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# General Physics II PHYSICS 240

UM

GPA 3.91

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This 11 page Class Notes was uploaded by Lamont Barton on Thursday October 29, 2015. The Class Notes belongs to PHYSICS 240 at University of Michigan taught by Staff in Fall. Since its upload, it has received 8 views. For similar materials see /class/231648/physics-240-university-of-michigan in Physics 2 at University of Michigan.

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Date Created: 10/29/15

Discussion Notes and Class Agenda Physics 240 February 15 2006 Problem Assignments Problem Groups 26 10 1 3 2652 5 7 2622 2 4 2674 6 8 Discussion Kirchoff s Rules Both the book and Mastering Physics provide good coverage of this topic Here are some key points to remember The Loop Rule The Loop Rule states that the sum of the potentials around any closed loop must be zero This just says that potential gains must equal potential losses in order to end up back at the initial potential at the starting point of the loop This is just another way of stating that the electric force is conservative that is that the potential is uniquely de ned at a point in the circuit The direction chosen for traversing the loop is arbitrary If the actual current in a loop ows opposite the direction initially assigned to it it will solve out to be negative An EMF counts aspositive in a loop sum if it is crossed from its negative lower potential to its positive higher potential pole It counts as negative if it is crossed from its positive higher potential pole to its negative lower potential pole o A resistor counts as negative in a loop sum if it is crossed in the direction of the current through it from higher to lower potential It counts as positive if it is crossed in the direction opposite the current through it The Junction Rule 0 The Junction rule states that the algebraic sum of the currents into and out of anyjunction must total zero This just says that the total current owing into the junction must equal the total current owing out of the junction that is that charge is conserved Take current owing toward the junction as positive and away from the junction as negative These problems are not particularly dif cult conceptually but can be tedious and sometimes laborious Great care needs to be exercised in assigning the algebraic signs to the terms in a loop calculation Often many quantities are to be solved for as unknowns and it is necessary to generate as many equations as unknowns byjudicious choice of loops andjunctions Obviously the loops and junctions must be chosen to include the unknown quantities to be determined One useful technique is to be sure and assign labels to the junctions I use lower case letters and also to label points around the circuit so that loops can easily be speci ed For example abca the loop from a to b to c and back to a The best way to understand Kirchoff s rules is to practice applying them by working problems which we will do today in class You may nd it helpful to commit the following sketches to memory or have them with you on your index card for an exam They might help you to double check your algebraic signs when working a Kirchoff s Rules problem particularly when applying the Loop Rule Discussion Notes and Class Agenda Physics 240 April 10 2006 Problem Assignments Problem Groups 326 1 3 324 5 7 3234 2 4 322 6 8 Discussion Electromagnetic Waves Motivation Displacement Current and Maxwell s Equations It was known in Maxwell s time that c speed oflight in vacuum3 gtlt108 ms V 8010 Maxwell rst realized that formulation of Ampere s Law was incomplete and required the addition of a term proportional to the time derivative of the electric eld called the displacement current With the displacement current included in Ampere s Law it is possible for a time varying electric eld to induce a magnetic eld in a manner analogous to Faraday Induction where a changing magnetic eld induces an electric eld 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 elds 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 Ampere s Law in the absence of charges or currents due to moving charges which are shown below 13 39di 2 5010 39 d a 39 0111 080 E01Disp a a d a a 1d E dl ECDBFaraday EB d c ZE EMaxwell With the displacement current Disp Included In Maxwell s equatlons the combination Eouo 2 naturally appears in the equation for C 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 where 0 is the charge density and J the current density We have already mentioned the divergence operation V for example V E which measures how the ux of a vector eld ows into or out ofa region of space A region with a nonzero divergence contains a source or sink for the eld charge in the case of electrostatic elds The curl operation V X for example V X B similarly measures the circulation of a vector eldits tendency to form closed loops If you think of the velocity eld for water owing in a river a region with a nonzero divergence contains either a spring source of water or a drain sink A region of a owing river with a nonzero curl will cause a paddle wheel to turn if placed in the ow 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 EandB 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 ofvector 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 ofthese 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 322 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 eld oscillates in the y direction and the magnetic eld in the z direction For this set of assumptions the resulting wave equations for E and B are 2 2 2 6E 6Ey1aEy 6sz 6sz 1 6sz 6x2 280 52 72 52 which have solutions Ex I Emax coskx 02 3x I Bmax coskx wt2 as illustrated below in Figure 3210 yicnmponent only B component only 3210 Representation of the electric and magnetic elds as functions ofx for 21 lin early polarized sinusoidal plane electro magnetic wave Onc wavelength of the wave is shown at time 0 The wave is traveling in the positive g diregtion the same as the direction ofE X B This solution describes transverse traveling wave with linear polarization in the y direction that is the electric eld is always aligned in the y direction Other polarization states are possible for example z polarization with the electric eld oriented in the z direction and the magnetic eld 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 21d where A is the wavelength The wave number can also be expressed as a vector called the wave vector which is oriented in the direction of propagation For the y polarized wave 27zEgtltB 27z A A k The electric and magnetic elds are related to each other as E CB B souocE 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 cf127rf a127rf2 k 1 For waves propagating in matter the speed of light in vacuum must be replaced with the propagation speed in matter v xi 0 v 127 f f2 k a27rf2 1622 139 xi V l l c E J VKKMVSOIUO VKKM n c n2 V where K and KM are the relative permiativity and permeability of the material The maximum electric and magnetic eld are then related by EvB 329ng 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 energyareatime iExB E 0 0 The total energy ow per time exiting a closed surface is therefore P 5 dZ1 Closed Surface Thus the ux ofthe Poynting vector out of a closed surface yields the total power energytime 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 elds 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 a E B E 2 l l I Z E SGV max max max 8 0Erilax 606Erax 2 uo 2 uoc 2 uo 2 To describe waves propagating in matter the quantities 80 uo C are replaced by 8D uv E B E2 I max max max 1 3E2 Zlngz 2 2 2 max 2 max 1vE2 lxESOCEZ 1 50c 2 max 2 max 2m max IEriax E1ax KKM801u0 l i 5 0 2 2uv 2KMu0 2 KM uo max Eriax Eilax K Eriax 2uv 2KMuoc KM 2uoc Finally electromagnetic waves carry momentum as well as energy The momentum density momentum per unit volume is dP EB dV uocz c2 The momentum transferred per unit area per unit time is Discussion Notes and Class Agenda Physics 240 January 23 2005 Problem Assignments Problem Groups 2254 1 3 2238 5 7 2224 2 4 2258 6 8 Discussion Gauss s Law Gauss s Law provides a powerful tool for calculating electric elds from charge distributions Gauss s law relates the electric ux through a surface to the charge enclosed within the surface an E dg 80 The double integral sign is to emphasize that the integration is performed over a closed surface which contains a net amount of charge QEndmd inside it The quantity qDE is called the electric ux through the surface It is important to understand that the area element dA is a vector It has magnitude 03914 and its direction is defined by a unit vector normal perpendicular to the area element on the surface Also note that the quantity E dA is a dot product between two vectors In order to properly perform the integral to evaluate the electric ux the vector nature of both the electric eld and the area element must be properly taken into account Gauss s Law can be used in two ways 0 The electric eld can be deduced from a charge distribution by use of a properly chosen Gaussian surface over which to evaluate the integral This usually involves picking a surface that exploits the symmetry of the charge distribution and makes the integral easy to perform 0 The electric ux through an arbitrary surface can be deduced by determining the charge contained within the surface Gauss s law permits the electric eld due to a point a line of charge or a plane of charge to be calculated quickly and easily So easily in fact that it is not necessary to memorize these electric elds Study examples 226 and 227 carefully You should be able to derive the electric eld for a point charge surrounded by a spherical Gaussian surface a line of charge surrounded by a cylindrical Gaussian surface and an infinite plane of charge using a small Gaussian pill box through the surface From Coulomb s law we can de ne the electric eld due to a point charge q at a point in space located at position I7 as 12 472290 r 57 Gauss s law gives this result almost by inspection Center a sphere of radius r about the point charge By symmetry the electric eld and the normal vector on the surface ofthe sphere are everywhere parallel and point radially outward everywhere on the sphere The magnitude of the electric eld is constant on the sphere Gauss s law then yields Q E 80 80 1 q Er r 47z 90r2 where the radial unit vector has been introduced to express Eas a vector pointed radially outward Be sure and be able to get the electric

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