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### 905-G: Photon Formulas 1

• Topic Cluster: Waves
• Topic: The Electromagnetic Spectrum: Quantitative
• Objective: Quantitatively relate wavelength, frequency, and photon energy of a wave; problems presented step by step
• Content: Physicists use two key formulas to relate the wavelength, frequency, and energy of an electromagnetic photon.
• Level: 3

#### BACK to Ladder The Electromagnetic Spectrum: Quantitative

All electromagnetic waves, including both visible light and invisible light, come in tiny pieces called photons. Each photon has three quantities that determine its physical behavior: wavelength, frequency, and energy. Wavelength, frequency, and energy determine if light is visible or invisible. For visible light, they determine what color it is. They determine whether a wave is dangerous, and how dangerous. They determine if a wave can travel through walls or water. Wavelength, frequency, and energy determine the properties of an electromagnetic photon. In this pod, we will learn the formulas that relate these three crucial quantities.

There is one quantity, however, that is identical for all waves: speed. All electromagnetic photons, whether red or violet, radio waves or gamma rays, move at exactly the same speed: 300,000,000 m/s, or 3 x 108 m/s.

The formulas involved are not long and complicated, but there are some mathematical challenges. Whenever we use formulas, we must always use the SI units: Joules for Energy, Hertz for frequency, and meters for wavelength. But these quantities are frequently posted in units different than the SI unit! Visible light photons are typically reported in electronVolts for energy, teraHertz for frequency, and nanometers for wavelength. As a result, many unit conversions are necessary to use the photon formulas. To understand these problems, you must first be adept at using conversion factors, completing metric conversions, and using scientific notation on a calculator.

In this pod you will focus exclusively on photons of visible light. In the next pod you will apply the formulas to all wavelengths of light, visible and invisible.

## Pre-Problem Information

#### Reference: Table of Wavelengths

Type of Wave
Wavelength
greater than 1 meter
Microwave
between 1 millimeter and 1 meter
Infrared Light
between 700 nanometers and 1 millimeter

Visible Light

Red
between 635 nanometers and 700 nanometers
Orange
between 590 nanometers and 635 nanometers
Yellow
between 560 nanometers and 590 nanometers
Green
between 520 nanometers and 560 nanometers
Blue
between 450 nanometers and 520 nanometers
Violet
between 400 nanometers and 450 nanometers
Ultraviolet Light
between 10 nanometers and 400 nanometers
X-ray
between 1 picometer and 10 nanometers
Gamma Ray
less than 1 picometer

#### Definitions

All light (electromagnetic waves) comes in tiny packets of energy called photons.

All photons, no matter the color, have the same speed: 3 x 108 m/s.

All photons consist of oscillating electric and magnetic energy. The frequency tells how many times per second these energies oscillations. Frequencies of visible light are huge numbers (on the order of 1014 oscillations per second.)

The wavelength of a photon determines how far it moves while undergoing a single oscillation.

The energy of a photon will determine how it interacts with your eye, giving it color. The energies of photons are on the orders of 1 - 10 electronVolts.

#### Formulas

$$c = \lambda f$$

Symbol
Quantity
SI Unit
$$c$$
Speed of Light (3e8)
m/s
$$\lambda$$
wavelength
meters
$$f$$
frequency
Hertz (1/s)

Note that frequency is also sometimes denoted by the Greek letter nu: $$\nu$$.

$$E = h f$$

Symbol
Quantity
SI Unit
$$E$$
Energy of Photon
Joules (J)
$$h$$
Planck's Constant (6.626e-34)
Joule-seconds
$$f$$
frequency
Hertz (1/s)

#### Fundamental Constants

Both of the equations above involve a fundamental constant. This is a number with a known value that never changes.

• The speed of light:
• The speed of light is a fundamental constant represented by $$c$$.
• It is equal to $$3.00 \cdot 10^8 \text{ m/s}$$.
• Planck's Constant:
• Plan's constant is a fundamental constant that represents the relationship between the frequency and energy of a photon.
• It is represented by $$h$$.
• It is equal to $$6.626 \cdot 10^{-34} \text{ J} \cdot \text{s}$$

#### Conversions and Other Units

• Wavelength:
• The SI unit for wavelength is meters.
• Whenever using a formula, wavelength must always be in units of meters.
• However, wavelength is frequently expressed in nanometers as well.
• Frequency:
• The SI unit for frequency is Hertz.
• Whenever using a formula, frequency must always be in units of Hertz.
• However, frequency is frequently expressed in teraHertz as well.
• Energy:
• The SI unit for energy is Joules.
• Whenever using a formula, energy must always be in units of Joules.
• However, energy is frequently expressed in electron Volts (eV) as well.

$$1 \text{ Joules} = 6.242 \cdot 10^{18} \text{ electronVolts}$$

## Problems

For full credit on the practice page, you must show your work in solving these problems. You must also use the conversion factor method for all non-metric conversions; you do not need to use it for metric conversions. Please write all answers to three significant figures of accuracy.

1. A photon of visible light has a wavelength of 600. nanometers.
1. Determine what color this photon will be.
2. Convert the the wavelength to meters in scientific notation. Because this is a metric conversion, you do not need to use conversion factors.
3. Determine the frequency of the photon in Hertz.
4.  Looking For Formula Already Know Answer in a complete sentence with unit.
5. Convert the frequency of the photon into teraHertz. Because this is a metric conversion, you do not need to use conversion factors.
6. Determine the energy of the photon in Joules. You must use the frequency in Hertz to complete this calculation.
7.  Looking For Formula Already Know Answer in a complete sentence with unit.
8. Convert the energy of the photon into electronVolts. For full credit, you must use the conversion factor method correctly.
2. A photon of visible light has a wavelength of 432. nanometers.
1. Determine what color this photon will be.
2. Convert the the wavelength to meters in scientific notation.
3. Determine the frequency of the photon in Hertz.
4.  Looking For Formula Already Know Answer in a complete sentence with unit.
5. Convert the frequency of the photon into teraHertz.
6. Determine the energy of the photon in Joules.
7.  Looking For Formula Already Know Answer in a complete sentence with unit.
8. Convert the energy of the photon into electronVolts. For full credit, you must use the conversion factor method correctly.
3. A photon of visible light has an energy of 2.48 electronVolts.
1. Convert the energy of the photon into Joules. For full credit, you must use the conversion factor method correctly. (Write your answer to 3 significant figures of accuracy, but then continue working with an unrounded number.)
2. Determine the frequency of the photon in Hertz.
3.  Looking For Formula Already Know Answer in a complete sentence with unit.
4. Convert the frequency of the photon into teraHertz.
5. Determine the wavelength of the photon in meters.
6.  Looking For Formula Already Know Answer in a complete sentence with unit.
7. Convert the wavelength of the photon into nanometers.
8. Determine what color this photon will be.
4. A photon of visible light has an energy of 2.30 electronVolts.
1. Convert the energy of the photon into Joules. For full credit, you must use the conversion factor method correctly. (Write your answer to 3 significant figures of accuracy, but then continue working with an unrounded number.)
2. Determine the frequency of the photon in Hertz.
3.  Looking For Formula Already Know Answer in a complete sentence with unit.
4. Convert the frequency of the photon into teraHertz.
5. Determine the wavelength of the photon in meters.
6.  Looking For Formula Already Know Answer in a complete sentence with unit.
7. Convert the wavelength of the photon into nanometers.
8. Determine what color this photon will be.
5. A photon of visible light has a frequency of 462 teraHertz.
1. Convert the frequency of the photon into Hertz.
2. Determine the energy of the photon into Joules.
3.  Looking For Formula Already Know Answer in a complete sentence with unit.
4. Convert the energy of the photon into electronVolts. For full credit, you must use the conversion factor method correctly.
5. Determine the wavelength of the photon in meters.
6.  Looking For Formula Already Know Answer in a complete sentence with unit.
7. Convert the wavelength of the photon into nanometers.
8. Determine what color this photon will be.
6. A photon of visible light has a frequency of 533 teraHertz.
1. Convert the frequency of the photon into Hertz.
2. Determine the energy of the photon into Joules.
3.  Looking For Formula Already Know Answer in a complete sentence with unit.
4. Convert the energy of the photon into electronVolts. For full credit, you must use the conversion factor method correctly.
5. Determine the wavelength of the photon in meters.
6.  Looking For Formula Already Know Answer in a complete sentence with unit.
7. Convert the wavelength of the photon into nanometers.
8. Determine what color this photon will be.