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Chapter 7: Problem 64

The optic nerve needs a minimum of \(2.0 \times 10^{-17} \mathrm{~J}\) of energy to trigger a series of impulses that eventually reach the brain. (a) How many photons of red light ( \(700 . \mathrm{nm}\) ) are needed? (b) How many photons of blue light (475 nm)?

Short Answer

Expert verified
About 70 photons of red light and about 48 photons of blue light are needed.

Step by step solution

01

Understand the problem

We need to find the number of photons of red and blue light required to deliver the specified energy to the optic nerve.
02

Write down the energy of one photon

Use the formula for the energy of a photon: \[ E = \frac{hc}{\text{wavelength}} \]
03

Calculate the energy of one photon for red light

For red light (wavelength = 700 nm): \(\text{E}_{\text{red}} = \frac{6.626 \times 10^{-34} \times 3.00 \times 10^8}{700 \times 10^{-9}} \approx 2.84 \times 10^{-19} \text{ J}\)
04

Calculate the number of photons for red light

Using \( \text{number of photons} = \frac{\text{required energy}}{\text{energy per photon}} \): \[ \text{number of red photons} = \frac{2.0 \times 10^{-17} \text{ J}}{2.84 \times 10^{-19} \text{ J}} \approx 70.42 \]
05

Calculate the energy of one photon for blue light

For blue light (wavelength = 475 nm): \(\text{E}_{\text{blue}} = \frac{6.626 \times 10^{-34} \times 3.00 \times 10^8}{475 \times 10^{-9}} \approx 4.19 \times 10^{-19} \text{ J}\)
06

Calculate the number of photons for blue light

Using \( \text{number of photons} = \frac{\text{required energy}}{\text{energy per photon}}\): \[ \text{number of blue photons} = \frac{2.0 \times 10^{-17} \text{ J}}{4.19 \times 10^{-19} \text{ J}} \approx 47.73 \]

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Optic Nerve Energy Threshold
The optic nerve is an essential part of the human visual system. It translates light signals into electrical impulses that the brain can interpret as images.
For this to happen, a certain amount of energy is needed to activate the optic nerve. This minimum energy requirement is known as the 'optic nerve energy threshold'. In this exercise, we are given that the threshold is \( 2.0 \times 10^{-17} \text{ J} \).
Understanding this concept helps us determine how many photons of light are required to trigger our vision.
Photon Energy Formula
The energy of a single photon can be calculated using the formula: \( E = \frac{hc}{\text{wavelength}} \).
Where:
  • \text{E}: Energy of the photon
  • \text{h}: Planck's constant ( \( 6.626 \times 10^{-34} \text{ J} \cdot\text{s} \) )
  • \text{c}: Speed of light ( \( 3.0 \times 10^8 \text{ m/s} \) )
  • \text{wavelength}: Wavelength of the light
This formula is important because it connects the wave-like nature of light with its particle-like properties.
By knowing the wavelength, we can find out how much energy a photon of that specific light contains.
Wavelength of Light
Light comes in various wavelengths, which determines its color.
The wavelength of light is the distance between successive crests of a wave.
In this exercise:
  • Red light has a wavelength of \( 700 \text{ nm} \) (nanometers)
  • Blue light has a wavelength of \( 475 \text{ nm} \)
Understanding wavelengths helps us determine how much energy a photon of that particular light will have.
Since energy is inversely proportional to wavelength, shorter wavelengths (blue light) have more energy compared to longer wavelengths (red light).
Red Light Photons
To calculate the energy of a single photon of red light, we use its wavelength: \( \text{E}_{\text{red}} = \frac{6.626 \times 10^{-34} \times 3.00 \times 10^8}{700 \times 10^{-9}} \approx 2.84 \times 10^{-19} \text{ J} \).
This is the energy of one photon of red light.
Next, to find out how many such photons are needed to meet the optic nerve energy threshold, we divide the required energy by the energy per photon:
\( \text{number of red photons} = \frac{2.0 \times 10^{-17} \text{ J}}{2.84 \times 10^{-19} \text{ J}} \approx 70.42 \).
Thus, approximately 70 photons of red light are required to trigger the optic nerve.
Blue Light Photons
Similarly, for blue light, we calculate the energy of a single photon using its wavelength:
\( \text{E}_{\text{blue}} = \frac{6.626 \times 10^{-34} \times 3.00 \times 10^8}{475 \times 10^{-9}} \approx 4.19 \times 10^{-19} \text{ J} \).
Then, we divide the required energy for the optic nerve by this energy per photon:
\( \text{number of blue photons} = \frac{2.0 \times 10^{-17} \text{ J}}{4.19 \times 10^{-19} \text{ J}} \approx 47.73 \).
Therefore, about 47 photons of blue light are needed to reach the optic nerve's energy threshold.
This shows that fewer photons of blue light are required because each blue photon carries more energy than a red photon.

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Most popular questions from this chapter

How fast must a \(56.5-g\) tennis ball travel to have a de Broglie wavelength equal to that of a photon of green light ( \(5400 \mathrm{~A}\) )?

A sodium flame has a characteristic yellow color due to emission of light of wavelength \(589 \mathrm{nm}\). What is the mass equivalence of one photon with this wavelength \(\left(1 \mathrm{~J}=1 \mathrm{~kg} \cdot \mathrm{m}^{2} / \mathrm{s}^{2}\right) ?\)

Use the Rydberg equation to find the wavelength (in \(\mathrm{A}\) ) of the photon absorbed when an electron in an \(\mathrm{H}\) atom undergoes a transition from \(n=1\) to \(n=3\).

Enormous numbers of microwave photons are needed to warm macroscopic samples of matter. A portion of soup containing \(252 \mathrm{~g}\) of water is heated in a microwave oven from \(20 .^{\circ} \mathrm{C}\) to \(98^{\circ} \mathrm{C}\), with radiation of wavelength \(1.55 \times 10^{-2} \mathrm{~m}\). How many photons are absorbed by the water in the soup?

A radio wave has a frequency of \(3.8 \times 10^{10} \mathrm{~Hz}\). What is the energy (in \(J\) ) of one photon of this radiation?
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