Question

What are the equations relating photon energy \(E\) to light's frequency \(\nu\) and wavelength \(\lambda\) ?

Short answer

Photon energy equations are: \( E = h \nu \) and \( E = \frac{hc}{\lambda} \).

Step-by-step solution

Understanding Photon Energy Equation

Photon energy is related to the frequency of light through the equation: \[ E = h u \] where \( E \) is the energy of the photon, \( h \) is Planck's constant \( (6.626 \times 10^{-34} \text{ Js}) \), and \( u \) is the frequency of the light.

Understanding the Relationship Between Frequency and Wavelength

The frequency \( u \) of light is related to its wavelength \( \lambda \) through the speed of light \( c \): \[ c = \lambda u \] where \( c \) is the speed of light \((3 \times 10^8 \, \text{m/s})\).

Substituting Frequency in Photon Energy Equation

To express photon energy in terms of wavelength, solve for \( u \) from the equation \( c = \lambda u \):\[ u = \frac{c}{\lambda} \] Substitute this into the photon energy equation: \[ E = h \left(\frac{c}{\lambda}\right) = \frac{hc}{\lambda} \] Now, energy is expressed in terms of wavelength.

Key concepts

Planck's Constant

Planck's constant is a fundamental constant in quantum mechanics that relates the energy of a photon to the frequency of light. This constant, represented by the symbol \( h \), has a value of approximately \( 6.626 \times 10^{-34} \text{ Js} \). The significance of Planck's constant is that it quantifies the smallest unit of energy possible due to the quantized nature of electromagnetic waves. The energy of a single photon can be calculated using Planck's equation \( E = hu \), where \( E \) is the energy, \( h \) is Planck's constant, and \( u \) (often denoted as \( u \)) is the frequency of the photon. This equation shows that the energy of a photon is directly proportional to its frequency. When frequency increases, the energy of the photon becomes higher. This relationship helps us understand various phenomena in nature and is crucial for fields like quantum physics, chemistry, and optics.

Frequency-Wavelength Relationship

The frequency and wavelength of light are inversely related through the speed of light. Light waves are electromagnetic waves and have a constant speed in a vacuum. The equation \( c = \lambda u \) describes the relationship between wavelength \( \lambda \) and frequency \( u \). Here, \( c \) is the speed of light, approximately \( 3 \times 10^8 \, \text{m/s} \).

• \( \lambda \) (wavelength): The distance between successive crests of a wave.
• \( u \) (frequency): The number of waves that pass a given point per second.

From the equation, you can see that if the wavelength increases, the frequency decreases, and vice versa. This inverse relationship is key to solving problems involving electromagnetic waves, including calculating the energy of photons at different wavelengths. To express the frequency in terms of the wavelength, you rearrange the equation to \( u = \frac{c}{\lambda} \). This allows you to see how a change in wavelength will affect frequency.

Speed of Light

The speed of light, denoted as \( c \), is a key constant in physics that is crucial in relating wavelength and frequency of electromagnetic waves. In vacuum, it remains constant at approximately \( 3 \times 10^8 \text{ m/s} \).

• It is this constant speed that allows us to convert between frequency and wavelength.
• It also means that all electromagnetic waves travel at this speed in a vacuum, regardless of their frequency or wavelength.

Understanding the speed of light helps us grasp the behavior of photons and other electromagnetic phenomena. In equations like \( c = \lambda u \), the speed of light enables the conversion and calculation of different wave properties. Moreover, it plays a crucial role in the special theory of relativity, where it remains a fundamental limit for the speed of any object with mass.