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Today... Speed of light Maxwell’s Displacement Current Charge conservation Modification of Ampere’s Law Modified Maxwell equations → waves!! Text Reference: Chapter 32.1, 2, and 4, 33.4 Examples: 32.1 and 2

Speed of Light An aside with foreshadowing < Galileo (1667): Light speed is infinite? Galileo’s hilltop experiment: 1675: Roemer looks at eclipse of moons of Jupiter, depending on where Earth is → v is finite ( v ≈ 2.3∙10 8 m/s ) 1860: Foucault (also Fizeau) - change mirror rotation speed - no signal unless mirror rotates 1/8 in ΔT c → 2.98∙10 8 m/s 3 km Response time ~0.1 s C > 3km / 1s > 30,000 m/s 2002: Physics 112 → Back to Galileo

Our story thus far: J. C. Maxwell (~1860) realizes that the equations of electricity & magnetism are inconsistent! Also there is a lack of symmetry → In the case of no sources: Why do these look different? Maybe a changing flux in E should → B ?

Maxwell’s Displacement Current 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 ! (i.e., there is no current between the plates of the capacitor) circuit Big Idea: In order to have for surface 1 = for surface 3 Maxwell proposed there was an extra “displacement current” in the region between the plates, equal to the current in the wire→ Modified Ampere’s law:

Maxwell’s Displacement Current Recall def’n of flux: But where does the “displacement current” come from?! Although there is no actual charge moving between the plates, nevertheless, something is changing – the electric field between them! The Electric Field E between the plates of the capacitor is determined by the charge Q on the plate of area A : E = Q/(A  0 ) → Q = E A  0 Because there is current flowing through the wire, there must be a change in the charge on the plates: Modified Ampere’s Law:

Lecture 21, 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)  II =  I (b)  II =  I /2 (c) impossible 1A  R C I  R/2 2C2C I I II I 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 ro/2ro/2 roro I I ro/2ro/2 I II

Lecture 21, 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)  II =  I (b)  II =  I /2 (c) impossible 1A Both circuits have the same time constant  = RC. Therefore, if the currents are equal at any time, they are equal at all times. At t=0, the capacitors look like wires. For circuit I :For circuit II : These currents are equal if:  R C I  R/2 2C2C I I II I

Lecture 21, 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: ro/2ro/2 ro/2ro/2 roro I I ro/2ro/2 I II

On to Waves!! Note the symmetry now of Maxwell’s Equations in free space, meaning when no charges or currents 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. How? The wave equation is contained in these equations. Do you remember the wave equation????

Step 1 Assume we have a plane wave propagating in z (i.e., E, B not functions of x or y ) 4-step Plane Wave Derivation Example: does this x z y z1z1 z2z2 ExEx ExEx ZZ xx ByBy Step 2 Apply Faraday’s Law to infinitesimal loop in x-z plane

 4-step Plane Wave Derivation x z y z1z1 z2z2 ByBy ZZ yy 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 Now use M. Eqn:

Review of Waves from Physics 111 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. h x A A = amplitude = wavelength f = frequency v = speed k = wave number A specific solution for harmonic waves traveling in the +x direction is:

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 (i.e., 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).

Lecture 21, ACT 2 Snapshots of a wave with angular frequency  are shown at 3 times: –Which 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? 2B2A x y t    2 t  0 t   

Lecture 21, ACT 2 Snapshots of a wave with angular frequency  are shown at 3 times: –Which 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 x y t    2 t  0 t   

Lecture 21, ACT 2 Snapshots of a wave with angular frequency  are shown at 3 times: –Which 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!! x y t    2 t  0 t   

Wave Examples 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 111 days... e.g., sqrt(tension/mass) is wave speed of a guitar string, proportional to frequency of fundamental

Velocity of Electromagnetic Waves We derived the wave equation for E x (Maxwell did it first, in ~1865!): 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 essentially identical to the speed of light measured by Foucault in 1860! –Maxwell identified light as an electromagnetic wave.

How is B related to E ? We derived the wave eqn for E x : 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 B y is in phase with E x B 0 = E 0 / c (Result from step 2 )

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

Lecture 21, ACT 3 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 ?

Lecture 21, ACT 3 Suppose the electric field in an e-m wave is given by: –In 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: Relative signs opposite means + direction

Lecture 21, ACT 3 Suppose the electric field in an e-m wave is given by: –In 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) 3A3B 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,

Properties of electromagnetic waves (e.g., light) Speed: in vacuum, always m/s, no matter how fast the source is moving (there is no “aether”!). In material, the speed can be reduced, usually only by ~1.5, but in 1999 to 17 m/s! In reality, light is often somewhat localized transversely (e.g., a laser) or spreading in a spherical wave (e.g., a star). A plane wave can often be a good approximation (e.g., the wavefronts hitting us from the sun are nearly flat). Direction: The wave described by cos(kx-wt) is traveling in the direction. This is a “plane” wave—extends infinitely in and.

Plane Waves (next time) x z y For any given value of z, the magnitude of the electric field is uniform everywhere in the x-y plane with that z value.

Summary Repaired Ampere’s Law Maxwell’s Displacement Current Combined Faraday’s Law and Ampere’s Law –time varying B-field induces E-field –time varying E-field induces B-field Electromagnetic waves that travel at c = 3 x 10 8 m/s Text Reference: Chapter 32.1, 2, and 4, 33.4 Examples: 32.1 and 2

Next time Properties of E&M waves How to change “colors” The Doppler shift Dipole radiation pattern Reading Assignment: Chapter Examples: