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**Electromagnetic Waves**

Chapter 24 Electromagnetic Waves

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**Unpolarized visible light X-ray**

Electromagnetic waves cover a broad spectrum of wavelengths, with waves in various wavelength ranges having distinct properties. These photos of the Crab Nebula show different structure for observations made with waves of various wavelengths. The photos (clockwise starting from the upper left) were taken with x-rays, unpolarized visible light, radio waves, and visible light passing through a polarizing filter. Radio waves Polarized visible light

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**24.1 Electromagnetic Waves, Introduction**

Electromagnetic (EM) waves permeate our environment EM waves can propagate through a vacuum Much of the behavior of mechanical wave models is similar for EM waves Maxwell’s equations form the basis of all electromagnetic phenomena

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Conduction Current A conduction current is carried by charged particles in a wire The magnetic field associated with this current can be calculated by using Ampère’s Law: The line integral is over any closed path through which the conduction current passes

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**Conduction Current, cont.**

Ampère’s Law in this form is valid only if the conduction current is continuous in space In the example, the conduction current passes through only S1 but not S2 This leads to a contradiction in Ampère’s Law which needs to be resolved

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James Clerk Maxwell 1831 – 1879 Developed the electromagnetic theory of light Developed the kinetic theory of gases Explained the nature of color vision Explained the nature of Saturn’s rings

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Displacement Current Maxwell proposed the resolution to the previous problem by introducing an additional term called the displacement current The displacement current is defined as

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**Displacement current The electric flux through S2 is EA**

S2 is the gray circle A is the area of the capacitor plates E is the electric field between the plates If q is the charge on the plates, then Id = dq/dt This is equal to the conduction current through S1

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Displacement Current The changing electric field may be considered as equivalent to a current For example, between the plates of a capacitor This current can be considered as the continuation of the conduction current in a wire This term is added to the current term in Ampère’s Law

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Ampère-Maxwell Law The general form of Ampère’s Law is also called the Ampère-Maxwell Law and states: Magnetic fields are produced by both conduction currents and changing electric fields

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**24.2 Maxwell’s Equations, Introduction**

In 1865, James Clerk Maxwell provided a mathematical theory that showed a close relationship between all electric and magnetic phenomena Maxwell’s equations also predicted the existence of electromagnetic waves that propagate through space Einstein showed these equations are in agreement with the special theory of relativity

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**Maxwell’s Equations Gauss’ Law (electric flux)**

Gauss’ Law for magnetism Faraday’s Law of induction Ampère-Maxwell Law The equations are for free space No dielectric or magnetic material is present

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Lorentz Force Once the electric and magnetic fields are known at some point in space, the force of those fields on a particle of charge q can be calculated: The force is called the Lorentz force

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**24.3 Electromagnetic Waves**

In empty space, q = 0 and I = 0 Maxwell predicted the existence of electromagnetic waves The electromagnetic waves consist of oscillating electric and magnetic fields The changing fields induce each other which maintains the propagation of the wave A changing electric field induces a magnetic field A changing magnetic field induces an electric field

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Plane EM Waves We assume that the vectors for the electric and magnetic fields in an EM wave have a specific space-time behavior that is consistent with Maxwell’s equations Assume an EM wave that travels in the x direction with the electric field in the y direction and the magnetic field in the z direction

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**Plane EM Waves, cont The x-direction is the direction of propagation**

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 We assume that at any point in space, the magnitudes E and B of the fields depend upon x and t only

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Figure 24.4: As a plane wave moving in the +x direction passes through a rectangular path of width dx lying in the xy plane, the electric field in the y direction varies from E→ to E→ + dE→. This construction allows us to evaluate the line integral E→ of over the perimeter of the rectangle.

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**Equations of the Linear EM Wave**

From Maxwell’s equations applied to empty space, E and B are satisfied by the following equations These are in the form of a general wave equation, with Substituting the values for mo and eo gives c = x 108 m/s

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**Solutions of the EM wave equations**

The simplest solution to the partial differential equations is a sinusoidal wave: E = Emax cos (kx – wt) B = Bmax cos (kx – wt) The angular wave number is k = 2 p / l l is the wavelength The angular frequency is w = 2 p ƒ ƒ is the wave frequency

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**Ratio of E to B The speed of the electromagnetic wave is**

Taking partial derivations also gives

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Properties of EM Waves The solutions of Maxwell’s are wave-like, with both E and B satisfying a wave equation Electromagnetic waves travel at the speed of light This comes from the solution of Maxwell’s equations

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Properties of EM Waves, 2 The components of the electric and magnetic fields of plane electromagnetic waves are perpendicular to each other and perpendicular to the direction of propagation The electromagnetic waves are transverse waves

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Properties of EM Waves, 3 The magnitudes of the fields in empty space are related by the expression This also comes from the solution of the partial differentials obtained from Maxwell’s Equations Electromagnetic waves obey the superposition principle

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**EM Wave Representation**

This is a pictorial representation, at one instant, of a sinusoidal, linearly polarized plane wave moving in the x direction E and B vary sinusoidally with x

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**Rays A ray is a line along which the wave travels**

All the rays for the type of linearly polarized waves that have been discussed are parallel The collection of waves is called a plane wave A surface connecting points of equal phase on all waves, called the wave front, is a geometric plane

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**Doppler Effect for Light**

Light exhibits a Doppler effect Remember, the Doppler effect is an apparent change in frequency due to the motion of an observer or the source Since there is no medium required for light waves, only the relative speed, v, between the source and the observer can be identified

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Doppler Effect, cont. The equation also depends on the laws of relativity v is the relative speed between the source and the observer c is the speed of light ƒ’ is the apparent frequency of the light seen by the observer ƒ is the frequency emitted by the source

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Doppler Effect, final For galaxies receding from the Earth, v is entered as a negative number Therefore, ƒ’<ƒ and the apparent wavelength, l’, is greater than the actual wavelength The light is shifted toward the red end of the spectrum This is what is observed in the red shift

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