The Speed of Light Equation and Planck's Equation

You'll be working with two fundamental equations that describe the behavior of light (also known as electromagnetic radiation).

1. The Speed of Light Equation: This formula connects the speed of light (c), its wavelength (λ), and its frequency (f).

2. Planck's Equation: This formula connects the energy (E) of a single photon of light to its frequency (ν) using Planck's constant (h).

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Equation 1: Speed of Light, Wavelength, and Frequency

Imagine waves passing you in the ocean. Wavelength is the distance from the peak of one wave to the peak of the next. Frequency is how many waves pass you every second. If the waves are close together (short wavelength), more of them will pass you per second (high frequency). If they are far apart (long wavelength), fewer will pass you (low frequency).

This inverse relationship is captured in the following equation:

c = (λ) (f)

• c (speed of light): This is a constant! Its value is approximately 3.00 x 10^8 m/s. Light always travels at this speed in a vacuum.

• λ (lambda): This is the wavelength. It's a measure of distance, so its standard unit is meters (m). You will often see it given in nanometers (nm), so you'll need to convert it (1 nm = 10 x 10^-9 m).

• f: This is the frequency. It's a measure of cycles per second, and its unit is Hertz (Hz), which is the same as 1 over a second. 

Example Calculation:

A green laser pointer emits light with a wavelength of 532 nm. What is its frequency?

1. Identify your variables:

   • λ = 532 nm

   • c = 3.00 x 10^8 m/s

   • f = ?

2. Convert units to match the constant: Wavelength must be in meters.

   532 nm ​ =5.32×10^-7 m

3. Rearrange the formula to solve for the unknown (f):

   c = (λ)(f) ⟹ f = c​/λ

4. Plug in the numbers and solve:

   f=5.64 x 10^14 Hz

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Equation 2: Energy and Frequency (Planck's Equation)

Max Planck discovered that energy is quantized, meaning it comes in discrete packets called photons. The energy of a single photon is directly proportional to its frequency. Higher frequency light (like UV or X-rays) has more energy per photon than lower frequency light (like radio waves or infrared).

This relationship is described by Planck's equation:

E = (h) (f) 

• E: This is the energy of a single photon. Its unit is Joules (J).

• h (Planck's constant): This is another very important constant. Its value is 6.63 x 10^-34 J·s (Joule-seconds).

• f: This is the frequency in Hertz (Hz). 

Example Calculation:

What is the energy of a photon from the green laser pointer in the previous example (frequency = 5.64 x 10^14 Hz)? 

1. Identify your variables:

   • f = 5.64 x 10^14 Hz

   • h = 6.63 x 10^-34 J·s 

   • E = ?

2. Plug in the numbers and solve:

   E = 3.74 x 10^-19 J