Wavelength and Energy Example Problem

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Human Eye
You can use this wavelength and energy example to calculate the number of photons needed to “see”. Credit: Petr Novák, Wikipedia

This wavelength and energy example problem will show how to find the energy of a photon from its wavelength.

First, let’s look at the physics of the problem. The energy of a single photon of light is dependent on its frequency. This relationship is expressed in the equation

E = hƒ

where
E is the energy of the photon
h is Planck’s constant = 6.626 x 10-34 m2kg/s
ƒ is the frequency of the photon

The wavelength of a photon is related to the frequency by the equation

c = ƒλ

where
c is the speed of light = 3.0 x 108 m/s
ƒ is the frequency
λ is the wavelength

Solve this for frequency, and you get

ƒ = c / λ

Substitute this equation into the energy equation and get

E = hc / λ

With this equation, you can now find the energy of a photon when the wavelength is known. You can also find the wavelength if the energy of the photon is known.

Wavelength and Energy Example Problem

Question: The human eye’s optic nerve needs 2 x 10-17 joules of energy to trigger a series of impulses to signal the brain there is something to see. How many photons of 475 nm blue light is needed to trigger this response?

Solution: We are given the amount of energy needed to trigger the optic nerve and the wavelength of light.

First, let’s figure out how much energy is in a single photon of the blue light. We are given the wavelength as 475 nm. Before we go any further, let’s convert this to meters.

1 nm = 10-9 m

Using this relationship, convert 475 nm to meters

nm to m conversion example

x m = 4.75 x 10-7 m

Now we can use the energy formula from above

E = hc / λ

Plug in the variables

E = (6.626 x 10-34 m2kg/s)(3 x 108 m/s) / 4.75 x 10-7 m

Solve for E

E = 4.18 x 10-19 J

This is the energy of a single photon of 475 nm blue light. We need 2 x 10-17 J of energy to begin the process.

x photons = 2 x 10-17 J

1 photon = 4.18 x 10-19 J

Divide one equation into the other to get

Solve for x

x = 47.8 photons

Since you can’t have partial photons, we need to round this answer up to the nearest whole photon count. 47 photons isn’t enough, so one more is needed to get over the threshold energy.

x = 48 photons

Answer: It takes 48 photons of 475 nm blue light to trigger the optic nerve.