today in physics 122: antennas - university of rochester

Today in Physics 122: antennas

Application of radiation pressure: solar sails (concluded)

Standing waves and resonance

Reception of electromagnetic waves

Simple antennas

4 December 2019 Physics 122, Fall 2019 1

Dan, installing a Yagi antenna on top of a utility pole at Mees Observatory, for Verizon Band 13 (750 MHz) LTE network signal reception.

Recap: electromagnetic plane waves

In vacuum with no charges or currents present, electric and magnetic fields can exist as travelling waves, which have E ⊥ B and in phase (peaks line up), which are transverse, and which travel in the direction of E × B:

This wave is polarized in the x direction. (Polarization is direction of E.)

Relations among quantities that determine the wave’s pattern:


( ) ( )

( ) ( )0

0

ˆ,ˆ propagates along .

ˆ,

i z t

i z t

z t E eE

z t ec

κ ω

κ ω

±

±

=

=

E xz

B y

8 12.99792458 10 m sec22 1

f c

f

ω κ λλ π κτ π ω

−= = = ×== =

Recap: electromagnetic plane waves (continued)

Electromagnetic waves carry energy and momentum with fluxes S and (energy or momentum per unit area per unit time) respectively given by

Time averages:

where the average is taken over an integer number of periods τ = 2π/ω.


( )

( )

220

02

202

0

0

00

ˆ

ˆ

1 i z t

i z t

Ee

c

Ee

cc

κ ω

κ ω

µ µ

µ

εµ

−

−

=

= =

= ×

= ×

S

zS E B

zE B

for a plane electromagnetic wave travelling in the +z direction

20

020

2

0

0 0

0 ˆ

1

1

ˆ22

22

Ec

Ec

µ µ

ε

µµ

= ×

=

=

× =

S E B*

E B

z

z*

Recap: electromagnetic plane waves (continued)

Radiation pressure F/A, for plane wave incident normally (perpendicularly) on a surface with area A, which reflects a fraction α of the incident power and absorbs the rest (fraction 1 - α):

Flux from isotropic light emitter (e.g. a star) with luminosity (total power) L:

Electromagnetic energy density:


( ) ( )1 2 1A

α α α= − + = +F

2 2ˆ ˆ

4 4L L

cr crπ π= = =

SS r r

( )2 20 0

12

u E Bε µ= +

Radiation pressure (continued)

Example. The International Space Station has mass

and is in orbit a distance a = 6738 km from the Earth’s center (altitude 360 km). Design a “solar sail” which could carry the ISS out of the Solar system.

The maximum gravitational force on the ISS is that exerted on it when it’s on the opposite side of the Earth from the Sun:

where


52.27 10 kg,ISSm = ×

( )max 2 2

11

30

ˆ ˆ ,

1 AU 1.5 10 m, and

2.0 10 kg.

ISSE ISS GM mGM ma r a

r

M

= − −+

= = ×

= ×

F r r

Radiation pressure (continued)

The radiation force must offset this. We need to find the area A which makes this work:

Solve this for A:

If square, it’s at least 469 km on a side.


( ) ( )

( )( )

max2

2 2 2

ˆ1 14

14

ISSE ISS

LAcr

GM mGM mLAcr a r a

α απ

απ

= + = + > −

+ > ++

F r F

( ) ( )

2

2 2

11 2

41

2.2 10 m for 1 (i.e. perfectly reflecting)

ISSE ISS GM mGM mcrAL a r a

πα

α

> + + +

> × =

Standing waves of light

So far we have considered travelling waves of light that extend over all space. What if we let them reflect back and forth between two parallel, planar mirrors, analogous to reflection of waves on a violin string?

E for the resultant wave is the vector sum of E for all the reflections forward and backward.

Unless there is a special relationship between the wavelength and the distance between the mirrors, the peaks and troughs rarely align …

… and the vector sum for E is close to zero (i.e. destructive interference).


▬ Moving to right▬ Moving to left▬ Total

Reflections:0

EE

z L

12410

1

0

-1

MirrorMirror

Standing waves of light (continued)

But if special conditions are met, the peaks and troughs of the reflections line up, and the pattern appears stationary (i.e. standing waves) …

… and the vector sum grows as time goes on (i.e. resonance; constructive interference).

As in the case of waves on a string, the special conditions are that the mirrors are separated by an integer number of half-wavelengths of light.



Reflections:0

EE

z L

12410

MirrorMirror

12

0

-12

Standing waves of light (continued)

In workshop module 12, you will demonstrate that such standing waves that have the form

are solutions to Maxwell’s equations, and study what happens to the energy in standing-wave patterns.



Reflections:0

EE

z L

12410

MirrorMirror

12

0

-12

0

0

ˆ sin sinˆ cos cos

E z tB z t

κ ωκ ω

==

E xB y

3 September 2009 Astronomy 102, Fall 2009 10

The electromagnetic spectrum

Infrared, microwave, radio wavelengths

Ultraviolet, X-ray, γ-ray wavelengths

λ = 0.4 µm 0.5 µm 0.6 µm

λ = 0.01 Å 1 Å 100 Å 1 µm 100 µm 1 cm 1 m

UV Infrared Microwave RadioX-raysγ rays

Visible light is a tiny part of the spectrum of light.

Reception of long-wavelength electromagnetic radiation

Consider the standard 4G wireless communication bands: they all have wavelengths of tens of centimeters. So devices for wireless reception and transmission tend to be on the order of that size.


Band f (MHz) λ (cm) Band f (MHz) λ (cm)1 2100 14.3 20 800 37.5

2 1900 15.8 25 1900 15.8

3 1800 16.7 26 800 37.5

5 850 35.3 28 700 42.8

7 2600 11.5 29 700 42.8

8 900 33.3 30 2300 13.0

12 700 42.8 38 2600 11.5

13 700 42.8 39 1900 15.8

17 700 42.8 40 2300 13.0

18 700 42.8 41 2500 12.0

19 800 37.5

Reception of long-wavelength electromagnetic radiation

How might we receive long-wavelength wireless signals? That is, how might we turn the signals into currents in circuits?

By aligning a conductor wire of suitable shape and length with the electromagnetic radiation’s E, and sensing the current driven thereby in the wire in the circuitry you put in series with it;

or

by aligning a loop of wire perpendicular to the electromagnetic radiation’s B, which will induce an emf in the loop; and sensing the current driven thereby in the following circuitry.

The simplest forms these take are the antennas respectively called half-wave (or dipole) antennas and loop antennas. Let’s calculate the current that a plane electromagnetic wave will induce, in each kind.


Half-wave antenna

Make a straight conductor, of length half the wavelength you want to receive, interrupted in the center and free on the ends. The center leads are connected to the circuits (amplifiers, etc.) with which the signal is processed.

With orient the half-wavelength antenna in the xdirection, along the incoming light’s polarization.


Ex

z

2λ

Antenna

To sensing circuits

I

( ) ( )0ˆ, ,i z tz t E e κ ω−=E x

R

Half-wave antenna (continued)

The amplitude of the voltage across the antenna is

The current driven in the antenna and circuit can’t go past the ends of the antenna; it “reflects” from these ends. Thus the electromagnetic wave drives a resonance: a standing wave of current in the antenna. With x = 0 in the center of the antenna,


Ex

z

2λ

Antenna

To sensing circuits

I

0 0 2 .V Eλ=

( ) 0 00cos cos cos .

2

i t i tV e E eI x x x I x

R R

ω ωλκ κ κ

− −= = =

R


Thus the current driven in the sensing circuit is the current at the center feed:


( ) 0 00 Re cos .

2 2

i tE e EI I t t

R R

ωλ λω

−= = =

Ex

z

2λ

Antenna

To sensing circuits

I

R



(from Chetvorno)

https://en.wikipedia.org/wiki/Dipole_antenna#/media/File:Dipole_receiving_antenna_animation_6_800x394x150ms.gif

Loop antenna

Make a loop with N turns of a conducting wire, each loop having area A, and with its ends connected to the circuits (amplifiers, etc.) with which the signal is processed. Make the loops small compared to λ in all dimensions.

With orient the loop in the x-z plane, perpendicular to the magnetic field, and parallel to the signal’s polarization (direction of E).


B

x

z

Antenna(N turns)

To sensing circuits

I

Ry

A

( ) ( )0ˆ, ,i z tz t E e cκ ω−=B y

Not to scale: loop is really much smaller than λ.

Loop antenna (continued)

With the loop much smaller than a wavelength, and centered at x = z = 0, the magnetic field is approximately uniform over the loop’s area, and the magnetic flux through one turn of the loop is


B

x

z

Antenna(N turns)

To sensing circuits

I

Ry

A

( )0 0 .i z t i tB

E A E ABA e e

c cκ ω ω− −Φ = = =

0

Loop antenna (continued)

According to Faraday’s Law,

whence


B

x

z

Antenna(N turns)

To sensing circuits

I

Ry

A

0 0 ,i t i tB E A E Ad dN N e i N edt c dt c

ω ωω− −Φ= − = − =

( )0 0Re Re cos sin sin .NE A NE A

I i t i t tR cR cR

ω ωω ω ω

= = − =

today in physics 122: antennas - university of rochester

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