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Differential Form of Gauss' Law (Sec. 22.8) Think about a region of space, enclosed by a box. Divide Gauss' law by the volume of the box: E || x Take the limit of a small box Work on the left hand side of the equation: For a general case where E can point in any direction: GAUSS' LAW Differential Form where “divergence“ (div)

Differential Form of Ampere's Law (Sec. 22.9) 3 Write I in terms of current density J: 4 2 Divide Ampere's Law by a very small ΔA: 1 Current I out of the board In our geometry, n = z

Differential Form of Ampere's Law (Sec. 22.9) 3 We divided Ampere's Law by a very small ΔA, and got this: 4 2 1 Current I out of the board Definition of derivative! Now work on the left hand side: "Crossed derivative"

Differential Form of Ampere's Law (Sec. 22.9) 3 We divided Ampere's Law by a very small ΔA, and got this: 4 2 1 Current I out of the board For a loop in any direction, this can be re-expressed as: AMPERE'S LAW Differential Form

Curl: Here's the Math + ( - ) set up the answer copy 1st two colums

Curl: Here's the Math + ( - ) Blast from the Past! Lecture 12

Maxwell's Equations – The Full Story Divergence Curl GAUSS' LAW (Magnetism) FARADAY'S LAW AMPERE'S LAW Flux Circulation

Maxwell's Equations – No Charges In the ABSENCE of "sources" = charges, currents: GAUSS' LAW (Magnetism) FARADAY'S LAW AMPERE'S LAW This says once a wave starts, it keeps going!

Maxwell's Equations – No Charges What happens if we feed one equation into the other? Use This

Maxwell's Equations – No Charges What happens if we feed one equation into the other?

Maxwell's Equations – No Charges How do you solve a Differential Equation? Know the answer! (Ask Wolfram Alpha)  This is a WAVE EQUATION, with speed c Using similar ideas, you can show that E obeys the same equation:

Wave Description – wavelength: distance between crests (meters) T – period: the time between crests passing fixed location (seconds) v – speed: the distance one crest moves in a second (m/s) f – frequency: the number of crests passing fixed location in one second (1/s or Hz)  – angular frequency: 2f: (rad/s)

Wave: Variation in Time and Space After one period T – we will get the point which was at coordinate -lambda ‘-’ sign: the point on wave moves to the right

Wave: Phase Shift But E @ t=0 and x =0, may not equal E0 After one period T – we will get the point which was at coordinate -lambda Two waves are ‘out of phase’ (Shown for x=0)

Wave: Amplitude and Intensity E0 is a parameter called amplitude (positive). Time dependence is in cosine function Often we detect ‘intensity’, or energy flux ~ E2. Intensity I (W/m2): What if we triple the amplitude of the wave? Works also for other waves, such as sound or water waves.

Interference Superposition principle: The net electric field at any location is vector sum of the electric fields contributed by all sources. Laser: source of radiation which has the same frequency (monochromatic) and phase (coherent) across the beam. Two slits are sources of two waves with the same phase and frequency. We have first considered electric field single charges and then moved to multiple charges. We have now considered single source of em wave – and now we will consider em field produced by multiple sources. Simplest case – two. Since E for each depends on time situation is not as trivial. Laser – source of radiation which has basicqally single wave (same frequency and phase across the beam). Note – more complex, intensity of bands decreases from center to sides. What can we expect to see on the screen? Can particle model explain the pattern?

Interference: Constructive Two emitters: E1 E2 Fields in crossing point Two emitters (radio antennas, or laser beams) Superposition: Amplitude increases twice: constructive interference

Interference: Energy E1 E2 Two emitters: What about the intensity (energy flux)? Energy flux increases 4 times while two emitters produce only twice more energy Two emitters (radio antennas, or laser beams) There must be an area in space where intensity is smaller than that produced by one emitter

Interference: Destructive Two emitters (radio antennas, or laser beams) Two waves are 1800 out of phase: destructive interference

Two-Slit Experiment with Bullets Bullets arrive in lumps We measure the probability of arrival of a lump (bullet) P1 = probability bullet went through slit 1 in arriving at x P12 = P1 + P2

Two-Slit Experiment with Waves We measure the Intensity of the wave motion at the detector (related to the square of the wave height)

Two-Slit Experiment with Electrons Electrons arrive in clicks - no “half-clicks” Clicks come erratically As detector moves to a different position, rate at which clicks occur is faster or slower Adding a second detector, we would note that either one or the other would click - never both at once P12 = |1 + 2|2 There are some points at which very few electrons arrive when both holes are open. Closing one hole increases the number arriving at these places! So closing one hole increases the number arriving from the other. On the other hand, the number arriving at the central max (with both holes open) is more than twice that arriving when a single hole is open. So here, it is as if closing one hole decreases the number arriving from the other!

Conclusion Electrons arrive in lumps like particles and the probability of arrival of these lumps is distributed like the distribution of intensity of a wave. It is in this sense that an electron behaves sometimes like a particle and sometimes like a wave. - R. Feynman Vol 3 Lecture Series

Energy and Wavelength Photon: a ‘wave packet’, EM wave which occupies a short region Experimental observations: h = 6.6.10-34 J.s Planck constant This relationship was first introduced by Max Planck in 1900 to explain the spectrum of black body radiation. Historic moment: birth of quantum theory In 1905 Einstein proposed this for explaining photoelectric effect Nobel price in 1918