WIRELESS POWER

ELECTROMAGNETISM

P R E A M B L E

The original form of the problem is a four week (15 credit) module in the IScience programme at the University of Leicester on optics, electromagnetic theory and the functioning of the eye. The problem presented here is the physics component run over two weeks. The missing link (not given here) is the transmission line model of the propagation of nerve impulses (the Hodgkin-Huxley model).

MODULE PACING

This part of the module is run with the following weekly pattern over 2 weeks

Facilitated workshop – problems for class discussion (2 hours)

Lecture

Facilitated workshop – problems for class discussion (1 hour)

Lecture

Tutorial on individual exercises

I N T E N D E D L E A R N I N G O U T C O M E S

By the end of the module students should be able to:

Recall information regarding simple electrical circuits from previous modules Recall the operation of capacitors; be able to calculate the capacitance of parallel

plate and cylindrical capacitor; be able to calculate the energy stored in a capacitor Recall how to calculate the equivalent capacitance (or resistance) of a set of

capacitors (or resistors) in parallel and series. Be able to apply Kirchoff's Loop rule to any circuit. Understand, explain and be able to sketch the time varying properties (current,

charge, potential difference) of RC, RL, LC and RLC circuits that are initially charged but are not connected to a power source.

Define inductance as a property of a component in an electrical circuit. Compare the time varying equations in RC, RL, LC and RLC circuits to equivalent

'mass on a spring' systems. Understand the origin of, and the equations describing, an a.c. current. Understand, explain and be able to sketch the time varying properties of RLC circuits

driven by an a.c. generator. To define and state the total reactance and impedance of a series RLC circuit. To define and state the complex impedance of a series RLC circuit. Describe the conditions under which a series RLC circuit is deemed to be resonance. Define and calculate the Q factor of an RLC circuit. Understand the theory behind transmission lines and signal transmission. Understand the Hodgekin-Huxley of nerve-impulse transmission. Recall and describe Maxwell's equations as applied in a general electric circuit and to

visible light. Derive the electric and magnetic wave equations for light. Recall information regarding reflection and refraction from previous modules. Understand and apply Huygen's Principle and Fermat's Principle. Describe total internal reflection (TIR). Describe how fibre optic cables work. State and apply the equations relating to transmission and reflection at a boundary

between two media when the incident ray strikes the surface at normal and oblique incidence.

Describe the four phenomena that produce polarized light from unpolarized sources. Define and be able to calculate the Brewster angle for a boundary between two

media. Describe reflection in simple mirror systems; calculate the focal length of a mirror;

state and apply the mirror equation; calculate the lateral magnification of an image. Draw ray diagrams for a variety of mirror shapes using the three Principle Rays. State and apply the following equations: simple lens, lens makers and thin lens. Draw ray diagrams for a variety of simple lenses using the three Principle Rays.

R E A D I N G L I S T

The reading list is that provided for the original module. Other equivalent textbooks are available.

ESSENTIAL

Tipler, P.A., Physics for Scientists and Engineers. Freeman.

Grant, I.S., & Phillips, W.R., Electromagnetism. Wiley. Transmission Line Tutorial http://www.amanogawa.com/archive/docs/C-tutorial.pdf

FURTHER

Seshadri, S.R. (1971) Fundamentals of Transmission Lines and Electro-magnetic Fields. Addison-Wesley.

Baden Fuller, A.J. (1993) Engineering Electromagnetism. Wiley. Blake, R. (1993) Basic Electronic Communication. West Publishing Company. Benson, F.A. (1921) Field, Waves and Transmission Lines. Chapman and Hall.

OTHER

http://en.wikipedia.org/wiki/Coaxial_cable : Background on coaxial cables.

P R O B L E M S T A T E M E N T S

Contact lenses to get built-in virtual graphics

A researcher holds one of the completed lenses.

A contact lens that harvests radio waves to power an LED is paving the way for a new kind of display. The lens is a prototype of a device that could display information beamed from a mobile device. The circuitry requires 330 microwatts but doesn't need a battery. Instead, a loop antenna picks up power beamed from a nearby radio source. The developers claim that the magnetic fields involved are entirely safe. Future versions will have an array of micro-lenses to focus the image so that it appears suspended in front of the wearer's eyes. A contact lens that allows virtual graphics to be overlaid on the real world could provide a compelling augmented reality experience.

Based on:

http://www.newscientist.com/article/dn18146-contact-lenses-to-get-builtin-virtual-graphics.html

S U G G E S T E D D E L I V E R A B L E S

As a group write a technical report (1000 – 1500 words) on

(i) the proposed method of powering the device and (ii) the specification for a suitable lens.

Q U E S T I O NS F O R C L A S S D I S C U SS I O N

CIRCUITS

1. Consider the circuit diagram below. The capacitor initial contains a charge Q0. If the switch S is closed what equation describes the charge remaining on the capacitor at any given time t? Describe the flow of electrical current during this process.

2. A battery is now added to the circuit. If the capacitor is initially uncharged what equation describes the charge gained on the capacitor at any given time t if switch S is closed? Describe the flow of electrical current during this process.

3. In the above circuit how work is done by the battery as it charges the capacitor? What happens to this energy in the circuit?

4. What is inductance? 5. Consider the circuit diagram below. If the switch S is closed what is the sum of the

potential differences around this circuit (Kirchoff’s Loop rule)?

6. State the equation for the current in the circuit shown above at any time t. 7. Consider the following two diagrams: compare the behaviour of the two systems.

8. Sketch the variation in charge versus time and current versus time in an LC circuit. 9. Describe the total energy, and the time variant energy, in the LC circuit. 10. The circuit below contains a resistor, capacitor and inductor (RLC circuit). Use

Kirchoff’s loop rule to derive an equation for the circuit in terms of charge only.

11. The equation you have just derived for a RLC circuit is similar to that of a damped oscillator. Describe the similarities of the RLC circuit and classic damped oscillator systems.

12. What is the natural frequency of an RLC circuit? What caveats apply in this case? 13. What is the rate at which electrical energy is dissipated in the resistor? 14. What is the emf of an ac generator as a function of time?

15. If a circuit consists of an ac generator and a single resistor, what is the voltage across and the current through the resistor as a function of time? Are these two quantities in phase with each other?

16. What is the power delivered to the resistor as a function of time? How does this differ from the average power delivered?

17. How does the presence of a capacitor and an inductor in a circuit connect to an ac supply differ from a circuit connected to a dc supply?

18. Show that for an inductor placed in a circuit powered by an a.c. supply that Ipeak =

Vpeak/ωL. 19. What is the a) instantaneous power and b) average power delivered to the inductor? 20. Sketch the current and potential drop across the components in the two circuits

shown below:

a) b)

21. Apply Kirchoff’s Loop Rule to the circuit below to find an equation that details the

potential drops across the circuit. You may assume that the potential drop applied by the generator is of the form Vapp = Vapp, peakcosωt.

22. Rearrange the equation from the previous question so that it is in terms of charge, Q. How does this relate to a similar equation of a mass on a spring?

23. The steady-state current for the circuit given above is I = Ipeakcos(ωt - δ). What is δ

and how can you calculate it? 24. What is the total reactance and impedance of a driven, series RLC circuit? 25. How is Ohm’s Law generalised to this type of circuit? 26. What is the current in a series RLC circuit in terms of the impedance of the circuit? 27. Under what conditions is a series RLC circuit deemed to be in resonance?

28. What is the average power dissipated in the series RLC circuit as a function of ω and ω0? In which component(s) is the power dissipated?

29. What is the Q factor for an RLC circuit? What does this quantity tell you about the circuit?

30. How do you send messages along a wire? 31. What would you detect if the wire is;

a) Short, or b) long?

32. Is the received message identical to the sent one? 33. How do we represent a transmission line? 34. How do complex impedances Z1 and Z2 add

a) in series?

b) in parallel?

35. What is the complex impedance of a) a resistor?

b) a capacitor?

c) an inductor?

36. What is the resonant frequency of the circuit in figure 1? What is its complex impedance, Z? What is the meaning of the real and imaginary parts of Z? For the circuit shown Z is purely imaginary: what does this imply?

TRANSMISSION LINES

1. What is the transfer function (T = Vout/Vin) of the circuit in figure 2 for frequencies such

that RC >> 1? Why is this circuit referred to as a high pass filter?

2. What is the complex impedance of the circuit in figure 3? Under what condition is it purely reactive for low frequencies (i.e. acts as a pure resistance)? Why is such a circuit referred to as a low pass filter?

3. Use the result of the previous question to show that the impedance of the circuit in figure 4 is L/C.

4. For the ladder circuit of figure 4, at frequencies < c = 2/(LC), each L-C section introduces a phase change of ~ 2/c. Explain why the circuit acts as a delay line at these frequencies. What is the significance for the transmission of a signal of the fact that the delay is independent of frequency?

5. Why is an ordinary extension cord not usually considered as a transmission line, while a television antenna cable of the same length would be? (The question requires you to state what is meant by a transmission line.)

6. One of the greatest engineering feats of the 19th century was the laying of submarine cables under the Atlantic Ocean. The first such cable was laid in 1858 and had a length of about 3600 km. The theoretical analysis of signal propagation along a cable was first carried out by William Thomson (later Lord Kelvin). He derived the following equation for the voltage V(x,t) on the line

where R and C are respectively the resistance and capacitance per unit length of the line (figure 5). See supplementary material for a derivation.

t

VRC

x

V

2

2

Note that Thomson regarded the inductance of the line as unimportant and ignored it. This equation has the wave solution:

a) Deduce that these waves are dispersive.

b) Speech requires frequencies in the range 1 to 4 kHz. Typical parameters for an early submarine cable are C = 7.5×1011 F m1 and R = 7 × 103 m1. By comparing the propagation times of two frequencies within the speech range, show that, according to Thomson's theory, the transmission of intelligible speech across the Atlantic via a cable would not be possible.

c) Show that the transmission of intelligible Morse code is possible according to the theory.

OPTICS

1. What is light? 2. How do we see anything at all? 3. How are the electric and magnetic fields in light related? 4. How fast does light travel? State this in terms of the electric and magnetic constants. 5. What is Ampere’s Law? Why is it a problem for circuits containing capacitors and how

did Maxwell correct for this? 6. A parallel-plate capacitor has been constructed from two circular plates each of radius

1.5 cm held 5mm apart. Each plate currently holds a charge of 3.7 C. a) Describe the electric field between the plates. b) Calculate the electric flux between the capacitor plates.

If the capacitor is in an electric circuit with a current of 4 A flowing through the circuit: c) Calculate the displacement current flowing between the plates.

7. State all of Maxwell’s Equations and explain what they mean. 8. Consider the general form of Ampere’s law and Faraday’s Law. What do they tell us

about changing magnetic and electric fields? 9. State the wave equations for the electric and magnetic fields in light. 10. Derive the wave equation for the electric field from first principles, assuming the

following conditions: a) The wave is propagating in the positive x-direction. b) The electric field of the wave is in the y-direction. c) The time varying magnetic field of the wave is in the z-direction.

The following diagrams may help your derivation.

])2/(cos[),( 2/1)2/( 2/1

xRCtetxV xRC

11.

12. Figure 1 Figure 2

13. Briefly describe Huygen’s geometric construction for the propagation of light. 14. What is Fermat Principle? 15. State, in the form of equations, the laws of reflection and refraction. 16. What is definition for the refractive index of a medium? 17. Describe how reflection and refraction works at an atomic level. 18. What is the relative intensity of reflected and transmitted light at a boundary between

two different media assuming the incoming ray hits the boundary at normal incidence? 19. What is total internal reflection (TIR)? 20. How do fibre optic cables work? 21. Describe the magnetic and electric fields for unpolarized, linearly polarized and

circularly polarized light. 22. What are the four phenomena that produce polarized EM waves from unpolarized ones? 23. You are sitting by the pool whilst on holiday in Singapore. At roughly what time of day

will the sunlight, which has been reflected off of the water’s surface, be completely polarized?

24. How do polarized sunglasses work? What would you see if you held your sunglasses at arms length and rotated them whilst looking at the pool in the question above?

25. The formulae for the reflected intensity, when the incoming ray hits a boundary between two different media at an oblique incidence, can be expressed in various equivalent forms. For an electric field polarised parallel to the plane of incidence:

2

21

21

2

coscos

coscos

)tan(

)tan(

incident)(

reflected)(

it

it

ti

ti

nn

nn

I

I

,

and

2

2121

2

coscos

2coscos

)cos()sin(

cossin2

incident)(

d)transmitte(

itti

titi

it

nnnn

I

I

.

For an electric field normal to the plane of incidence:

2

21

21

2

coscos

coscos

)sin(

)sin(

incident)(

reflected)(

ti

ti

ti

ti

nn

nn

I

I

,

and

2

2121

2

coscos

2coscos

)sin(

cossin2

incident)(

d)transmitte(

titi

ti

it

nnnn

I

I

.

For unpolarised light we take the averages.

The diagram below shows a ray of light striking a glass block with an angle of incidence, θ1, of 32o. Draw the resultant light rays indicating their angles and the intensity of light.

MIRRORS

1. Why are we able to model rays of light as straight lines when considering mirrors and

lenses? 2. You are sitting beside a river and see the reflection of a hot air balloon in the water. If

you know that the balloons are being launched from a field 2 km away and you are 3.5 m from the river, how high is the balloon?

3. What is a virtual image and where would it appear to be in the balloon question above? 4. What equation would you use to calculate the focal length of a mirror?

5. State the mirror equation and explain what each term means. 6. A point source is 15 cm from a concave mirror and 2.0 cm above the axis of the mirror. If

the radius of curvature of the mirror is 10 cm find: o The focal length of the mirror. o The image distance. o The position of the image relative to the axis of the mirror.

7. Three Principle Rays are used to construct ray diagrams for mirrors: what are they and draw a diagram showing all three.

8. Complete the diagram below.

9. State the equation used to calculate the lateral magnification of an image reflected in a mirror.

10. State the lens equation describing refraction at a single surface. 11. How can you calculate the magnification of an image at a refracting boundary? 12. State the lens makers and thin lens equations. 13. The three Principle Rays are also used to construct ray diagrams for thin lenses, how are

they modified? 14. How does the eye focus on objects at different distances?

I N D I V I D U A L E X E R C I S E S

1. Calculate the transfer function (Vout/Vin) for the circuit in figure 1. Hence show that it acts as a low pass filter. [5]

2. Calculate the transfer function (Vout/Vin) for the circuit in figure 2. Hence show that it acts as a low pass filter. [5]

3. Derive the wave equation for the LC lossless transmission line (figure 3) as follows:

Consider the voltage drop across a length dx along the line with inductance Ldx:

dt

dILdxdV

and the current through the capacitor, capacitance Cdx,

dt

dVCdxdI .

Now eliminate I.

4. Repeat question 3 for Thomson’s model of a transmission line (inductance in figure 3 replaced by a resistance) and hence derive the cable equation (Thomson’s diffusion equation).

5. The inductance and capacitance per unit length of a coaxial transmission line are;

L= 2

1μ0μrln(

a

b)

C =

)ln(

2 0

a

br

where a and b are the radii of the inner and outer conductors respectively.

a) Find the characteristic impedance of an air-filled line with b/a = 2.3 and

b) show that the phase velocity on an air-filled line is the speed of light. (For an air-filled line μr=1 and εr=1).

6. Letting conductances depend on the membrane potential leads to a (simplified) form of the Hodgkin-Huxley equation:

)1)((2

2

VsVV

VV

(1)

Surprisingly the non-linear right hand side means that this no longer behaves as a diffusion

equation. Show that the condition for (1) to have a wave-like solution V = V( /c), with speed c, is

)1)((1

2 VsVVVV

c

where y = /c and ./ dydVV Show that a solution is 1)1( yeV where

21 s and )(2 2

1 sc . Hence show (by

writing V as a function of x and t) that the speed of a pulse is proportional to a, where a is the diameter of the axon.

7. A concave mirror has a radius of curvature of 1m. An object is placed 5m away. Find the image distance.

8. A thin lens of index of refraction 1.5 has one convex side with a radius of magnitude 20cm. When an object 1cm is height is placed 50cm from this lens, an upright image 2.15cm in height is formed.

a) Calculate the radius of the second side of the lens.

b) Is the second side of the lens concave or convex?

9. An object is 17.5cm to the left of a lens of focal length 8.5cm. A second lens of focal length -30cm is 5cm to the right of the first lens. Find the distance between the object and the final image formed by the second lens.

10. An object placed 8cm from a concave spherical mirror produces a virtual image 10cm behind the mirror.

a) If the object is moved back to 20cm from the mirror, where is the image located?

b) Is it real or virtual?

11. Parallel light from a distant object strikes the large mirror and is reflected by the small mirror that is 2m from the large mirror. The small mirror is actually spherical, not planar as shown. The light is focussed at the vertex of the large mirror.

a) What is the radius of curvature of the small mirror?

b) Is it convex or concave?

12. A plane wave propagating in the z-direction has an electric field Ex =E0sin (t – kz). Show

that the magnetic field is By = (E0/c) sin (t – kz).

13. Show that in a plane electromagnetic wave E = cB. Show that in vacuum E and B are in phase and that the energy in a wave is shared equally between E and B. (In a conductor, E and B are 900 out of phase and the energy is carried by the B field.)

14. The curves show the intensity of light reflected from a surface with the polarisation parallel and perpendicular to the plane of incidence. Deduce the refractive index of the medium.