Electromagnetic Waves

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1 Lecture 20 Chapter 34 Physics II Electromagnetic Waves Course website:

2 Let s finish climbing our EM mountain. Maxwell s equations

Let s revisit Ampere s Law for current I

Let s apply Ampere s law for both surfaces

(flat) Amperian loop Surface S 2 (curved)

3 Let s revisit Ampere s Law for current I and a capacitor Let s consider a wire with current I and a capacitor: Surface S 1 (flat) +Q Q I Amperian loop Surface S 2 Let s apply Ampere s law for both surfaces (S1 and S2): Amperian loop Surface S 1 (flat) Amperian loop Surface S 2 (curved) The LH sides are the same, but the RH sides are different!!?? Something is missing in Ampere s law. So! Ampere s Law needs to be adjusted!

Displacement current/ Ampere-Maxwell Law

units of current and use it to generalize

4 Displacement current/ Ampere-Maxwell Law Let s get somehow an additional term with units of current and use it to generalize Ampere s Law +Q Q I=dQ/dt E I But we need something which has units of current. So let s take a derivative: Maxwell interpreted as being equivalent current and called it d a Displacement current Ampere-Maxwell Law

Displacement current Displacement current 1) The displacement current is only between the plates

current in the wire charging the capacitor).

It is equivalent to a real current in that it creates the same magnetic field Let s apply

5 Displacement current Displacement current 1) The displacement current is only between the plates since is zero outside 2) The way I D was introduced allows us to say that numerically I D =I (real current in the wire charging the capacitor). In some sense current is conserved all the way through the capacitor 3) I D is not a flow of charge. It is equivalent to a real current in that it creates the same magnetic field Let s apply Ampere-Maxwell Law for the capacitor system I D I E I D Amperian Surface S 1 Amperian Surface S 2 Now it works. Each surface gives us the same answer as it should be. I

electric flux (particularly, changing electric field) Another amazing

6 Induced Magnetic Field Ampere-Maxwell Law Thus, the magnetic field B can be generated by: 1) An ordinary electric current, I in 2) Changing electric flux (particularly, changing electric field) Another amazing thing!!! Changing electric field inside a capacitor produces a magnetic field

7 Induced Fields An increasing solenoid current causes an increasing magnetic field, which induces a circular electric field. An increasing capacitor charge causes an increasing electric field, which induces a circular magnetic field.

8 The last (4 th ) Maxwell s equation

Gauss s Law for Magnetic Fields Gauss s law for the

enclosing total charge Q in, the net electric flux

a magnetic flux Magnetic field lines form continuous

9 Gauss s Law for Magnetic Fields Gauss s law for the electric field says that for any closed surface enclosing total charge Q in, the net electric flux through the surface is: There is a similar equation for a magnetic flux Magnetic field lines form continuous curves; every field line leaving a surface at some point must reenter it at another. Gauss s law for the magnetic field states that the net magnetic flux through a closed surface is zero:

Maxwell s Equations Electric and magnetic fields are

meaning An electric field is produced by a charge No

changing magnetic field A magnetic field is produced by

Maxwell s equations, which describes the fields, a fifth

10 Maxwell s Equations Electric and magnetic fields are described by the four Maxwell s Equations: the physical meaning An electric field is produced by a charge No magnetic monopoles An electric field is produced by a changing magnetic field A magnetic field is produced by a changing electric field or by a current In addition to Maxwell s equations, which describes the fields, a fifth equation is needed to tell us how matter responds to these fields:

11 There are a total of 11 fundamental equations describing classical physics: 1. Newton s first law 2. Newton s second law Physics I 3. Newton s third law 4. Newton s law of gravity 5. Gauss s law 6. Gauss s law for magnetism Physics II 7. Faraday s law 8. Ampère-Maxwell law 9. Lorentz force law 10. First law of thermodynamics 11. Second law of thermodynamics Physics III

12 Electromagnetic waves

13 Light is an electromagnetic wave I From Maxwell s equations we can get wave equations for E and B: Traveling waves: y y, Which can y be solved:, z z z Amplitude Wavenumber According to the wave theory, this factor is v 2 The magnitude of this speed is 3.0 x 10 8 m/s precisely equal to the measured speed of light. Thus, it means that light is an electromagnetic wave (shocking conclusion) Maxwell was the first to understand that light is an oscillation of the electromagnetic field. Angular frequency It travels in +x direction The speed of light can be found in two ways: The oscillation amplitudes are related These propagating fields are called electromagnetic waves.

Light is an electromagnetic wave (II) The electric and magnetic waves

Since a changing electric field produces a magnetic field, and a

14 Light is an electromagnetic wave (II) The electric and magnetic waves are perpendicular to each other, and to the direction of propagation. Since a changing electric field produces a magnetic field, and a changing magnetic field produces an electric field, they can propagate on their own. Maxwell was able to predict that electromagnetic waves can exist at any frequency, not just at the frequencies of visible light.

15 Production of Electromagnetic Waves Oscillating charges will produce electromagnetic waves: nd half cycle Far from the source, the waves are plane waves _ Wave travel direction

ConcepTest Before the days of cable, televisions often had two antennae on them, one straight and one circular. Which antenna picked up the magnetic oscillations?

16 ConcepTest Before the days of cable, televisions often had two antennae on them, one straight and one circular. Which antenna picked up the magnetic oscillations? TV Antennas A) the circular one B) the straight one C) both equally; they were straight and circular for different reasons The varying B field in the loop means the flux is changing and therefore an emf is induced.

17 Polarization

Polarization The plane of the electric field vector and

18 Polarization The plane of the electric field vector and the direction of propagation is called the plane of polarization. The electric field in the figure oscillates vertically, so this wave is vertically polarized. The electric field in the figure above is horizontally polarized. Most natural sources of light are unpolarized, emitting waves whose electric fields oscillate randomly with all possible orientations. Quantum cascade lasers emit vertically polarized light.

19 Polarizers The most common way of artificially generating polarized visible light is to send unpolarized light through a polarizing filter. Polarizer axis

Crossed polarizers No Light Polarizer axis Polarizer axis Two polarizing filters with

The polarizing direction of the first polarizer is oriented vertically to the incident

The wave passing through the first polarizer is subsequently blocked by the second

20 Crossed polarizers No Light Polarizer axis Polarizer axis Two polarizing filters with perpendicular axes, called crossed polarizers, block all the light. The polarizing direction of the first polarizer is oriented vertically to the incident beam so it will pass only the waves having vertical electric field vectors. The wave passing through the first polarizer is subsequently blocked by the second polarizer, because this polarizer is oriented horizontally with respect to the electric field vector in the light wave.

Polarizing Sunglasses Glare the reflection of the

horizontal surfaces has a strong horizontal

Vertically polarizing sunglasses can cut glare

Polarizer axis Polarizer axis Wearing polarized

21 Polarizing Sunglasses Glare the reflection of the sun and the skylight from roads, water and other horizontal surfaces has a strong horizontal polarization. Vertically polarizing sunglasses can cut glare without affecting the main scene you wish to see. Polarizer axis Polarizer axis Wearing polarized sun glasses helps to reduce glare This light is almost completely blocked by a vertical polarizing filter.

22 What you should read Chapter 34 (Knight) Sections (skip derivations of wave equations) 34.7

23 Thank you See you on Tuesday

Lecture 22 Chapter 31 Maxwell s equations

Lecture 22 Chapter 31 Maxwell s equations Finally, I see the goal, the summit of this Everest Today we are going to discuss: Chapter 31: Section 31.2-4 Let s revisit Ampere s Law a straight wire with current