Problem 5 - University Physics with Modern Physics

Question

A 75 -W light source consumes 75 W of electrical power. Assume all this energy goes into emitted light of wavelength 600 \(\mathrm{nm}\) . (a) Calculate the frequency of the emitted light. (b) How many photons per second does the source emit? (c) Are the answers to parts (a) and (b) the same? Is the frequency of the light the same thing as the number of photons emitted per second? Explain.

Step-by-Step Solution

Answer

(a) Frequency: \(5 \times 10^{14}\) Hz. (b) \(2.27 \times 10^{20}\) photons/second. (c) No, frequency is not the same as photon count.

Step 1: Calculate the Frequency of the Emitted Light

To calculate the frequency, we use the formula \( c = \lambda u \), where \( c \) is the speed of light, \( \lambda \) is the wavelength, and \( u \) is the frequency. Given that \( \lambda = 600 \ nm = 600 \times 10^{-9} \ m \) and \( c = 3 \times 10^8 \ m/s \), we rearrange to find \( u = \frac{c}{\lambda} \). \[ u = \frac{3 \times 10^8}{600 \times 10^{-9}} \]Upon calculating, \( u = 5 \times 10^{14} \ Hz \).

Step 2: Calculate the Energy of a Single Photon

To find the energy of a single photon, use the formula \( E = h u \), where \( h = 6.626 \times 10^{-34} \ Js \) is Planck's constant and \( u = 5 \times 10^{14} \ Hz \) is the frequency calculated in Step 1.\[ E = 6.626 \times 10^{-34} \times 5 \times 10^{14} \]Thus, the energy \( E \approx 3.31 \times 10^{-19} \ Joules \).

Step 3: Calculate Number of Photons Emitted Per Second

The total energy consumed per second by the light source is 75 Joules (since 75 Watts = 75 Joules/second). To find the number of photons emitted per second, divide the total energy by the energy of one photon.\[ N = \frac{75}{3.31 \times 10^{-19}} \]Calculating this gives \( N \approx 2.27 \times 10^{20} \) photons per second.

Step 4: Explanation of the Relationship between Frequency and Photon Emission Rate

The answers to (a) and (b) are not the same because they address different concepts. Frequency refers to the number of wave cycles per second (in Hertz), while the number of photons emitted per second is simply a count of photons. The frequency is a property of individual light waves, independent of the total number of photons.

Key Concepts

Frequency of Light

The frequency of light refers to the number of oscillations or cycles a light wave completes per second. It's measured in Hertz (Hz). You can think of it as how fast the light wave is "wiggling" as it travels. This concept is crucial when studying photon emission.

To find the frequency, we often use the equation:

• \( c = \lambda u \), where:
  • \( c \) = Speed of light (approximately \( 3 \times 10^8 \ m/s \))
  • \( \lambda \) = Wavelength
  • \( u \) = Frequency

An example of this is calculating the frequency of a light with a wavelength of 600 nm, which gives us a frequency of \( 5 \times 10^{14} \ Hz \).

The frequency tells us how "color" or energy-laden a light wave is. Higher frequencies correspond to bluer light, while lower frequencies correspond to redder light.

Photon Energy

Photon energy is the energy carried by a single photon, the fundamental particle of light. This concept is pivotal in understanding how energy is absorbed, emitted, or transferred in light interactions. The energy of a photon is given by the formula:

• \( E = h u \), where:
  • \( E \) = Energy of the photon
  • \( h \) = Planck's constant (\( 6.626 \times 10^{-34} \ Js \))
  • \( u \) = Frequency of the photon

By using the frequency calculated from our previous step, \( 5 \times 10^{14} \ Hz \), we find the energy of a single photon to be approximately \( 3.31 \times 10^{-19} \) Joules.

Photon energy is crucial in applications like photovoltaics, where light energy is converted into electrical energy, or in analyzing the efficiency of different light sources.

Planck's Constant

Planck's constant is a fundamental quantity in quantum mechanics, represented by \( h \). It relates the energy of photons to their frequency. This tiny constant, approximately \( 6.626 \times 10^{-34} \ Js \), bridges the often distinct worlds of wave-like and particle-like behaviors of light.

When calculating photon energy, Planck's constant is key. Without it, linking frequency and energy in the electromagnetic realm would be impossible.

This constant is named after Max Planck, who was a pioneering physicist in the development of quantum theory. Understanding and utilizing Planck's constant allows scientists to predict the energy outcomes of photons based on frequency and to delve deeper into the mysteries of atomic and subatomic processes.

Wavelength and Frequency Relationship

The relationship between wavelength and frequency is foundational in the study of light and photons. Essentially, as one increases, the other decreases. This inversely proportional relationship is defined by the speed of light equation:

• \( c = \lambda u \)

• Where \( c \) is constant, any increase in wavelength \( \lambda \) must result in a decrease in frequency \( u \), and vice versa.
• This relationship impacts how we perceive light, as different frequencies correspond to different colors in the visible spectrum.

For example, in our exercise, a wavelength of 600 nm corresponds to a frequency of \( 5 \times 10^{14} \ Hz \). Comprehending this relationship helps when adjusting variables like wavelength or frequency during experiments and when designing optical equipment like lenses and lasers.