physicsa.com
 class code: SPS22 teacher: Mr. Elert classroom: A314 office: A214 test day: Friday phone: (718) 724–8500 ext. 2141 email: elert@midwoodscience.org

## Physics A: Problem Set 21: Electromagnetic Waves

### recommended reading

 Barron's Let's Review: n/a physics.info: n/a Wikipedia: Electromagnetic radiation, Electromagnetic spectrum

### in class

1. Sound is a longitudinal mechanical wave and radio is a transverse electromagnetic wave. Humans and other animals have organs called ears for receiving sound waves. They do not have organs for receiving radio waves. Such a device is called a radio receiver or just a radio. Analog radio relies one of two basic techniques to encode sound information into a radio signal — by varying the amplitude of the radio signal (amplitude modulation or AM) or by varying the frequency of the radio signal (frequency modulation or FM).
1. AM radio broadcasts in the US are allocated frequencies from 530 to 1710 kHz. What range of wavelengths correspond to these frequencies? About how big is this?
2. FM radio broadcasts in the US are allocated frequencies from 88 to 108 MHz. What range of wavelengths correspond to these frequencies? About how big is this?

Wavelength is computed using the wave speed equation and the speed of light in a vacuum (c = 3.00 × 108 m/s).

 λ = c ⇐ c = fλ f

Do this four times. Recall that a kilohertz is one thousand hertz (103 Hz) and a megahertz is one million hertz (106 Hz).

1. AM broadcast wavelengths are a few city blocks long. They are neighborhood sized.

λ =
 c f

λ =  3.00 × 108 m/s
530 × 103 Hz
λ =  566 m

λ =
 c f

λ =  3.00 × 108 m/s
1710 × 103 Hz
λ =  175 m

2. FM broadcast wavelengths would probably fit inside most of the rooms in a house. They are room-sized.

λ =
 c f

λ =  3.00 × 108 m/s
88 × 106 Hz
λ =  3.41 m

λ =
 c f

λ =  3.00 × 108 m/s
108 × 106 Hz
λ =  2.78 m

2. Mobile phone companies use many frequencies, but across the Americas most of them are in the 800 MHz or 1900 MHz bands. The 800 MHz band is the original "Cellular" band, introduced in 1983. The 1900 MHz band is the newer "PCS" band, introduced in 1994.
1. The Cellular band spans frequencies from 824 to 894 MHz. What range of wavelengths correspond to these frequencies? About how big is this?
2. The PCS band spans frequencies from 1850 to 1990 MHz. What range of wavelengths correspond to these frequencies? About how big is this?

Wavelength is computed using the wave speed equation and the speed of light in a vacuum (c = 3.00 × 108 m/s).

 λ = c ⇐ c = fλ f

Do this four times. Recall that a megahertz is one million hertz (106 Hz).

1. Mobile phone signals in the Cellular band have wavelengths that are about as long as a large adult foot. They are foot-sized.

λ =
 c f

λ =  3.00 × 108 m/s
824 × 106 Hz
λ =  0.364 m = 36 cm

λ =
 c f

λ =  3.00 × 108 m/s
894 × 106 Hz
λ =  0.336 m = 34 cm

2. Mobile phone signals in the PCS band have wavelengths that are somewhere between the length and width of a large adult hand. They are hand-sized.

λ =
 c f

λ =  3.00 × 108 m/s
1850 × 106 Hz
λ =  0.162 m = 16 cm

λ =
 c f

λ =  3.00 × 108 m/s
1990 × 106 Hz
λ =  0.151 m = 15 cm

3. The wireless networking standard known as Wi-Fi encompasses several frequency bands in the microwave part of the electromagnetic spectrum. The two most common are the 2.4 GHz and 5 GHz bands.
1. Wi-Fi channels in the 2.4 GHz band range from 2.401 GHz to 2.473 GHz in the US. What range of wavelengths correspond to these frequencies? About how big is this?
2. Wi-Fi channels in the 5 GHz band range from 5.170 GHz to 5.835 GHz in the US. What range of wavelengths correspond to these frequencies? About how big is this?

Wavelength is computed using the wave speed equation and the speed of light in a vacuum (c = 3.00 × 108 m/s).

 λ = c ⇐ c = fλ f

Do this four times. Recall that a gigahertz is one billion hertz (109 Hz).

1. Wi-Fi wavelengths in the 2.4 GHz band would fit inside a pint glass. They are pint-sized.

λ =
 c f

λ =  3.00 × 108 m/s
2.401 × 109 Hz
λ =  0.125 m = 12.5 cm

λ =
 c f

λ =  3.00 × 108 m/s
2.473 × 109 Hz
λ =  0.121 m = 12.1 cm

2. Wi-Fi wavelengths in the 5 GHz band would fit inside a coffee cup. They are cup-sized.

λ =
 c f

λ =  3.00 × 108 m/s
5.170 × 109 Hz
λ =  0.058 m = 5.8 cm

λ =
 c f

λ =  3.00 × 108 m/s
5.835 × 109 Hz
λ =  0.051 m = 5.1 cm

4. Bluetooth is a wireless technology standard for exchanging data over short distances using ultrahigh frequency radio waves (UHF) in one of the internationally reserved industrial, scientific, and medical radio bands (ISM). Equipment operating in an ISM band must tolerate interference generated by other users, which limits its use to short range, low power applications like Bluetooth, Wi-Fi, microwave ovens, and and medical diathermy (therapeutic heating). The Bluetooth band is divided into 79, 1 MHz channels that start at 2402 MHz and end at 2480 MHz. What range of wavelengths correspond to these frequencies? About how big is this?

Wavelength is computed using the wave speed equation and the speed of light in a vacuum (c = 3.00 × 108 m/s).

 λ = c ⇐ c = fλ f

Do this two times. Recall that a megahertz is one million hertz (106 Hz).

Bluetooth signals are about the same size as a compact disc. Remember compact discs? They were invented in 1982 and lasted about 30 years as the supreme physical storage medium before they were replaced with nothing. Given that Bluetooth was invented in 1992, logic dictates it should disappear around 2022.

λ =
 c f

λ =  3.00 × 108 m/s
2402 × 106 Hz
λ =  0.125 m

λ =
 c f

λ =  3.00 × 108 m/s
2480 × 106 Hz
λ =  0.121 m

5. Police, fire, and other emergency services in New York City communicate mostly using ultra high frequency (UHF) narrowband frequency modulation (NFM) push-to-talk two way radios. Frequencies used range from 451.3 MHz to 487.5 MHz. What range of wavelengths correspond to these frequencies? About how big is this?

Wavelength is computed using the wave speed equation and the speed of light in a vacuum (c = 3.00 × 108 m/s).

 λ = c ⇐ c = fλ f

Do this two times. Recall that a megahertz is one million hertz (106 Hz).

First responders in New York City use radios that broadcast at wavelengths that are as long as a stepladder is tall.

λ =
 c f

λ =  3.00 × 108 m/s
451.3 × 106 Hz
λ =  0.668 m

λ =
 c f

λ =  3.00 × 108 m/s
487.5 × 106 Hz
λ =  0.615 m

6. The satellites in the US Global Positioning System (GPS) broadcast microwave signals that enable GPS receivers on or near the Earth's surface to determine their location and time and to derive their velocity (speed and bearing). All GPS satellites currently broadcast on two frequencies: L1 at 1.57542 GHz and L2 at 1.22760 GHz. (L3 is a legacy system for detonating nuclear warheads, L4 is currently experimental, and L5 is for more demanding tasks like landing airplanes.) Determine the corresponding wavelengths of the L1 and L2 GPS frequencies. About how big is this?

Wavelength is computed using the wave speed equation and the speed of light in a vacuum (c = 3.00 × 108 m/s).

 λ = c ⇐ c = fλ f

Do this two times. Recall that a gigahertz is one billion hertz (109 Hz).

GPS signals are hand sized. The L1 wavelengths are close to the distance from the heel of my hand to the tip of my middle finger (a hand length). The L2 wavelengths are slightly longer than the width of my outspread hand from the tip of my thumb to the tip of my little finger (a hand span).

λ =
 c f

λ =  3.00 × 108 m/s
1.57542 × 109 Hz
λ =  0.190 m

λ =
 c f

λ =  3.00 × 108 m/s
1.22760 × 109 Hz
λ =  0.244 m

### at home

1. Two simple facts
1. What is the source of all magnetism?
2. What is the source of all electromagnetic waves?
1. The source of all magnetism is moving electric charge.
2. The source of all electromagnetic waves is accelerating electric charge.
2. A typical household microwave oven operates at a frequency of 2.45 GHz.
1. What is the wavelength of this radiation?
2. Hotspots in a microwave oven are caused by the formation of standing waves within the cooking chamber. What is the separation between hotspots in a 2.45 GHz microwave oven?
3. How is the problem of hotspots dealt with in the design of microwave ovens or in the cooking techniques employed by the people that use them?
1. Use the wave speed equation (c = fλ) and the speed of light in a vacuum (c = 3.00 × 108 m/s) to compute the wavelength of these microwaves.

λ =
 c f

λ =  3.00 × 108 m/s
2.45 × 109 Hz
λ =  0.12 m = 12 cm

2. Hotspots in a microwave oven correspond to antinodes on a standing wave (locations with maximal amplitude). Adjacent antinodes are half a wavelength apart.

 ℓ = ½λℓ = ½(12 cm)ℓ = 6 cm
3. To reduce the effect of hotspots in microwave cooking, the food should be rotated, stirred, or otherwise moved around. This is a practice that makes sense whenever heat is applied to food. It is a technique that is as old as cooking. The controlled use of fire goes back 1–2 million years. Modern humans (Homo sapiens) appeared 100–200 thousand years ago. Countertop microwave ovens first became popular in the 1970s.