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Transcript

Discussion Notes and Class Agenda Physics 240 December 5, 2005 Problem Assignments Problem Groups 32.52 1, 3 32.17 5, 7 32.10 2, 4 32.13 6, 8 Special Event The U of M Society of Physics Students Presents: The Golden Age of Ballooning: Particle Astrophysics Aboard High Altitude Balloons Dr. Andrew Tomasch Tuesday December 6, 2005 7:00 pm in the Don Meyer Commons (Next to 335 West Hall) Refreshments will be served Class Agenda Physics 240 December 7, 2005 Problem Assignments Problem Groups 32.7 1, 3 32.15 5, 7 32.51 2, 4 32.41 6, 8 Discussion: Electromagnetic Waves Motivation: Displacement Current and Maxwell’s Equations It was known in Maxwell’s time that 1 0 0 c speed of light in vacuum=3 108 m/s Maxwell first realized that formulation of Ampere’s Law was incomplete and required the addition of a term proportional to the time derivative of the electric field called the displacement current. With the displacement current included in Ampere’s Law, it is possible for a time varying electric field to induce a magnetic field in a manner analogous to Faraday Induction, where a changing magnetic field induces an electric field. This process is now called Maxwell Induction. Once the displacement current is included, the two induction equations become symmetric and imply that electric and magnetic fields can propagate in the absence of charges and currents. Once the displacement current had been included in the equations governing electromagnetism, Maxwell was able to derive a traveling wave equation from Faraday’s Law and 2 Ampere’s Law in the absence of charges or currents due to moving charges which are shown below E dl d d B dA B dt dt d 1 d d E dA E dA 0 0 E 0 I Disp dt c 2 dt dt B dl 0 0 E dl d B (Faraday) dt B dl 1 d E (Maxwell) c 2 dt With the displacement current I Disp included in Maxwell’s equations, the combination 0 0 1 c2 naturally appears in the equation for Maxwell induction. We have studied Maxwell’s equation in integral form. To derive a traveling wave equation, Maxwell actually worked with the equations in differential form. While we will not study this formulation in this course, it is still informative to display Maxwell’s equations in differential form E E B t 0 B 0 E B 0 J 0 t J the current density. We have already mentioned the divergence operation ( ), for example ( E ), where is the charge density and which measures how the flux of a vector field flows into or out of a 3 region of space. A region with a nonzero divergence contains a source or sink for the field (charge in the case of electrostatic fields). The curl operation ( ), for example ( B ), similarly measures the circulation of a vector field--its tendency to form closed loops. If you think of the velocity field for water flowing in a river, a region with a nonzero divergence contains either a spring (source) of water or a drain (sink). A region of a flowing river with a nonzero curl will cause a paddle wheel to turn if placed in the flow. Whirlpools have a nonzero curl. It is very easy to obtain the integral form of Maxwell’s equations from the differential form by integrating both sides of each equation over a volume (Gauss’ Law for E and B ) or over a surface (Faraday induction and Ampere’s Law with displacement current) and then applying two theorems from vector calculus, the Divergence Theorem and Stokes’ Theorem respectively. You will soon learn this approach in your study of vector calculus. The point to emphasize is that this is very straightforward to learn and offers powerful new techniques and insights for the further study of Maxwell’s equation and electromagnetism. Electromagnetic Waves Having stated that a wave equation can be derived from Maxwell’s equations and that solutions to this equation propagate in vacuum at the speed of light, we will now study the properties of these waves. The discussion thus far has been restricted to waves propagating in vacuum. We will also include the correct prescription to describe electromagnetic waves traveling in matter. Section 32.2 of Young and Freedman demonstrates the derivation of the wave equation from the integral formulation of Maxwell’s equations. The derivation involves taking limits where the paths over which the loop integrals for Faraday’s Law and Ampere’s Law approach zero size. This amounts to employing the differential form of Maxwell’s equations. The waves are assumed to be plane polarized, traveling in the +x direction. The electric field oscillates in the +y direction and the magnetic field in the +z direction. For this set of assumptions, the resulting wave equations for E and B are 4 2 E y 2 E y 2 1 Ey 0 0 2 x 2 t 2 c t 2 2 Bz 2 Bz 1 2 Bz 0 0 2 2 x 2 t c t 2 which have solutions E x, t Emax cos(kx t ) yˆ B x, t Bmax cos(kx t ) zˆ as illustrated below in Figure 32.10 This solution describes transverse traveling wave with linear polarization in the y direction, that is, the electric field is always aligned in the y direction. Other polarization states are possible, for example z polarization, with the electric field oriented in the z direction and the magnetic field in the –y direction. In fact, any general electromagnetic wave can be described as a superposition of these two linear polarization states. The quantity k is called the wave number and has value 2/ where is the wavelength. The wave number can also be expressed as a 5 vector, called the wave vector, which is oriented in the direction of propagation. For the y-polarized wave k 2 ( Eˆ Bˆ ) 2 xˆ The electric and magnetic fields are related to each other as E cB B 0 o cE The assigned problems demand complete familiarity with the relationships between frequency, period, wavelength, angular frequency, speed of propagation and wave number for traveling waves. For electromagnetic waves propagating in vacuum c f (2 f ) 2 2 f 2 k k 2 1 c 0 0 For waves propagating in matter, the speed of light in vacuum must be replaced with the propagation speed in matter v v f (2 f ) 2 f v 1 2 2 k k 2 1 KK M 0 0 n c v 6 c c KK M n where K and KM are the relative permiativity and permeability of the material. The maximum electric and magnetic field are then related by E vB B vB The quantity n is called the refractive index of the material. The fact that electromagnetic waves travel with a lower speed in matter than in vacuum as characterized by the refractive index gives rise to many of the effects in the science of optics. It is the refractive index that enables lenses to focus light and produce images. Energy and Momentum for Electromagnetic Waves The Poynting vector describes the instantaneous power per unit area transmitted by a traveling electromagnetic wave (energy/area-time) S 1 0 EB S EB 0 The total energy flow per time exiting a closed surface is therefore P S dA Closed Surface Thus the flux of the Poynting vector out of a closed surface yields the total power (energy/time) exiting the surface in the form of electromagnetic radiation. The Poynting vector is time dependant, and we must substitute the time varying forms of the electric and magnetic fields to evaluate it at any instant in time. The time average of the Poynting vector is called the intensity and is given in vacuum by 7 2 Emax Bmax Emax 1 0 2 1 2 I S Sav Emax 0 cEmax 2 0 2 0 c 2 0 2 To describe waves propagating in matter, the quantities replaced by , ,v 0 , 0 ,c are 2 Emax Bmax Emax 1 2 1 2 I Emax vEmax 2 2v 2 2 1 1 K 0c 2 1 0c 2 2 I vEmax Emax Emax 2 2 KK M 2 KM 2 2 Emax KK M 0 0 1 K Emax I 2v 2 K M 0 2 KM 2 2 Emax KK M Emax I 2v 2 K M 0 c 0 2 Emax 0 2 K Emax K M 2 0 c Finally, electromagnetic waves carry momentum as well as energy. The momentum density (momentum per unit volume) is dp EB S dV 0 c 2 c 2 The momentum transferred per unit area per unit time is 8 1 dp S EB A dt c 0 c The average rate of momentum transfer per area is obtained by substituting the intensity (average value of S) S 1 dp I av A dt c c Since momentum transfer per time is equivalent to a force, the traveling electromagnetic wave will exert a radiation pressure (force/area=momentum transfer/time-area) on a surface if it is absorbed or reflected on a surface. For a surface perpendicular to the propagation direction Sav I total absorbtion c c 2S 2I av total reflection c c prad prad The factor of two difference between reflection and absorption is easy to see. Recall from mechanics that in a perfectly inelastic collision a particle with momentum p transfers its total momentum to the target. If however the collision is perfectly elastic, the projectile recoils from the target with momentum –p and the momentum transferred to the target is -(pf - pi)=2p and twice the momentum is transferred in the collision. Total absorption is therefore equivalent to a perfectly inelastic collision, and the total reflection is equivalent to a perfectly elastic collision. 9