Question

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

Short answer

Equations are \( E = h \nu \) and \( E = \frac{hc}{\lambda} \).

Step-by-step solution

Understanding Photon Energy

Photon energy, denoted as \( E \), is calculated using Planck's equation, which relates energy to the frequency \( u \) of the light. The equation is given by \( E = h u \), where \( h \) represents Planck's constant (\(6.626 \times 10^{-34} \text{ Js}\)).

Connecting Frequency with Wavelength

The frequency \( u \) of light is related to its wavelength \( \lambda \) and the speed of light \( c \) (approximately \(3.00 \times 10^8 \text{ m/s}\)) by the equation \( c = \lambda u \). Solving for \( u \), we get \( u = \frac{c}{\lambda} \).

Expressing Photon Energy in terms of Wavelength

By substituting the expression for \( u \) from the second step (\( u = \frac{c}{\lambda} \)) into Planck's equation (\( E = h u \)), we obtain \( E = h \frac{c}{\lambda} \). This equation relates the photon's energy directly with its wavelength \( \lambda \).

Summarizing the Equations

There are two key equations: 1) \( E = h u \) relates energy to frequency, and 2) \( E = \frac{hc}{\lambda} \) relates energy to wavelength. These equations are fundamental for understanding the energy of photons in terms of light's frequency and wavelength.

Key concepts

Planck's Constant

Planck's constant, denoted by the symbol \(h\), is a crucial element in the realm of quantum mechanics. It plays a vital role in understanding the behavior of photons and their energies. Planck's constant is approximately equal to \(6.626 \times 10^{-34} \, \text{Js}\). It is a very small number, highlighting the minuscule energy levels at the atomic and subatomic scale. This constant is used in Planck's equation, \(E = h u\), where \(E\) represents the energy of a photon, and \(u\) is the frequency of the light.

To put it simply:

• Every photon carries energy proportional to its frequency.
• Planck's constant is the factor of proportionality.

Understanding Planck's constant helps us bridge the classical and quantum worlds, offering insight into how energy is quantized. This quantization means energy levels are discrete, not continuous, which is fundamental to quantum theory.

Frequency and Wavelength Relationship

The relationship between frequency and wavelength is pivotal in the study of electromagnetic waves. Frequency, \(u\), refers to how many wave crests pass a point in one second, measured in Hertz (Hz). Wavelength, \(\lambda\), is the distance between consecutive wave crests. These two are related through the speed of light, \(c\), given by the formula \(c = \lambda u\).

This tells us:

• Light always travels at a constant speed in a vacuum \(\sim 3.00 \times 10^8 \, \text{m/s}\).
• If wavelength increases, frequency decreases, and vice versa.

To express frequency solely in terms of wavelength, you can rearrange the equation to \(u = \frac{c}{\lambda}\). Understanding this link provides deeper insights into how different electromagnetic waves behave and interact with matter.

Speed of Light

The speed of light, symbolized as \(c\), is one of the fundamental constants in physics. It defines how fast light travels in a vacuum, approximately \(3.00 \times 10^8 \, \text{m/s}\). This speed is crucial for calculations involving the properties of light and electromagnetic waves.Here are some interesting features:

• The speed of light is constant and does not change unless a medium other than a vacuum is involved.
• This constant plays a key role in equations that relate energy to both frequency and wavelength.

For instance:

• In \(c = \lambda u\), it shows the direct relationship between frequency and wavelength.
• In the context of photon energy, \(E = h \frac{c}{\lambda}\) incorporates light speed to connect energy with wavelength.

Understanding the speed of light enables us to explore how light behaves across different mediums and its role in the universe's physics.