’s this all about!

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Physics 1304: Lecture 17, Pg 4 How Do We Understand? l How can we understand the results? çDC current … no detected fields çAC current … fields detected l Answer? çActually we do not yet have all that we need to know to understand this phenomenon. çOne of our basic laws (Ampere’s Law) needs to be modified!! l J. C. Maxwell (~1860) realizes that the equations of electricity & magnetism are inconsistent with the conservation of charge! It is probably not be obvious to you that these equations are inconsistent with conservation of charge, but there is a lack of symmetry here!

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Physics 1304: Lecture 17, Pg 5 Ampere’s Law is the Culprit! l Gauss’ Law: both equations have the same symmetry (the difference is that magnetic charge does not exist!) Ampere’s Law and Faraday’s Law: If Ampere’s Law were correct, the right hand side of Faraday’s Law should be equal to zero -- since no magnetic current. Therefore(?), maybe there is a problem with Ampere’s Law. In fact, Maxwell proposes a modification of Ampere’s Law by adding another term (the “displacement” current) to the right hand side of the equation! ie

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Physics 1304: Lecture 17, Pg 6 Maxwell’s Displacement Current l Can we understand why this “displacement current” has the form it does? Consider applying Ampere’s Law to the current shown in the diagram. If the surface is chosen as 1, 2 or 4, the enclosed current = I If the surface is chosen as 3, the enclosed current = 0! (ie there is no current between the plates of the capacitor) Big Idea: The added term is non-zero in this case, since the current I causes the charge Q on the capacitor to change in time which causes the Electric field in the region between the plates to change in time. The “displacement current” I D = 0 (d E /dt) in the region between the plates = the real current I in the wire. circuit

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Maxwell’s Displacement Current The Electric Field E between the plates of the capacitor is determined by the chrg Q on the plate of area A: E = Q/(A 0 ) What we want is a term that relates E to I without involving A. The answer: the time derivative of the electric flux!! Therefore, if we want I D = I, we need to identify: In order to have for surface 2 to be equal to for surface 3, we want the displacement current in the region between the plates to be equal to the current in the wire. Recall flux:

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Physics 1304: Lecture 17, Pg 8 Lecture 22, ACT 1 At t=0, the switches in circuits I and II are closed and the capacitors start to charge. What must be the relation between I and II such that the current in the circuits are the same at all times? (a) I = II (b) I = 2 II (c) impossible Suppose at time t, the currents flowing into C and 2C are identical and that 2C has twice the area of C as shown. Compare the magnetic fields at the points shown B I (r o /2) and B II (r o /2): (a) B I < B II (b) B I = B II (c) B I > B II 1A I II 1B r o /2 roro I I I II

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Physics 1304: Lecture 17, Pg 9 Lecture 22, ACT 1 At t=0, the switches in circuits I and II are closed and the capacitors start to charge. What must be the relation between I and II such that the current in the circuits are the same at all times? (a) I = II (b) I = 2 II (c) impossible 1A I II Both circuits have the same time constant = RC. Therefore, if the currents are equal at any time, they are equal at all times. The easiest time to evaluate the currents is at t=0. For circuit I : For circuit II : These currents are equal if:

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Lecture 22, ACT 1 Suppose at time t, the currents flowing into C and 2C are identical and that 2C has twice the area of C as shown. Compare the magnetic fields at the points shown B I (r o /2) and B II (r o /2): (a) B I < B II (b) B I = B II (c) B I > B II 1B If the currents flowing to the capacitors are equal, the total displacement currents are also equal. The displacement currents are distributed uniformly throughout the volume between the two plates. Therefore, when we apply our new form of Ampere’s Law to find the magnetic field, the fraction of the displacement current with r<r o /2 is smaller in case II than it is in case I. Therefore, B II < B I. Here are the equations behind these words: r o /2 roro I I I II

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Physics 1304: Lecture 17, Pg 11 On to Waves!! l Note the symmetry of Maxwell’s Equations in free space, i.e. when no charges are present h is the variable that is changing in space (x) and time (t). v is the velocity of the wave We can now predict the existence of electromagnetic waves. Why? The wave equation is contained in these equations. Do you remember the wave equation????

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Review of Waves from Phys 1303 h x A A = amplitude = wavelength f = frequency v = speed k = wave number The one-dimensional wave equation: has a general solution of the form: where h 1 represents a wave traveling in the +x direction and h 2 represents a wave traveling in the -x direction. A specific solution for harmonic waves traveling in the +x direction is:

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Physics 1304: Lecture 17, Pg 13 Movies from 111 Transverse Wave: Note how the wave pattern definitely moves to the right. However any particular point (look at the blue one) just moves transversely (ie up and down) to the direction of the wave. Wave Velocity: The wave velocity is defined as the wavelength divided by the time it takes a wavelength (green) to pass by a fixed point (blue).

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Physics 1304: Lecture 17, Pg 14 Lecture 22, ACT 2 Snapshots of a wave with angular frequency are shown at 3 times: çWhich of the following expressions describes this wave? (a) y = sin(kx- t) (b) y = sin(kx+ t) (c) y = cos(kx+ t) (a) +x direction (b) -x direction In what direction is this wave traveling? 2B 2A

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Physics 1304: Lecture 17, Pg 15 Lecture 22, ACT 2 Snapshots of a wave with angular frequency are shown at 3 times: çWhich of the following expressions describes this wave? (a) y = sin(kx- t) (b) y = sin(kx+ t) (c) y = cos(kx+ t) 2A The t=0 snapshot at t=0, y = sinkx At t= /2 and x=0, (a) y = sin(- /2) = -1 At t= /2 and x=0, (b) y = sin(+ /2) = +1

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Physics 1304: Lecture 17, Pg 16 Lecture 22, ACT 2 Snapshots of a wave with angular frequency are shown at 3 times: çWhich of the following expressions describes this wave? (a) y = sin(kx- t) (b) y = sin(kx+ t) (c) y = cos(kx+ t) In what direction is this wave traveling? (a) +x direction (b) -x direction 2A2B We claim this wave moves in the -x direction. The orange dot marks a point of constant phase. It clearly moves in the -x direction as time increases!!

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Physics 1304: Lecture 17, Pg 17 Wave Examples l Wave on a String: Electromagnetic Wave What is waving?? The Electric & Magnetic Fields !! Therefore, we’ll rewrite Maxwell’s eqns to look for equations of the form: And then identify the velocity of the wave, v, in terms of the available constants 0 and 0. The good old phys 1303 days... e.g., sqrt(tension/mass) is wave speed of a guitar string, proportional to frequency of fundamental

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Physics 1304: Lecture 17, Pg 18 Step 1 Assume we have a plane wave propagating in z (ie E, B not functions of x or y) 4-step Plane Wave Derivation Example: does this x z y z1z1 z2z2 ExEx ExEx ZZ xx ByBy Step 2 Apply Faraday’s Law to infinitesimal loop in x-z plane

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Physics 1304: Lecture 17, Pg 19 4-step Plane Wave Derivation x z y z1z1 z2z2 ByBy ZZ yy ByBy ExEx Step 3 Apply Ampere’s Law to an infinitesimal loop in the y-z plane: Step 4 Combine results from steps 2 and 3 to eliminate B y

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Velocity of Electromagnetic Waves l We derived the wave equation for E x : Therefore, we now know the velocity of electromagnetic waves in free space: Putting in the measured values for 0 & 0, we get: This value is identical to the measured speed of light! We identify light as an electromagnetic wave.

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How is B related to E? l We derived the wave eqn for E x : B y is in phase with E x B 0 = E 0 / c We could have also derived for B y : How are E x and B y related in phase and magnitude? Consider the harmonic solution: where (Result from step 2)

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E & B in Electromagnetic Wave l Plane Harmonic Wave: where: x z y where are the unit vectors in the (E,B) directions. Nothing special about (E x,B y ); eg could have (E y,-B x ) Note: the direction of propagation is given by the cross product Note cyclical relation:

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Physics 1304: Lecture 17, Pg 23 Lecture 22, ACT 3 l Suppose the electric field in an e-m wave is given by: (a) + z direction (b) -z direction Which of the following expressions describes the magnetic field associated with this wave? 3B (a) B x = -(E o /c)cos(kz + t) (b) B x = +(E o /c)cos(kz - t) (c) B x = +(E o /c)sin(kz - t) 3A In what direction is this wave traveling ?

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Physics 1304: Lecture 17, Pg 24 Lecture 22, ACT 3 l Suppose the electric field in an e-m wave is given by: çIn what direction is this wave traveling ? (a) + z direction (b) -z direction 3A To determine the direction, set phase = 0: Therefore wave moves in + z direction! Another way:

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Lecture 22, ACT 3 l Suppose the electric field in an e-m wave is given by: çIn what direction is this wave traveling ? (a) + z direction (b) -z direction Which of the following expressions describes the magnetic field associated with this wave? (a) B x = -(E o /c)cos(kz + t) (b) B x = +(E o /c)cos(kz - t) (c) B x = +(E o /c)sin(kz - t) 3A 3B B is in phase with E and has direction determined from: At t=0, z=0, E y = -E o Therefore at t=0, z=0,

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Energy in Electromagnetic Waves l Electromagnetic waves contain energy. We know already expressions for the energy density stored in E and B fields: The Intensity of a wave is defined as the average power transmitted per unit area = average energy density times wave velocity: For ease in calculation, define Z 0 as: In an e-m wave, B = E/c Therefore, the total energy density in an e-m wave = u, where

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The Poynting Vector l The direction of the propagation of the electromagnetic wave is given by: This wave carries energy. This energy transport is defined by the Poynting vector S as: The direction of S is the direction of propagation of the wave The magnitude of S is directly related to the energy being transported by the wave: The intensity for harmonic waves is then given by:

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