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 a nocturnal bird's eye can detect?

Short answer

Answer: A nocturnal bird's eye can detect about 61 photons per second.

Step-by-step solution

Write down the given values

We have the following given values: Frequency of light (f) = \(5.8 \cdot 10^{14} Hz\) Minimum power detected (P) = \(2.333 \cdot 10^{-17} W\)

Calculate the energy per photon

We can calculate the energy per photon (E) using Planck's equation: E = hf Where E is the energy per photon, h is Planck's constant (\(6.626 \cdot 10^{-34} Js\)), and f is the frequency of the light. Substituting the given values, we get: E = \((6.626 \cdot 10^{-34} Js)(5.8 \cdot 10^{14} Hz)\) E = \(3.84308 \cdot 10^{-19} J\)

Calculate the number of photons per second

Now, we can use the equation: Number of photons per second = (Power detected) / (Energy per photon) Substituting the given values and the calculated energy per photon, we get: Number of photons per second = \(\frac{2.333 \cdot 10^{-17} W}{3.84308 \cdot 10^{-19} J}\) Number of photons per second = \(6.072 \cdot 10^1\) So, a nocturnal bird's eye can detect about 61 photons per second.

Key concepts

Energy per Photon

In the realm of light and photons, energy per photon is a fundamental concept. Simply put, each photon, which is a particle of light, carries a specific amount of energy. This energy can be calculated if we know the frequency of the light.
For any light-dependent calculations, this energy attribute becomes important. It helps us determine how much energy is being transported by light.
In our everyday observations, more energetic photons correspond to brighter or more intense light sources.

• Photon: Basic particle of light
• Energy carried by photon: A key attribute helping in calculations

Understanding this concept allows us to delve deeper into how light interacts with matter.

Planck's Equation

Planck's Equation is the tool we use to calculate the energy of a photon. It bridges the gap between light's wave properties and its quantum characteristics. **Planck's Equation** is written as:\[E = hf\]where:- \(E\) is the energy per photon,- \(h\) is Planck's constant \((6.626 \, \times 10^{-34} \, \text{Js})\),- \(f\) is the frequency of light.This formula shows that energy is proportional to the frequency of the light. Higher frequency means higher energy. This relationship is foundational in quantum mechanics and helps in calculating how much energy is emitted or absorbed in light interactions.Whether you're studying stars or measuring light in a lab, Planck's Equation is indispensable. It applies across the spectrum, from radio waves to gamma rays. Keeping \(h\) constant binds the equation universally, while changes in \(f\) adjust \(E\), demonstrating the dynamic nature of light's energy.

Frequency of Light

Every color in the spectrum has its unique frequency. When we talk about the frequency of light, we mean how many wave cycles pass a reference point per second, measured in Hertz (Hz). It's like the heartbeat of light.
Frequency is tied to both color and energy. Higher frequencies are found in colors like violet and blue, which are more energetic compared to lower frequencies like red or infrared light.

• Frequency: Number of cycles per second
• Carries information about light color and energy

This concept helps us explain why certain lights are more energetic and directly affects calculations for energy each photon carries.