Electromagnetic Waves Chapter 34

James Clerk Maxwell Maxwell’s Theory Electricity and magnetism were originally thought to be unrelated Maxwell’s theory showed a close relationship between all electric and magnetic phenomena and proved that electric and magnetic fields play symmetric roles in nature Maxwell hypothesized that a changing electric field would produce a magnetic field He calculated the speed of light – 3x10 8 m/s – and concluded that light and other electromagnetic waves consist of fluctuating electric and magnetic fields

James Clerk Maxwell Maxwell’s Theory Stationary charges produce only electric fields Charges in uniform motion (constant velocity) produce electric and magnetic fields Charges that are accelerated produce electric and magnetic fields and electromagnetic waves A changing magnetic field produces an electric field A changing electric field produces a magnetic field These fields are in phase and, at any point, they both reach their maximum value at the same time

Modifications to Ampère’s Law Ampère’s Law is used to analyze magnetic fields created by currents But this form is valid only if any electric fields present are constant in time Maxwell modified the equation to include time-varying electric fields and added another term, called the displacement current, I d This showed that magnetic fields are produced both by conduction currents and by time-varying electric fields

Maxwell’s Equations In his unified theory of electromagnetism, Maxwell showed that the fundamental laws are expressed in these four equations:

Maxwell’s Equations Gauss’ Law relates an electric field to the charge distribution that creates it The total electric flux through any closed surface equals the net charge inside that surface divided by  o

Maxwell’s Equations Gauss’ Law in magnetism: the net magnetic flux through a closed surface is zero The number of magnetic field lines that enter a closed volume must equal the number that leave that volume If this wasn’t true, there would be magnetic monopoles found in nature

Maxwell’s Equations Faraday’s Law of Induction describes the creation of an electric field by a time-varying magnetic field The emf (the line integral of the electric field around any closed path) equals the rate of change of the magnetic flux through any surface bounded by that path

Maxwell’s Equations Ampère-Maxwell Law describes the creation of a magnetic field by a changing electric field and by electric current The line integral of the magnetic field around any closed path is the sum of  o times the net current through that path and    o times the rate of change of electric flux through any surface bounded by that path

Maxwell’s Equations Once the electric and magnetic fields are known at some point in space, the force acting on a particle of charge q can be found Maxwell’s equations with the Lorentz Force Law completely describe all classical electromagnetic interactions

Maxwell’s Equations In empty space, q = 0 and I = 0 The equations can be solved with wave-like solutions (electromagnetic waves), which are traveling at the speed of light This result led Maxwell to predict that light waves were a form of electromagnetic radiation

Electromagnetic Waves From Maxwell’s equations applied to empty space, the following relationships can be found: The simplest solutions to these partial differential equations are sinusoidal waves – electromagnetic waves: The speed of the electromagnetic wave is:

Plane Electromagnetic Waves The vectors for the electric and magnetic fields in an em wave have a specific space- time behavior consistent with Maxwell’s equations Assume an em wave that travels in the x direction We also assume that at any point in space, the magnitudes E and B of the fields depend upon x and t only The electric field is assumed to be in the y direction and the magnetic field in the z direction

Plane Electromagnetic Waves The components of the electric and magnetic fields of plane electromagnetic waves are perpendicular to each other and perpendicular to the direction of propagation Thus, electromagnetic waves are transverse waves Waves in which the electric and magnetic fields are restricted to being parallel to a pair of perpendicular axes are said to be linearly polarized waves

John Henry Poynting 1852 – 1914 Poynting Vector Electromagnetic waves carry energy As they propagate through space, they can transfer that energy to objects in their path The rate of flow of energy in an em wave is described by a vector, S, called the Poynting vector defined as: Its direction is the direction of propagation and its magnitude varies in time The SI units: J/(s. m 2 ) = W/m 2 Those are units of power per unit area

Poynting Vector Energy carried by em waves is shared equally by the electric and magnetic fields The wave intensity, I, is the time average of S (the Poynting vector) over one or more cycles When the average is taken, the time average of cos 2 (kx - ωt) = ½ is involved

Energy Density The energy density, u, is the energy per unit volume It can be shown that The instantaneous energy density associated with the magnetic field of an em wave equals the instantaneous energy density associated with the electric field and in a given volume this energy is shared equally by E and B The total instantaneous energy density is the sum of the energy densities associated with each field: u =u E + u B = ε o E 2 = B 2 / μ o

Energy Density When this is averaged over one or more cycles, the total average becomes u avg = ε o (E 2 ) avg = ½ ε o E 2 max = B 2 max / 2μ o The intensity of an em wave equals the average energy density multiplied by the speed of light I = S avg = cu avg Electromagnetic waves transport linear momentum as well as energy As this momentum is absorbed by some surface, pressure is exerted on the surface

Chapter 34 Problem 6 An electron moves through a uniform electric field E = (2.50 i^ j^) V/m and a uniform magnetic field B = (0.400 k^) T. Determine the acceleration of the electron when it has a velocity v = 10.0 i^ m/s.

Hertz’s Experiment In 1887 Hertz was the first to experimentally generate and detect electromagnetic waves An induction coil was connected to two large spheres forming a capacitor Oscillations were initiated by short voltage pulses by the coil As the air in the gap is ionized, it becomes a better conductor At a very high frequencies the discharge between the electrodes exhibited an oscillatory behavior Heinrich Rudolf Hertz 1857 – 1894

Hertz’s Experiment The inductor and capacitor formed the transmitter, equivalent to an LC circuit from a circuit viewpoint Several meters away from the transmitter was the receiver (a single loop of wire connected to two spheres) with its own inductance and capacitance When the resonance frequencies of the transmitter and receiver matched, energy transfer occurred between them Heinrich Rudolf Hertz 1857 – 1894

Hertz’s Results Hertz hypothesized the energy transfer was in the form of waves (now known to be electromagnetic waves) Hertz confirmed Maxwell’s theory by showing the waves existed and had all the properties of light waves (with different frequencies and wavelengths) Hertz measured the speed of the waves from the transmitter (used the waves to form an interference pattern and calculated the wavelength) The measured speed was very close to 3 x 10 8 m/s, the known speed of light, which provided evidence in support of Maxwell’s theory

Electromagnetic Waves Produced by an Antenna Neither stationary charges nor steady currents can produce electromagnetic waves The fundamental mechanism responsible for this radiation: when a charged particle undergoes an acceleration, it must radiate energy in the form of electromagnetic waves Electromagnetic waves are radiated by any circuit carrying alternating current An alternating voltage applied to the wires of an antenna forces the electric charge in the antenna to oscillate

Electromagnetic Waves Produced by an Antenna Half-wave antenna: two rods are connected to an ac source, charges oscillate between the rods (a) As oscillations continue, the rods become less charged, the field near the charges decreases and the field produced at t = 0 moves away from the rod (b) The charges and field reverse (c) and the oscillations continue (d)

Electromagnetic Waves Produced by an Antenna Because the oscillating charges in the rod produce a current, there is also a magnetic field generated As the current changes, the magnetic field spreads out from the antenna The magnetic field lines form concentric circles around the antenna and are perpendicular to the electric field lines at all points The antenna can be approximated by an oscillating electric dipole

The Spectrum of EM Waves Types of electromagnetic waves are distinguished by their frequencies (wavelengths): c = ƒ λ There is no sharp division between one kind of em wave and the next – note the overlap between types of waves

The Spectrum of EM Waves Radio waves are used in radio and television communication systems Microwaves (1 mm to 30 cm) are well suited for radar systems + microwave ovens are an application Infrared waves are produced by hot objects and molecules and are readily absorbed by most materials

The Spectrum of EM Waves Visible light (a small range of the spectrum from 400 nm to 700 nm) – part of the spectrum detected by the human eye Ultraviolet light (400 nm to 0.6 nm): Sun is an important source of uv light, however most uv light from the sun is absorbed in the stratosphere by ozone

The Spectrum of EM Waves X-rays – most common source is acceleration of high-energy electrons striking a metal target, also used as a diagnostic tool in medicine Gamma rays: emitted by radioactive nuclei, are highly penetrating and cause serious damage when absorbed by living tissue

Chapter 34 Problem 11 In SI units, the electric field in an electromagnetic wave is described by E y = 100 sin (1.00 × 10 7 x – ωt). Find (a) the amplitude of the corresponding magnetic field oscillations, (b) the wavelength λ, and (c) the frequency f.

Answers to Even Numbered Problems Chapter 34: Problem nT

Properties of em Waves, 3 Electromagnetic waves obey the superposition principle