1. Current in an LC Circuit
Period:
Frequency:

2. Question (Chap. 23)
A capacitor C was charged and contains charge +Q0 and –Q0 on each of its plates, respectively. It is then connected to an inductor (coil) L.
Assuming the ideal case (wires have no resistance) which is true? Q0
There will be no current in the circuit at any time because of the opposing emf in the inductor.
The current in the circuit will maximize at time t when the capacitor will have charge Q(t)=0.
The current in the circuit will maximize at time t when capacitor will have full charge Q(t)=Q0.
The current will decay exponentially.
Answer: B ; charge goes like cosine, current goes like sine

3. Question
Two metal rings lie side-by-side on a table. Current in the left ring runs clockwise and is increasing with time. This induces a current in the right ring. This current runs
Clockwise
Counterclockwise
when viewed from above
(A), B field is increasing upwards through second loop … thumb downward

4. Faraday’s Law: Applications
Single home current: 100 A service
Vwires=IRwires
Transformer: emfHV IHV = emfhomeIhome
Single home current in HV: <0.1 A
Power loss in wires ~ I2

5. Faraday’s Law: Applications

6. Faraday’s Law: Applications
Induction
microphone

7. Classical Theory of Electromagnetic Radiation
Chapter 24
Classical Theory of Electromagnetic Radiation

8. Maxwell’s Equations Gauss’s law for electricity
Gauss’s law for magnetism
𝐸 ∙𝑑 𝑙 =− 𝑑 𝑑𝑡 𝐵 ∙ 𝑛 𝑑𝐴
Complete Faraday’s law
Ampere’s law
(Incomplete Ampere-Maxwell law)
Lets work with the last one…

9. Ampere’s Law Current pierces surface No current inside
Lets work with the last one…

10. Maxwell’s Approach
Time varying magnetic field leads to curly electric field.
Time varying electric field leads to curly magnetic field? I
Current in wire I – causes change in E flux, should cause the same effect in curly B
‘equivalent’ current
combine with current in Ampere’s law

11. The Ampere-Maxwell Law
Works!
This law cannot be derived, but all experimental facts prove it, especially phenomena of radio and light-waves

12. Maxwell’s Equations Four equations (integral form) : Gauss’s law
Gauss’s law for magnetism
Faraday’s law
Ampere-Maxwell law
Add Lorentz to complete the list of fundamental equations of electricity and magnetism
Lorentz eq – defines the meaning of electric and magnetic field in terms of their effect on charge
+ Lorentz force

13. Fields Without Charges
Time varying magnetic field makes electric field
Time varying electric field makes magnetic field
Do we need any charges around to sustain the fields?
Is it possible to create such a time varying field configuration which is consistent with Maxwell’s equation?
Solution plan:
Propose particular configuration
Check if it is consistent with Maxwell’s eqs
Show the way to produce such field
Identify the effects such field will have on matter
Analyze phenomena involving such fields

14. A Simple Configuration of Traveling Fields
Key idea: Fields travel in space at certain speed
Disturbance moving in space – a wave?
1. Simplest case: a pulse (moving slab)
very high and deep slab – fields only inside
Is it consistent with Maxwell’s law?

15. A Pulse and Gauss’s Laws
Pulse is consistent with Gauss’s law
very hign and deep slab – fields only insude
Is it consistent with Maxwell’s law?
Pulse is consistent with Gauss’s law
for magnetism

16. A Pulse and Faraday’s Law
Area does
not move
Since pulse is ‘moving’, B depends on time and thus causes E
emf
E=Bv
Direction is correct since B increases out of page so –dBdt points into page. RHR gives E in correct direction (up; along only part
in Bfield of length h) .
Is direction right?

17. A Pulse and Ampere-Maxwell Law=0

18. A Pulse: Speed of Propagation
E=Bv
E=cB
Spectacular result: Maxwell predicts speed of light for electromagnetic pulse in mid-1800 – but no one at this time could guess the electromagnetic nature of light, though the speed of light was already known.
He got speed of light theoretically from constants epsilon and mu which are easy to measure in experiment
Based on Maxwell’s equations, pulse must propagate at speed of light

19. Question
At this instant, the magnetic flux Fmag through the entire rectangle is: D
B; B) Bx; C) Bwh; D) Bxh; E) Bvh

20. Question In a time Dt, what is DFmag?c
A) 0; B) BvDt; C) BhvDt; D) Bxh; E) B(x+vDt)h

21. Question
emf = DFmag/Dt = ? B
A) 0; B) Bvh; C) Bv; D) Bxh; E) B(x+v)h

22. Question What is 𝐸 ∙𝑑 𝑙 around the full rectangular path?
Eh; B) Ew+Eh; C) 2Ew+2Eh; D) Eh+2Ex+2EvDt; E)2EvDt

23. Question 𝐸 ∙𝑑 𝑙 =𝐸ℎ What is E?
𝐸 ∙𝑑 𝑙 =𝐸ℎ
What is E? B
A) Bvh; B) Bv; C) Bvh/(2h+2x); D) B; E) Bvh/x

24. Exercise 𝐹 𝑚𝑎𝑔 𝐹 𝑒𝑙 = 𝑣𝐵 𝐸 = 𝑣𝐵 𝑐𝐵 = 𝑣 𝑐
If the magnetic field in a particular pulse has a magnitude of 1x10-5 tesla (comparable to the Earth’s magnetic field), what is the magnitude of the associated electric field?
Force on charge q moving with velocity v perpendicular to B:
𝐹 𝑚𝑎𝑔 𝐹 𝑒𝑙 = 𝑣𝐵 𝐸
= 𝑣𝐵 𝑐𝐵 = 𝑣 𝑐

25. Direction of Propagation
Direction of speed is given by vector product
Convenient to remember direction is E x B

26. Electromagnetic Radiation
Last shown

27. Accelerated Charges Electromagnetic pulse can propagate in space
How can we initiate such a pulse?
Short pulse of transverse
electric field

28. Accelerated Charges Transverse pulse propagates at speed of light
Since E(t) there must be B
Direction of v is given by: E B v
No magnetic field out of the circle
Ordinary (steady) magnetic field out of page inside – same as radiative.
But this is does not need to be the case – acceleration matters – if it moves up and then slows down – different direction.

29. Accelerated Charges: 3D

30. Magnitude of the Transverse Electric Field
We can qualitatively predict the direction.
What is the magnitude?
Magnitude can be derived from Gauss’s law
Field ~ -qa
Derivation is in not difficult, but long and mostly geometry.
Kink in the electric field
Much slower than 1/r2 – makes it possible to affect matter that is very far from the accelerated charges – that is why we see very distant stars.
1. The direction of the field is opposite to qa
2. The electric field falls off at a rate 1/r

31. Exercise a
An electron is briefly accelerated in the direction shown. Draw the electric and magnetic vectors of radiative field. E B
1. The direction of the field is opposite to qa
2. The direction of propagation is given by
Electron so E same direction as aperp. Velocity in direction of r so B into page.

32. Exercise
An electric field of 106 N/C acts on an electron for a short time. What is the magnitude of electric field observed 2 cm away?
2 cm
E=106 N/C B
Erad a
1. Acceleration a=F/m=qE/m= m/s2
2. The direction of the field is opposite to qa
3. The magnitude:
E= N/C
4. The direction of propagation is given by
Derivation is too complex for this course
Kink in the electric field
Much slower than 1/r2 – makes it possible to affect matter that is very far from the accelerated charges – that is why we see very distant stars.
Quali
What is the magnitude of the Coulomb field at the same location?

33. Question
A proton is briefly accelerated as shown below. What is the direction of the radiative electric field that will be detected at location A? B A A D C + C

34. Question
A narrow collimated pulse of radiation propagates in the -x direction. There is an electron at location A. What is the direction of the radiative electric field observed at location B? B A C B D e-

35. Stability of Atoms v Circular motion: Is there radiation emitted? a
Classical physics says “YES”
orbiting particle must lose energy!
speed decreases
particle comes closer to center
Classical model of atom:
Electrons should fall on nucleus!
Puzzle was solved by introducing QM
Synchrotron ratiation
To explain the facts - introduction of
quantum mechanics:
Electrons can move around certain orbits only and emit E/M radiation only when jumping from one orbit to another

36. Sinusoidal Electromagnetic Radiation
Acceleration:
Sinusoidal E/M field

37. Sinusoidal E/M Radiation: Wavelength
Instead of period can
use wavelength:
Freeze picture in time:
Example of sinusoidal E/M radiation:
atoms
radio stations
E/M noise from AC wires

38. Electromagnetic Spectrum
Microwave: not because it is used in microwave, but because it has ~micron wavelength
Visible – the whole spectrum from blue to red – the frequency barely changes twice!

39. Color Vision
Three types of receptors (cones) in retina which incorporate three different organic molecules which are in resonance with red, green and blue light frequencies (RGB-vision):
Response spectra for three types of receptors
Max response wavelengths:
S – 440 nm ( Hz)
M – 540 nm ( Hz)
L – 560 nm ( Hz)
Microwave: not because it is used in microwave, but because it has ~micron wavelength
Visible – the whole spectrum from blue to red – the frequency barely changes twice!
Three types of rhodopsin receptors contain three different opsin molecules (red, green, blue)
Cone absorption spectra: peaks at 440, 540, 560 nm (Short, Medium and Long – type cones, three types of opsin in rhodopsin)
Refers to length of cone

40. E/M Radiation Transmitters
How can we produce electromagnetic radiation of a desired frequency?
Need to create oscillating motion of electrons
Radio frequency
LC circuit: can produce oscillating motion of charges
To increase effect: connect to antenna
Visible light
Heat up atoms, atomic vibration can reach visible frequency range
Transitions of electrons between different quantized levels

41. Polarized E/M Radiation
AC voltage
(~300 MHz)
E/M radiation can be polarized along one axis… no
light
At every single time E is of course in just one direction, but it changes randomly and rapidly over time
Light from the sun is not polarized!
…and it can be unpolarized:

42. Polarized Light Making polarized light Turning polarization
Polaroid sunglasses and camera filters:
Show trick with three polarizers – two crossed ones, add one at 45 degrees in the middle
Sheet of special plastic whose long molecules are aligned with each other
Sunglasses
reflected light is highly polarized: can block it
Considered: using polarized car lights and polarizers-windshields

43. Field of an Accelerated Charge