Physics 1502: Lecture 26 Today s Agenda
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1 Phsics 1502: Lecture 26 Toda s Agenda Announcements: Midterm 2: NOT Nov. 6 About Monda Nov. 16 Homework 07: due Frida this week Electromagnetic Waves Mawell s Equations - Revised Energ and Momentum in Waves f( f( ) 1
2 Mawell s Equations These equations describe all of Electricit and Magnetism. The are consistent with modern ideas such as relativit. The describe light! Mawell s Equations - Revised In free space, outside the wires of a circuit, Mawell 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. 2
3 Step 1 Plane Wave Derivation Assume we have a plane wave propagating in (ie E, B not functions of or ) Eample: does this Step 2 Appl Farada s Law to infinitesimal loop in - plane E E Δ B 1 2 ΔZ Plane Wave Derivation Step 3 Appl Ampere s Law to an infinitesimal loop in the - plane: B E ΔZ 1 2 Δ B Step 4 Combine results from steps 2 and 3 to eliminate B!! 3
4 Plane Wave Derivation We derived the wave eqn for E : We could have also derived for B : How are E and B related in phase and magnitude? Consider the harmonic solution: (Result from step 2) where B is in phase with E B 0 = E 0 / c Review of Waves from last semester The one-dimensional wave equation: has a general solution of the form: where h 1 represents a wave traveling in the + direction and h 2 represents a wave traveling in the - direction. A specific solution for harmonic waves traveling in the + direction is: h λ A A = amplitude λ = wavelength f = frequenc v = speed k = wave number 4
5 E & B in Electromagnetic Wave Plane Harmonic Wave: where: Note: the direction of propagation is given b the cross product where are the unit vectors in the (E,B) directions. Nothing special about (E,B ); eg could have (E,-B ) Note cclical relation: Lecture 26, ACT 1 Suppose the electric field in an e-m wave is given b: 5A In what direction is this wave traveling? (a) + direction (c) + direction (b) - direction (d) - direction 5
6 Lecture 26, ACT 2 Suppose the electric field in an e-m wave is given b: 5B Which of the following epressions describes the magnetic field associated with this wave? (a) B = -(E o /c)cos(k + ωt) (b) B = +(E o /c)cos(k - ωt) (c) B = +(E o /c)sin(k - ωt) Velocit of Electromagnetic Waves The wave equation for E : (derived from Mawell s Eqn) Therefore, we now know the velocit of electromagnetic waves in free space: Putting in the measured values for µ 0 & ε 0, we get: This value is identical to the measured speed of light! We identif light as an electromagnetic wave. 6
7 The EM Spectrum These EM waves can take on an wavelength from angstroms to miles (and beond). We give these waves different names depending on the wavelength. Gamma Ras X Ras Ultraviolet Visible Light Infrared Microwaves Short Wave Radio TV and FM Radio AM Radio Wavelength [m] Long Radio Waves Lecture 26, ACT 3 Consider our favorite radio station. I will assume that it is at 100 on our FM dial. That means that it transmits radio waves with a frequenc f=100 MH. What is the wavelength of the signal? A) 3 cm B) 3 m C) ~0.5 m D) ~500 m 7
8 Energ in EM Waves / review Electromagnetic waves contain energ which is stored in E and B fields: = Therefore, the total energ densit in an e-m wave = u, where The Intensit of a wave is defined as the average power transmitted per unit area = average energ densit times wave velocit: 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 quantit. If a surface completel absorbs the incident light, the momentum gained b the surface is, We use the above epression plus Newton s Second Law in the form F=dp/dt to derive the following epression for the Pressure, If the surface completel reflects the light, conservation of momentum indicates the light pressure will be double that for the surface that absorbs. 8
9 The Ponting Vector The direction of the propagation of the electromagnetic wave is given b: This wave carries energ. This energ transport is defined b the Ponting vector S as: The direction of S is the direction of propagation of the wave The magnitude of S is directl related to the energ being transported b the wave: The intensit for harmonic waves is then given b: The Ponting Vector Thus we get some useful relations for the Ponting vector. 1. The direction of propagation of an EM wave is along the Ponting vector. 2. The Intensit of light at an position is given b the magnitude of the Ponting vector at that position, averaged over a ccle. I = S avg 3. The light pressure is also given b the average value of the Punting vector as, P = S/c Absorbing surface P = 2S/c Reflecting surface 9
10 Generating E-M Waves Static charges produce a constant Electric Field but no Magnetic Field. Moving charges (currents) produce both a possibl 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. V(t)=V o cos(ωt) + + A Dipole Antenna - - E - - E + + time t=0 time t=π/2ω time t=π/ω one half ccle later 10
11 dipole radiation pattern proportional to sin(ωt) oscillating electric dipole generates e-m radiation that is polaried in the direction of the dipole radiation pattern is doughnut shaped & outward traveling ero amplitude directl above and below dipole maimum amplitude in-plane Receiving E-M Radiation receiving antenna Speaker One wa to receive an EM signal is to use the same sort of antenna. Receiving antenna has charges which are accelerated b the E field of the EM wave. The acceleration of charges is the same thing as an EMF. Thus a voltage signal is created. 11
12 Lecture 26, ACT 4 Consider an EM wave with the E field POLARIZED to lie perpendicular to the ground. In which orientation should ou turn our receiving dipole antenna in order to best receive this signal? a) Along S b) Along B C) Along E Loop Antennas Magnetic Dipole Antennas The electric dipole antenna makes use of the basic electric force on a charged particle Note that ou 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 b eploiting Farada s Law. For a changing B field through a fied loop 12
13 Lecture 26, ACT 5 Consider an EM wave with the E field POLARIZED to lie perpendicular to the ground. In which orientation should ou turn our receiving loop antenna in order to best receive this signal? a) â Along S b) â Along B C) â Along E 13
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