91Ó°ÊÓ

In the study of photon emission, understanding the relationship between wavelength and other photon parameters is crucial. Wavelength, denoted by \( \lambda \), is inversely related to the energy of a photon. This means that as the wavelength increases, the energy decreases, and vice versa. Mathematically, this relationship is expressed by the formula:

• \( E = \frac{hc}{\lambda} \)

where \( h \) is Planck's constant and \( c \) is the speed of light.
This relationship impacts the number of photons emitted per second by a source. For sources with the same power, if one emits photons of a longer wavelength, it will emit more photons to maintain the same total energy output as a source with shorter wavelengths.

Photons per second, denoted as \( n \), indicates the number of photons emitted by a light source each second.
In our problem, we have two sources, \( A \) and \( B \), which emit light with wavelengths \( \lambda_a \) and \( \lambda_b \), respectively.
Both sources have the same power, meaning the total energy output per unit time is equal.The expression that helps us compare the number of photons emitted by these two sources is:

• \( \frac{n_a}{n_b} = \frac{\lambda_a}{\lambda_b} \)

Here, \( n_a \) represents the number of photons per second from source \( A \), and \( n_b \) from source \( B \).
This equation tells us that if source \( A \) has a longer wavelength than \( B \), then \( A \) must emit more photons per second to match the energy output, as given by the ratio \( \frac{\lambda_a}{\lambda_b} \).

The energy of a photon is a fundamental concept in photon emission.
Each photon carries a certain amount of energy, defined by its wavelength, and the equation for this energy is:

• \( E = \frac{hc}{\lambda} \)

Here:

• \( E \) is the energy
• \( h \) is Planck's constant (\( 6.626 \times 10^{-34} \) Joule seconds)
• \( c \) is the speed of light (\( 3 \times 10^8 \) meters per second)
• \( \lambda \) is the wavelength in meters

Understanding this relationship helps explain why, for the same power, a source emitting longer wavelengths needs more photons. This is because each photon has less energy, requiring a greater number to deliver the same overall power or energy per second.

Power in the context of photon emission relates to how much energy is emitted per unit time. For a light source, power \( P \) is the product of the number of photons emitted per second \( n \) and the energy of one photon \( E \). Thus, power can be expressed as:

• \( P = n \cdot E = n \cdot \frac{hc}{\lambda} \)

In scenarios where two sources emit with the same power, the number of photons and the energy of each photon will adjust depending on the wavelength to maintain this equality.
For sources \( A \) and \( B \):

• \( n_a \frac{hc}{\lambda_a} = n_b \frac{hc}{\lambda_b} \)

Simplifying this tells us how the number of photons and wavelengths relate: when \( \lambda_a < \lambda_b \), for instance, \( n_a < n_b \), meaning fewer photons are needed with higher energy (shorter wavelengths) to maintain the same power.