Chapter 23: Electromagnetic Waves Chapter 31 opener. These circular disk antennas, each 25 m in diameter, are pointed to receive radio waves from out in space. Radio waves are electromagnetic (EM) waves that have frequencies from a few hundred Hz to about 100 MHz. These antennas are connected together electronically to achieve better detail; they are a part of the Very Large Array in New Mexico searching the heavens for information about the Cosmos. We will see in this Chapter that Maxwell predicted the existence of EM waves from his famous equations. Maxwell’s equations themselves are a magnificent summary of electromagnetism. We will also examine how EM waves carry energy and momentum.

1. Changing Magnetic Fields Produce Electric Fields Chapter Outline 1. Changing Magnetic Fields Produce Electric Fields Faraday’s Law! (Ch. 21)

Chapter Outline Maxwell’s Displacement Current 1. Changing Magnetic Fields Produce Electric Fields 2. Modification of Ampère’s Law: Maxwell’s Displacement Current (Changing Electric Fields Produce Magnetic Fields) Faraday’s Law! (Ch. 21)

Chapter Outline Maxwell’s Displacement Current 1. Changing Magnetic Fields Produce Electric Fields 2. Modification of Ampère’s Law: Maxwell’s Displacement Current (Changing Electric Fields Produce Magnetic Fields) 3. Gauss’s Law for Magnetic Fields Magnetic “Charge” Doesn’t Exist! Faraday’s Law! (Ch. 21)

Chapter Outline Maxwell’s Displacement Current 1. Changing Magnetic Fields Produce Electric Fields 2. Modification of Ampère’s Law: Maxwell’s Displacement Current (Changing Electric Fields Produce Magnetic Fields) 3. Gauss’s Law for Magnetic Fields Magnetic “Charge” Doesn’t Exist! 4. Gauss’s Law for Electric Fields Faraday’s Law! (Ch. 21)

Chapter Outline “Maxwell’s Equations” 1. Changing Magnetic Fields Produce Electric Fields 2. Modification of Ampère’s Law: Maxwell’s Displacement Current (Changing Electric Fields Produce Magnetic Fields) 3. Gauss’s Law for Magnetic Fields Magnetic “Charge” Doesn’t Exist! 4. Gauss’s Law for Electric Fields Faraday’s Law! (Ch. 21) “Maxwell’s Equations”

“Maxwell’s Equations” 1. Changing Electric Fields Produce Magnetic Fields 2. Modification of Ampère’s Law: Maxwell’s Displacement Current (Changing Electric Fields Produce Magnetic Fields) 3. Gauss’s Law for Magnetic Fields 4. Gauss’s Law for Electric Fields “Maxwell’s Equations” From these come production of Electromagnetic Waves & their Speed And Light is an Electromagnetic Wave!

Derived from Maxwell’s Equations:

Derived from Maxwell’s Equations: Light is an Electromagnetic Wave

Derived from Maxwell’s Equations: Light is an Electromagnetic Wave The Electromagnetic Spectrum

Derived from Maxwell’s Equations: Light is an Electromagnetic Wave The Electromagnetic Spectrum Measuring The Speed of Light

Derived from Maxwell’s Equations: Electromagnetic Waves Light is an Electromagnetic Wave The Electromagnetic Spectrum Measuring The Speed of Light Energy & Momentum in Electromagnetic Waves

Derived from Maxwell’s Equations: Electromagnetic Waves Light is an Electromagnetic Wave The Electromagnetic Spectrum Measuring The Speed of Light Energy & Momentum in Electromagnetic Waves The Poynting Vector

Derived from Maxwell’s Equations: Electromagnetic Waves Light is an Electromagnetic Wave The Electromagnetic Spectrum Measuring The Speed of Light Energy & Momentum in Electromagnetic Waves The Poynting Vector Radiation Pressure

Derived from Maxwell’s Equations: Electromagnetic Waves Light is an Electromagnetic Wave The Electromagnetic Spectrum Measuring The Speed of Light Energy & Momentum in Electromagnetic Waves The Poynting Vector Radiation Pressure Radio & Television

Derived from Maxwell’s Equations: Electromagnetic Waves Light is an Electromagnetic Wave The Electromagnetic Spectrum Measuring The Speed of Light Energy & Momentum in Electromagnetic Waves The Poynting Vector Radiation Pressure Radio & Television Wireless Communication

Electromagnetic Theory The theoretical understanding of electricity & magnetism seemed complete by around 1850: Coulomb’s Law & Gauss’ Law explained electric fields & forces Ampère’s Law & Faraday’s Law explained magnetic fields & forces These laws were verified in many experiments

Unanswered Questions (1850) What is the nature of electric & magnetic fields? What is the idea of action at a distance? How fast do the field lines associated with a charge react to a movement in the charge? James Clerk Maxwell studied some of these questions in the mid-1800’s His work led to the discovery of electromagnetic waves

Discovery of EM Waves A time-varying magnetic field causes the creation of an electric field

Discovery of EM Waves A time-varying magnetic field causes the creation of an electric field A magnetic field can produce an electric field

Discovery of EM Waves A time-varying magnetic field causes the creation of an electric field A magnetic field can produce an electric field Maxwell proposed a modification to Ampère’s Law such that a time-varying electric field produces a magnetic field

Discovery of EM Waves A time-varying magnetic field causes the creation of an electric field A magnetic field can produce an electric field Maxwell proposed a modification to Ampère’s Law such that a time-varying electric field produces a magnetic field This gives a new way to create a magnetic field

Discovery of EM Waves A time-varying magnetic field causes the creation of an electric field A magnetic field can produce an electric field Maxwell proposed a modification to Ampère’s Law such that a time-varying electric field produces a magnetic field This gives a new way to create a magnetic field It also gives the equations of electromagnetism some symmetry

A changing electric flux produces a magnetic field. Symmetry of E and B The correct form of Ampère’s Law (due to Maxwell): A changing electric flux produces a magnetic field. Since a changing electric flux can be caused by a changing E, this was an indication that a changing electric field produces a magnetic field

A changing magnetic field produces an electric field Symmetry of E and B Faraday’s Law says that a changing magnetic flux produces an induced emf, & an emf is always associated with an electric field Since a changing magnetic flux can be caused by a changing B, we can also say that A changing magnetic field produces an electric field

Symmetry of E and B

Maxwell’s Ideas & Reasoning Consider Faraday’s Law: Figure 31-2. Ampère’s law applied to two different surfaces bounded by the same closed path.

Maxwell’s Ideas & Reasoning Consider Faraday’s Law: A time dependent Magnetic Field induces an Electric Field. Figure 31-2. Ampère’s law applied to two different surfaces bounded by the same closed path.

Maxwell’s Ideas & Reasoning Consider Faraday’s Law: A time dependent Magnetic Field induces an Electric Field. Question: In an analogy with Faraday’s Law, does a time dependent Electric Field induce a Magnetic Field? Figure 31-2. Ampère’s law applied to two different surfaces bounded by the same closed path.

Maxwell’s Ideas & Reasoning Consider Faraday’s Law: A time dependent Magnetic Field induces an Electric Field. Question: In an analogy with Faraday’s Law, does a time dependent Electric Field induce a Magnetic Field? The answer, based on experiment, is YES!!! Figure 31-2. Ampère’s law applied to two different surfaces bounded by the same closed path.

Maxwell’s Ideas & Reasoning Consider Faraday’s Law: A time dependent Magnetic Field induces an Electric Field. Question: In an analogy with Faraday’s Law, does a time dependent Electric Field induce a Magnetic Field? The answer, based on experiment, is YES!!! So, Ampère’s Law needs to be modified to account for this!!! Figure 31-2. Ampère’s law applied to two different surfaces bounded by the same closed path.

Maxwell’s Ideas & Reasoning Consider Faraday’s Law: A time dependent Magnetic Field induces an Electric Field. Question: In an analogy with Faraday’s Law, does a time dependent Electric Field induce a Magnetic Field? The answer, based on experiment, is YES!!! So, Ampère’s Law needs to be modified to account for this!!! The modification is to add a time dependent Electric Field to the right side of Ampère’s Law Figure 31-2. Ampère’s law applied to two different surfaces bounded by the same closed path.

Also: Consider Gauss’s Law: Figure 31-2. Ampère’s law applied to two different surfaces bounded by the same closed path.

Also: Consider Gauss’s Law: Applies to Electric Fields & is about the E field produced by electric charges. Figure 31-2. Ampère’s law applied to two different surfaces bounded by the same closed path.

Also: Consider Gauss’s Law: Applies to Electric Fields & is about the E field produced by electric charges. Question: Is there an analogous Law for Magnetic Fields? Figure 31-2. Ampère’s law applied to two different surfaces bounded by the same closed path.

Also: Consider Gauss’s Law: Applies to Electric Fields & is about the E field produced by electric charges. Question: Is there an analogous Law for Magnetic Fields? The answer, based on experiment, is YES!!! Figure 31-2. Ampère’s law applied to two different surfaces bounded by the same closed path.

Also: Consider Gauss’s Law: Applies to Electric Fields & is about the E field produced by electric charges. Question: Is there an analogous Law for Magnetic Fields? The answer, based on experiment, is YES!!! However, experiments also show that there are no “Magnetic Charges” (Magnetic Monopoles) which are analogous to electric charges. Figure 31-2. Ampère’s law applied to two different surfaces bounded by the same closed path.

B = 0 For any CLOSED surface! Also: Consider Gauss’s Law: Applies to Electric Fields & is about the E field produced by electric charges. Question: Is there an analogous Law for Magnetic Fields? The answer, based on experiment, is YES!!! However, experiments also show that there are no “Magnetic Charges (Magnetic Monopoles)” which are analogous to electric charges. So, Gauss’s Law for Magnetic Fields is B = 0 For any CLOSED surface! Figure 31-2. Ampère’s law applied to two different surfaces bounded by the same closed path.

B(2r) = 0Iencl for any closed loop This relates the magnetic field B Changing Electric Fields Produce Magnetic Fields: Ampère’s Law & Displacement Current Maxwell’s Generalization of Ampère’s Law A wire is carrying current I. Recall Ampère’s Law: B(2r) = 0Iencl for any closed loop This relates the magnetic field B around a current to the current Iencl through a surface Figure 31-2. Ampère’s law applied to two different surfaces bounded by the same closed path.

Faraday’s Law: So, changing Magnetic Fields produce currents Changing Electric Fields Produce Magnetic Fields: Ampère’s Law & Displacement Current Maxwell’s Generalization of Ampère’s Law Faraday’s Law: “The emf induced in a circuit is equal to the time rate of change of magnetic flux through the circuit.” So, changing Magnetic Fields produce currents & thus Electric Fields Figure 31-2. Ampère’s law applied to two different surfaces bounded by the same closed path.

Maxwell’s reasoning about Ampère’s Law: For Ampere’s Law to hold, it can’t matter which surface is chosen. But look at a discharging capacitor; there is current through surface 1 but there is none through surface 2: Figure 31-3. A capacitor discharging. A conduction current passes through surface 1, but no conduction current passes through surface 2. An extra term is needed in Ampère’s law.

Maxwell’s reasoning about Ampère’s Law: Therefore, Maxwell modified Ampère’s Law to include the creation of a magnetic field by a changing electric field. This is analogous to Faraday’s Law which says that electric fields can be produced by changing magnetic fields. In the case shown, the electric field between the plates of the capacitor is changing & so a magnetic field must be produced between the plates. Figure 31-3. A capacitor discharging. A conduction current passes through surface 1, but no conduction current passes through surface 2. An extra term is needed in Ampère’s law.

Maxwell’s reasoning about Ampère’s Law: Results in a new version of Ampère’s Law: B(2r) = E B(2r) = Figure 31-3. A capacitor discharging. A conduction current passes through surface 1, but no conduction current passes through surface 2. An extra term is needed in Ampère’s law. B(2r) = B(2r) = B(2r) =

Example: Charging A Capacitor. A C = 30-pF air-gap capacitor has circular plates of area A = 100 cm2. It is charged by a V = 70-V battery through a R = 2.0-Ω resistor. At the instant the battery is connected, the electric field between the plates is changing most rapidly. At this instant, calculate (a) The current into the plates, and (b) The rate of change of the E field between the plates. (c) The B field induced between the plates. Assume that E is uniform between the plates at any instant & is zero at all points beyond the edges of the plates. Solution: a. At t = 0, all the voltage is across the resistor, so the current is V/R = 35 A. b. The field at any instant is (Q/A)/ε0. So dE/dt = (dQ/dt)/(Aε0) = (I/A)/ε0 = 4.0 x 1014 V/m·s. c. Due to symmetry, the lines of B are circular and perpendicular to E. If E is constant over the area of the plates, Ampere’s law gives B = μ0ε0r0/2 dE/dt = 1.2 x 10-4 T.

Capacitor with Circular Plates

Solution: Circular Plate Capacitor A C = 30-pF air-gap capacitor has circular plates of area A = 100 cm2. Charged by a V0 = 70-V battery through a R = 2.0-Ω resistor. At the instant the battery is connected, the E field between the plates is changing most rapidly. At this instant, calculate (a) The current into the plates. We know, for charge Q on capacitor plates, time dependence is: I = (Q/t). At t = 0, this must be given by Ohm’s Law: I = (V0/R) = 35A

Solution: Circular Plate Capacitor A C = 30-pF air-gap capacitor has circular plates of area A = 100 cm2. Charged by a V0 = 70-V battery through a R = 2.0-Ω resistor. At the instant the battery is connected, the E field between the plates is changing most rapidly. At this instant, calculate (b) The rate of change of the E field between the plates. The E field between two closely spaced conductors is: E = /0 = Q(t)/(0A). So (E/t) = (Q/t)/(0A) at t = 0 (E/t) = I/(0A) = 4  1014 V/(m s)

Solution: Circular Plate Capacitor A C = 30-pF air-gap capacitor has circular plates of area A = 100 cm2. Charged by a V0 = 70-V battery through a R = 2.0-Ω resistor. At the instant the battery is connected, the E field between the plates is changing most rapidly. At this instant, calculate (c) The B field induced between the plates. Use Ampere’s Law with Displacement Current to solve this. B(2πr) = 0μ0(E/t) = 0μ0(πr2)(E/t) B = 0μ0(½)r (E/t) at outer radius r = 5.6 cm B = 1.2  10-4 T

The second term in Ampere’s Law, first written by Maxwell, has the dimensions of current (after factoring out μ0), & is sometimes called the Displacement Current: B(2r) = B(2r) = B(2r) = E  where t

Gauss’s Law for Magnetism Gauss’s law relates the electric field on a closed surface to the net charge enclosed by that surface. The analogous law for magnetic fields is different, as there are no single magnetic point charges (monopoles): BA BA

Maxwell’s Equations Maxwell’s Equations. In the absence of We now have a complete set of equations that describe electric and magnetic fields, called Maxwell’s Equations. In the absence of dielectric or magnetic materials, they are: EA BA B Eℓ = t B(2r) = E B(2r) = t

Production of Electromagnetic Waves Since a changing electric field produces a magnetic field, and a changing magnetic field produces an electric field, once sinusoidal fields are created, they can propagate on their own. These propagating fields are called electromagnetic waves.

Oscillating charges will produce electromagnetic waves: Close to the antenna, the fields are complicated, and are called the near field: Figure 31-6. Fields produced by charge flowing into conductors. It takes time for the and fields to travel outward to distant points. The fields are shown to the right of the antenna, but they move out in all directions, symmetrically about the (vertical) antenna.

Far from the source, the waves are plane waves: Figure 31-8. (a) The radiation fields (far from the antenna) produced by a sinusoidal signal on the antenna. The red closed loops represent electric field lines. The magnetic field lines, perpendicular to the page, also form closed loops. (b) Very far from the antenna the wave fronts (field lines) are essentially flat over a fairly large area, and are referred to as plane waves.

The electric & magnetic waves are perpendicular to each other, & to the propagation direction. Figure 31-9. Electric and magnetic field strengths in an electromagnetic wave. E and B are at right angles to each other. The entire pattern moves in a direction perpendicular to both E and B.