Question

Human color vision is "produced" by the nervous system based on how three different cone receptors interact with photons of light in the eye. These three different types of cones interact with photons of different frequency light, as indicated in the following chart: $$ \begin{array}{|lc|} \hline \text { Cone Type } & \begin{array}{c} \text { Range of Light } \\ \text { Frequency Detected } \end{array} \\ \hline \mathrm{S} & 6.00-7.49 \times 10^{14} \mathrm{~s}^{-1} \\ \mathrm{M} & 4.76-6.62 \times 10^{14} \mathrm{~s}^{-1} \\ \mathrm{~L} & 4.28-6.00 \times 10^{14} \mathrm{~s}^{-1} \\ \hline \end{array} $$ What wavelength ranges (and corresponding colors) do the three types of cones detect?

Short answer

The S cone detects wavelengths ranging from 400 nm to 500 nm, which corresponds to blue-violet light. The M cone detects wavelengths ranging from 453 nm to 630 nm, which corresponds to green-red light. The L cone detects wavelengths ranging from 500 nm to 701 nm, which corresponds to orange-red light.

Step-by-step solution

Identify the formula for the relationship between frequency and wavelength

The formula we will use to find the wavelength ranges for each cone is: c = λf Where: -c is the speed of light (2.998 x 10^8 m/s) -λ is the wavelength (in meters) -f is the frequency (in Hz, or 1/s)

Rearrange the formula to find the wavelength

We will rearrange the formula to get the wavelength (λ) on one side: λ = c/f

Calculate the wavelength range for the S cone

We will use the formula λ = c/f to find the wavelength range for the S cone: Lowest frequency: \(f = 6.00 \times 10^{14} Hz\) λ = (2.998 x 10^8 m/s) / (6.00 x 10^14 Hz) ≈ 4.997 x 10^(-7) m ≈ 500 nm Highest frequency: \(f = 7.49 \times 10^{14} Hz\) λ = (2.998 x 10^8 m/s) / (7.49 x 10^14 Hz) ≈ 4.00 x 10^(-7) m ≈ 400 nm The S cone detects wavelengths ranging from 400 nm to 500 nm (blue-violet light).

Calculate the wavelength range for the M cone

We will use the formula λ = c/f to find the wavelength range for the M cone: Lowest frequency: \(f = 4.76 \times 10^{14} Hz\) λ = (2.998 x 10^8 m/s) / (4.76 x 10^14 Hz) ≈ 6.30 x 10^(-7) m ≈ 630 nm Highest frequency: \(f = 6.62 \times 10^{14} Hz\) λ = (2.998 x 10^8 m/s) / (6.62 x 10^14 Hz) ≈ 4.53 x 10^(-7) m ≈ 453 nm The M cone detects wavelengths ranging from 453 nm to 630 nm (green-red light).

Calculate the wavelength range for the L cone

We will use the formula λ = c/f to find the wavelength range for the L cone: Lowest frequency: \(f = 4.28 \times 10^{14} Hz\) λ = (2.998 x 10^8 m/s) / (4.28 x 10^14 Hz) ≈ 7.01 x 10^(-7) m ≈ 701 nm Highest frequency: \(f = 6.00 \times 10^{14} Hz\) λ = (2.998 x 10^8 m/s) / (6.00 x 10^14 Hz) ≈ 4.997 x 10^(-7) m ≈ 500 nm The L cone detects wavelengths ranging from 500 nm to 701 nm (orange-red light).

Report the wavelength ranges and corresponding colors for each cone type

The S cone detects wavelengths ranging from 400 nm to 500 nm, which corresponds to blue-violet light. The M cone detects wavelengths ranging from 453 nm to 630 nm, which corresponds to green-red light. The L cone detects wavelengths ranging from 500 nm to 701 nm, which corresponds to orange-red light.

Key concepts

Frequency and Wavelength Relationship

The intricate dance between frequency and wavelength is an essential aspect of physics that plays a crucial role in understanding light and color vision. This relationship is beautifully simplified in the equation \( c = \lambda f \), where \( c \) represents the constant speed of light, \( \lambda \) stands for wavelength, and \( f \) denotes the frequency. In a medium like a vacuum, light travels at approximately \( 2.998 \times 10^8 \) meters per second (m/s), linking wavelength and frequency as inversely proportional entities. This means that as one increases, the other must decrease for the speed of light to remain constant.

Take for example a high-frequency photon of light: it will exhibit a shorter wavelength, which is typical for blue or violet light in the visible spectrum. Conversely, a low-frequency photon corresponds to a longer wavelength, such as that of red light. Knowing this relationship allows us to unlock the secrets of the colors that our eyes perceive, giving insight into the ranges of light that different cone receptors can detect.

Visible Light Spectrum

At the heart of human color vision lies the visible light spectrum, a tiny, vibrant segment of the electromagnetic spectrum to which our eyes are sensitive. Ranging from about 380 to 750 nanometers (nm), this spectrum includes all the colors of the rainbow. Each color has its own characteristic wavelength. Violet light has the shortest wavelength, typically around 380 nm, and as we move through the spectrum, each color gradually increases in wavelength: blue, green, yellow, orange, and finally red, which has the longest wavelength, around 750 nm.

When we discuss color vision, we refer to the ability of different cone receptors in the eye to detect certain ranges of these wavelengths. For instance, the S cone mainly responds to short wavelengths—violet and blue—while the L cone is more sensitive to longer wavelengths like yellow, orange, and red. It is this interplay of cones and the visible light spectrum that allows us to enjoy the vast array of colors in our world.

Human Color Vision

Human color vision is a marvel of biological engineering, involving three types of cone receptors in the eye, each fine-tuned to specific segments of the visible light spectrum. The S cones respond to short wavelengths, which correspond to blue-violet colors; the M cones react to medium wavelengths, covering greens and some reds; and the L cones are attuned to long wavelengths, responsible for orange-red hues. The perception of color happens when light enters the eye and stimulates these cones to various degrees based on its wavelength.

The diversity in color perception among humans results from variations in the wavelengths these cones are sensitive to. In the exercise, by calculating the wavelength ranges for each cone type using the relationship between frequency and wavelength, we uncovered that S cones detect the range between blue and violet, M cones capture the transition from green to red, and L cones pick up on hues from orange to red. This fundamental understanding of cone responses explains not just how we distinguish colors, but also why certain colors may appear more vibrant or nuanced depending on the individual's unique set of cones and their sensitivity to light.