Maxwell’s Equations These equations describe all of Electricity and Magnetism. They are consistent with modern ideas such as relativity. They describe light !
Maxwell’s Equations - Revised In free space, outside the wires of a circuit, Maxwell’s equations reduce to the following. These can be solved (see notes) to give the following differential equations for E and B. These are wave equations. Just like for waves on a string. But here the field is changing instead of the displacement of the string.
Step 1 Assume we have a plane wave propagating in z (ie E, B not functions of x or y) Plane Wave Derivation x z y z1z1 z2z2 ExEx ExEx ZZ xx ByBy Step 2 Apply Faraday’s Law to infinitesimal loop in x-z plane Example: does this
Plane Wave Derivation x z y z1z1 z2z2 ByBy ZZ yy ByBy ExEx Step 3 Apply Ampere’s Law to an infinitesimal loop in the y-z plane: Step 4 Combine results from steps 2 and 3 to eliminate B y
Plane Wave Derivation We derived the wave eqn for E x : B y is in phase with E x B 0 = E 0 / c How are E x and B y related in phase and magnitude? (Result from step 2) We could have also derived for B y : –Consider the harmonic solution: where
Review of Waves from last semester The one-dimensional wave equation: A specific solution for harmonic waves traveling in the +x direction is: has a general solution of the form: where h 1 represents a wave traveling in the +x direction and h 2 represents a wave traveling in the -x direction. h x A A = amplitude = wavelength f = frequency v = speed k = wave number
E & B in Electromagnetic Wave Plane Harmonic Wave: where: y x z Nothing special about (E y,B z ); eg could have (E y,-B x ) Note: the direction of propagation is given by the cross product where are the unit vectors in the (E,B) directions. Note cyclical relation:
Lecture 26, ACT 1 Suppose the electric field in an e-m wave is given by: –In what direction is this wave traveling ? 5A (a) + z direction (b) -z direction (c) +y direction (d) -y direction
Lecture 26, ACT 2 Suppose the electric field in an e-m wave is given by: Which of the following expressions describes the magnetic field associated with this wave? (a) B x = -(E o /c)cos(kz + t) (b) B x = +(E o /c)cos(kz - t) (c) B x = +(E o /c)sin(kz - t) 5B
Velocity of Electromagnetic Waves The wave equation for E x : (derived from Maxwell’s Eqn) Therefore, we now know the velocity of electromagnetic waves in free space: Putting in the measured values for 0 & 0, we get: This value is identical to the measured speed of light! –We identify light as an electromagnetic wave.
The EM Spectrum These EM waves can take on any wavelength from angstroms to miles (and beyond). We give these waves different names depending on the wavelength. Wavelength [m] 10 -14 10 -10 10 -6 10 -2 110 2 10 6 10 10 Gamma Rays Infrared Microwaves Short Wave Radio TV and FM Radio AM Radio Long Radio Waves Ultraviolet Visible Light X Rays
Lecture 26, ACT 3 Consider your favorite radio station. I will assume that it is at 100 on your FM dial. That means that it transmits radio waves with a frequency f=100 MHz. What is the wavelength of the signal ? A) 3 cmB) 3 mC) ~0.5 mD) ~500 m
Energy in EM Waves / review Electromagnetic waves contain energy which is stored in E and B fields: The Intensity of a wave is defined as the average power transmitted per unit area = average energy density times wave velocity: Therefore, the total energy density in an e-m wave = u, where =
Momentum in EM Waves Electromagnetic waves contain momentum. The momentum transferred to a surface depends on the area of the surface. Thus Pressure is a more useful quantity. If a surface completely absorbs the incident light, the momentum gained by the surface is, We use the above expression plus Newton’s Second Law in the form F=dp/dt to derive the following expression for the Pressure, If the surface completely reflects the light, conservation of momentum indicates the light pressure will be double that for the surface that absorbs.
The Poynting Vector The direction of the propagation of the electromagnetic wave is given by: This wave carries energy. This energy transport is defined by the Poynting vector S as: –The direction of S is the direction of propagation of the wave –The magnitude of S is directly related to the energy being transported by the wave: The intensity for harmonic waves is then given by:
The Poynting Vector Thus we get some useful relations for the Poynting vector. 1.The direction of propagation of an EM wave is along the Poynting vector. 2.The Intensity of light at any position is given by the magnitude of the Poynting vector at that position, averaged over a cycle. I = S avg 3.The light pressure is also given by the average value of the Puynting vector as, P = S/cAbsorbing surface P = 2S/cReflecting surface
Generating E-M Waves Static charges produce a constant Electric Field but no Magnetic Field. Moving charges (currents) produce both a possibly changing electric field and a static magnetic field. Accelerated charges produce EM radiation (oscillating electric and magnetic fields). Antennas are often used to produce EM waves in a controlled manner.
A Dipole Antenna V(t)=V o cos( t) x z y time t=0 ++++ ---- E time t= /2 E time t= / one half cycle later ---- ++++
dipole radiation pattern oscillating electric dipole generates e-m radiation that is polarized in the direction of the dipole radiation pattern is doughnut shaped & outward traveling –zero amplitude directly above and below dipole –maximum amplitude in-plane proportional to sin( t)
Receiving E-M Radiation receiving antenna One way to receive an EM signal is to use the same sort of antenna. Receiving antenna has charges which are accelerated by the E field of the EM wave. The acceleration of charges is the same thing as an EMF. Thus a voltage signal is created. Speaker y x z
Lecture 26, ACT 4 Consider an EM wave with the E field POLARIZED to lie perpendicular to the ground. y x z In which orientation should you turn your receiving dipole antenna in order to best receive this signal? C) Along E a) Along S b) Along B
Loop Antennas Magnetic Dipole Antennas The electric dipole antenna makes use of the basic electric force on a charged particle Note that you can calculate the related magnetic field using Ampere’s Law. We can also make an antenna that produces magnetic fields that look like a magnetic dipole, i.e. a loop of wire. This loop can receive signals by exploiting Faraday’s Law. For a changing B field through a fixed loop
Lecture 26, ACT 5 Consider an EM wave with the E field POLARIZED to lie perpendicular to the ground. y x z In which orientation should you turn your receiving loop antenna in order to best receive this signal? a) â Along S b) â Along B C) â Along E