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Maxwell’s Equations & Electromagnetic Waves We now want to explore how the principles of Electric & Magnetic Fields lead to an understanding of light & other electromagnetic waves. The connection between EM Theory and Optics may be a surprise. It was a surprise to physicists also.

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Towards a complete description of EM phenomena... The century from ~1750 to ~ 1850 saw intensive investigation of electricity & magnetism by Coulomb, Galvani, Volta, Oersted, Ampere, Faraday and many others. (You should recognize most of these names. Also, there was the mathematician Gauss, whose theorems were crucial to the new theory of fields.) Finally there was James Clerk Maxwell, a theoretician who showed that all EM phenomena could be described by a succinct but comprehensive theory. This theory is known as Maxwell’s Equations. In addition to developing the theoretical foundation of the subject, Maxwell was the first to recognize the connection between EM fields and light.

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Towards a complete description of EM phenomena... We have essentially been following the historical development in this course. Just like the physicists, we have discussed: Gauss’s Law (E-flux ~ Charge inside a volume ) Gauss’ Law for Magnetism (B-flux always = 0 since there are no magnetic charges – no sources of magnetic fields. It is sources that provide starting or ending points for field lines.) Ampere’s Law (B-field ~ Current through a surface) Faraday’s Law (EMF around a loop ~ time rate of change of B-flux) These are written out below...

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Up to now we have... Gauss’ Law Gauss’ Law for Magnetism Ampere’s Law Faraday’s Law

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We also need... I out Δt + Δq = 0 Equation of Continuity –The only way the charge density in a region can change is that a current flows out. –This is just a re-writing of the definition of current I = Δq/Δt in terms of current out of the region. –It also expresses the fact that the total charge in the universe does not change. F EM = q(E + v x B) Lorentz Force Law -Once we know the fields, we can find the force on charged objects. -This is a combination of electric force & magnetic force. + + + + Current out

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Towards a complete description of EM phenomena... Following Maxwell, we now take two steps to refine these principles: 1.First, let’s rewrite Faraday’s Law in terms of the electric field. Doing this reveals a near symmetry between E and B. 2.Second, correct an ambiguity in Ampere’s Law. Doing this will complete the symmetry.

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Faraday’s Law in terms of E An EMF implies an electric field. If a current “i” flows in a conducting loop around a changing B-flux, there must be a ΔV around the loop. Recall that Or:

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Faraday’s Law in terms of E That is, for each small segment we travel around the loop, the potential changes by. We can identify this with an increment in the EMF: If we sum up around the loop we get the full EMF:

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Faraday’s Law in terms of E So: Faraday’s Law But this implies the E-field lines form loops around a region of changing B-flux. (Diagram next slide). The loops do not start and end on charges. This is surprising. We have not seen this before. Also, nothing in Faraday’s Law says there has to be a conductor there: The loops exist even in empty space.

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Electric Field loops from changing B

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Now, reconsider Ampere’s Law What about a charging capacitor?

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Note: Charging capacitor There are B-fields around the currents (i C ). But no current runs through the gap. Ampere’s Law implies there is no B-field around the gap. But as the capacitor charges, there is a changing E-field, ΔE/Δt, between the plates. Should there not also be a B-field around the gap? ( The diagram shows such a field.)

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Maxwell’s postulate Maxwell postulated that there must such a B-field, and that the “current” that creates it is a changing Electric flux. We can use Gauss’ Law to see this. Set up a Gaussian cylinder with one capacitor plate at the end (yellow in diagram): q inside = charge on capacitor plate. E = field between plates. Gauss: But q inside is changing in time. The field E must be changing also. Take (Δ/Δt) of each side: I into E

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Maxwell’s postulate Sum over the Gaussian volume and simplify to: ε Δ(EA)/Δt = I into the volume (The long subscript is necessary, because the current does not run through the area A. A is area of an Ampere loop between the plates.) This shows that the quantity on the left is equivalent to a current. Maxwell called it a displacement current. It is not a physical current. He then broke the current “I” in Ampere’s law into two parts: I = I physical + I displacement : And he rewrote the law using the above:

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Maxwell’s postulate This is Maxwell’s revision of Ampere’s Law: A current or a changing Electric Flux implies a B-field. To see the symmetry between B and E, compare Faraday and Ampere when there is no current (I = 0): The term shows the relation between E and B (See below)

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Maxwell’s Equations Here’s the full set of Maxwell’s Equations These, plus the equation of continuity and the Lorentz force law, give a complete description of EM phenomena.

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Maxwell’s Prediction of EM Waves These equations imply that E-fields and B-fields are coupled. We call this coupling the Electro-magnetic field. The equations also make a prediction: A disturbance in the EM field will propagate through space at speed c, where c is: This has the value of 3.00 x 10^8 m/s. This is the speed of light. In this way, it was determined that light is an EM wave.

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An EM Wave This is an ideal case, corresponding to what is known as a plane wave (wave fronts are planes) Note that both the E- and B-fields can be described by sine waves.

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EM Waves For a real-time example, run this applet. Note these points: The E-fields & B-fields are at right angles to each other; both are at right angles to the direction of propagation (x) Remember that the E & B vectors do not occupy space. (The applet is misleading in this.) New E & B vectors are being generated at new positions x according to x = ct. http://facstaff.gpc.ed u/~dpenly/1112/em Wave/emWave.htmlhttp://facstaff.gpc.ed u/~dpenly/1112/em Wave/emWave.html Here an oscillating charge generates a changing E- field. This induces a changing B- field, which induces a changing E-field, and so on to infinity.

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Generating EM Waves How do Maxwell’s equations imply EM waves? An explicit demonstration requires vector calculus. (We need theorems on sums over curves & volumes; also, the equations are inherently 3-dimensional.) But think of it this way: Faraday’s Law: A variation of E in space implies a variation of B in time. Ampere-Maxwell: A variation of B in space implies a variation of E in time. Combine the two and we get wave equations for E and B:

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Generating EM Waves Waves on strings, sound waves, and other waves are described by equations of the same form. Equations with this same form (but using different variables) are called wave equations – they are well known in other branches of physics. The term “c” is always the speed of the wave. What do such equations do? –They specify the form that a description of the wave must have. –For instance a description of an EM plane wave traveling along the x-axis would be: –These satisfy Maxwell’s Equations. They state the way the magnitudes of E and B vary with x and with t.

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Traveling Wave Applet http://facstaff.gpc.edu/~dpenly/1112/wav e_point_move01.gif

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Notes on Traveling E-Wave The diagram could represent the Electric Field component of an EM wave. Vertical scale: Amplitude, or magnitude of E. (The diagram is scaled to (+1, -1)) E 0 is the maximum amplitude of E. Horizontal scale: position, x. λ is the wavelength c is the wave speed Note thatc = λ/T = λf (T is the period.) The small circles in the diagram show that at any point x, the amplitude of the E-field simply oscillates between maximum and minimum values.

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Notes on Traveling E-Wave -The terms inside the sine function simply give the phase of the wave at that point in x and t. -Here are equivalent ways to write this:

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Notes on Traveling E-Wave -You can verify these alternate forms by using: -We also define the wave number k and the angular frequency ω:

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Examples 1.The E-field in a plane wave varies as : E = 150sin(ax-bt), with: a = 1.9 per meterb = 566 megahertz Describe this wave. 150 is E0: It must have units of Newtons/Coulomb or Volts/meter. 1.9 is the wavenumber: So the wavelength is 3.3 meters. 566 x 10^6 is the angular frequency: So f = 90.1 MHz. This is radio station WABE!

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Examples 2.The E-field in a plane wave has a maximum amplitude of 3.2 milli-volts/meter and a wavelength of 520 nanometers. What is its value at x= 150 meters and t = 2.0 microseconds? The argument of the sine function is 2π(x-ct)/λ = -5.54 x 10^9 The sine function returns the value (-.866) So the E-field has the value 3.2mV x (-.866) = -2.77 mV. Why does not x = ct in this? Because 2 microseconds is long after the leading edge of the wave went through this position. The leading edge got there at.5 microseconds, if we assume the wave started at t = 0. The wave train continues to pass through after this time.

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Real EM Waves... Are not so simple. Below is a diagram of EM fields produced by a radio antenna. The “near fields” are very complex. Far away, “Franuhoffer Region”, they become plane waves. Note: H = μ 0 B

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EM Spectrum Wavelengths in EM Waves can have any value. Only a tiny portion of the spectrum is visible to our eyes.

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Maxwell’s Equations. - Gauss’s law, which is generalized form of Coulomb’s law, that relates electric fields to electric charges.

Maxwell’s Equations. - Gauss’s law, which is generalized form of Coulomb’s law, that relates electric fields to electric charges.

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