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MSU PHY 184 - PHY184-Physics for Scientists & Engineers 2

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March 18, 2005 Physics for Scientists&Engineers 2 1Physics for Scientists &Physics for Scientists &EngineersEngineers 22Spring Semester 2005Lecture 31March 18, 2005 Physics for Scientists&Engineers 2 2Electromagnetic WavesElectromagnetic Waves! This week we will study electromagnetic waves! We will see that light is an electromagnetic wave! Electromagnetic waves have electric and magnetic fields! We will see Maxwell’s Equations that describeelectromagnetic phenomena! We will see that the speed of light is constant and can berelated to !0 and µ0! We will see that electromagnetic waves can transportenergy and momentum! Electromagnetic waves can be polarizedMarch 18, 2005 Physics for Scientists&Engineers 2 3InducedInduced MagneticMagnetic FieldsFields! We have seen that a changing magnetic field induces and electricfield! Faraday’s Law of Induction tells us! In a similar manner, a changing electric field induces a magneticfield! Maxwell’s Law of Induction tells us! where B is the magnetic field induced in a closed loop by achanging magnetic field by a changing electric flux "E in that loop !E • d!s"!= "d#Bdt !B • d!s"!=µ0"0d#EdtMarch 18, 2005 Physics for Scientists&Engineers 2 4Circular CapacitorCircular Capacitor! To illustrate induced magnetic fields, consider the exampleof a circular parallel plate capacitor! We charge the capacitor and disconnect the battery! The charge is constant and the electric field between theplates is constant! There is no magnetic fieldMarch 18, 2005 Physics for Scientists&Engineers 2 5Circular Capacitor (2)Circular Capacitor (2)! Now let’s increase the charge as a function of time! A magnetic field is induced as indicated by the blue lines inthe direction indicated! The magnitude of the induced magnetic field is the samealong each line and the direction is tangential to the lineMarch 18, 2005 Physics for Scientists&Engineers 2 6Circular Capacitor (3)Circular Capacitor (3)! Now let’s consider a constant magneticfield! Now let’s increase the magnitude of themagnetic while keeping the magnetic fielduniform in space and in the same direction! An electric field is induced as shown bythe red loops! The magnitude of the electric field isconstant along each loop and thedirection is tangential to each loop! Note that the induced electric fieldpoints in the opposite direction from themagnetic field induced by a changingelectric fieldMarch 18, 2005 Physics for Scientists&Engineers 2 7Circular Capacitor (4)Circular Capacitor (4)! We now recall Ampere’s Law! relating the integral around a loop of the dot product of the magnetic fieldand the integration direction to the current flowing through the loop! We see that we can combine Maxwell’s Law of Induction and Ampere’s Lawto produce a description of magnetic fields created by moving charges andby changing electric fields! Which is called the Maxwell-Ampere Law (not surprisingly!)• For the case of constant current, such as current flowing in a conductor, thisequation reduces to Ampere’s Law• For the case of a changing electric field without current flowing, such as theelectric field between the plates of a capacitor, this equation reduces to theMaxwell Law of Induction !B • d!s"!=µ0ienc !B • d!s"!=µ0"0d#Edt+µ0iencMarch 18, 2005 Physics for Scientists&Engineers 2 8Displacement CurrentDisplacement Current! Looking at the Maxwell-Ampere Law! one can see that the quantity must have the units of current! This term has been called the displacement current! Note however that no actual current is being displaced! We can then rewrite the Maxwell-Ampere Law as !B • d!s"!=µ0"0d#Edt+µ0ienc!0d"Edtid=!0d"Edt !B • d!s"!=µ0(id+ ienc)March 18, 2005 Physics for Scientists&Engineers 2 9! Now let’s consider a parallel plate capacitor with circular plates as wedid earlier! We place the capacitor in a circuit in which a current i is flowing whilethe capacitor is charging! For a parallel plate capacitor with area A we can relate the charge q tothe electric field EDisplacement Current (2)Displacement Current (2)q =!0AEMarch 18, 2005 Physics for Scientists&Engineers 2 10! We can get the current by taking the time derivative of the charge! Assuming that the electric field between the plates of the capacitor isuniform we can obtain an expression for the displacement current! The current in the circuit is the same as the displacement current id! Although there is no actual current flowing between the plates of thecapacitor in the sense that no actual charges flow across the capacitorgap from one plate to the other, we can use the concept ofdisplacement current to calculate the induced magnetic fieldDisplacement Current (3)Displacement Current (3)i =dqdt=!0AdEdtid=!0d"Edt=!0d AE( )dt=!0AdEdtMarch 18, 2005 Physics for Scientists&Engineers 2 11! To calculate the magnetic field between the two plates of thecapacitor, we assume that the volume between the two plates can bereplaced with a conductor of radius R carrying current id! Thus from chapter 27 we know that the magnetic field at a distancefrom the center of the capacitor is given by! Outside the capacitor we can treat thesystem as a current-carrying wire! The magnetic field isDisplacement Current (4)Displacement Current (4)B =µ0id2!R2"#$%&'rB =µ0id2!rMarch 18, 2005 Physics for Scientists&Engineers 2 12MaxwellMaxwell’’ss EquationsEquations! The Maxwell-Ampere Law completes the explanation of the four equationsknown as Maxwell’s Equations that describe electromagnetic phenomena! We have used these equations to describe electric fields, magnetic fields, andcircuits! We now will apply these equations to electromagnetic wavesName Equation Description Gauss’ Law for Electric Fields !E • d!A =qenc!0"" Relates the net electric flux to the net enclosed electric charge Gauss’ Law for Magnetic Fields !B • d!A""= 0 States that the net magnetic flux is zero (no magnetic charge) Faraday’s Law !E • d!s""= #d$Bdt Relates the induced electric field to the changing magnetic flux Ampere-Maxwell Law !B • d!s""=µ0!0d$Edt+µ0ienc Relates the induced magnetic field to the changing electric flux and to the currentMarch 18, 2005 Physics for Scientists&Engineers 2 13Wave Solutions to MaxwellWave Solutions to Maxwell’’ss EquationsEquations! It is possible to derive a general wave equation from Maxwell’sEquations! Here we will assume that electromagnetic waves propagating in vacuum(no moving charges or currents) have


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MSU PHY 184 - PHY184-Physics for Scientists & Engineers 2

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