Question

A nocturnal bird's eye can detect monochromatic light of frequency \(5.8 \cdot 10^{14} \mathrm{~Hz}\) with a power as small as \(2.333 \cdot 10^{-17} \mathrm{~W}\). What is the corresponding number of photons per second that the nocturnal bird's eye can detect?

Short answer

Answer: The nocturnal bird's eye can detect approximately 606 photons per second at the given power and frequency.

Step-by-step solution

Calculate the energy of a single photon

The energy of a single photon can be determined using the formula, E = h * f Where E is the energy of the photon, h is the Planck's constant (6.626 * 10^{-34} Js), and f is the frequency of the light (5.8 * 10^{14} Hz). E = (6.626 * 10^{-34}) * (5.8 * 10^{14}) E = 3.8428 * 10^{-19} J So, the energy of a single photon is 3.8428 * 10^{-19} Joules.

Calculate the number of photons per second

We are given the power corresponding to the minimum detection threshold as 2.333 * 10^{-17} W. Since power is energy per unit time, we can find the number of photons by dividing this power by the energy of a single photon and simplifying. Number of photons per second = Power / Energy of a single photon Number of photons per second = (2.333 * 10^{-17}) / (3.8428 * 10^{-19}) After calculating the division, we get: Number of photons per second ≈ 606 The nocturnal bird's eye can detect approximately 606 photons per second at the given power and frequency.

Key concepts

Energy of a Photon

Understanding the energy of a photon is fundamental in grasping how light interacts with matter. A photon, the smallest unit or quantum of light, carries energy that is directly proportional to its frequency. This relationship is described by the formula:

equation This implies that higher frequency light, like ultraviolet rays, packs more energy per photon than lower frequency light, such as infrared. This concept becomes incredibly relevant when discussing phenomena like the photoelectric effect, where electrons are ejected from a metal's surface upon absorbing a photon's energy. Knowing the energy per photon can also help us comprehend how light sensors, like the nocturnal bird's eye in the given exercise, are sensitive to various intensities of light.

Planck's Constant

Planck's constant is a crucial number in quantum mechanics, symbolized by and valued at approximately [js]. It was introduced by Max Planck, and its discovery was pivotal to the development of quantum theory. Planck's constant serves as a bridge linking the energy of a photon to its frequency; the smaller its value, the finer the scale at which energy can be quantized. Without this constant, we couldn't accurately describe the discrete nature of energy exchange in quantum processes.

Planck's constant also comes into play in technologies such as LEDs and solar panels, where understanding energy transfer at a quantum level is essential. In the exercise, we use Planck's constant to calculate the energy of a photon, which then allows us to determine the number of photons a bird's eye can detect.

Light Frequency

Light frequency, measured in Hertz (Hz), indicates how many wave cycles pass a given point per second. Visible light is a small part of the electromagnetic spectrum, and each color corresponds to a different frequency. For instance, red light has a lower frequency than blue light.

The frequency of light is directly proportional to the energy of its photons, which is critical when analyzing the detection capabilities of an optical sensor such as a nocturnal bird's eye. The given exercise specifies a detection frequency for the bird's eye, and through the energy-frequency relation, this becomes the basis for determining the energy of individual photons the eye is capable of sensing.

Power and Photon Relationship

The relationship between power and photon detection is an intriguing aspect of photonic interactions. Power, measured in Watts (W), is the rate at which energy is transferred or converted. When it comes to photons, power can be thought of as the amount of energy carried by photons striking a surface per second.

By understanding the power emitted by a source and knowing the energy of individual photons, we can calculate the number of photons arriving at a detector every second, as demonstrated in the exercise. This principle is widely used in designing light-detecting systems, including those used in biological organisms like our nocturnal bird, and in technological applications such as photodetectors and communication systems.