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Relationship Between Energy and Light Frequency

# Relationship Between Energy and Light Frequency

The relationship between energy and light frequency is one of the fundamental concepts in the field of optics and quantum mechanics. It is a principle that applies to all electromagnetic radiation, of which visible light is one example.

Light behaves both as a wave and as a particle, a phenomenon known as wave-particle duality. As a wave, light has a frequency, which is the number of wave cycles that pass a point in space per second. This frequency is directly proportional to the energy of the light photons, which are the particle aspect of light.

## The Photoelectric Effect

The relationship between energy and light frequency was first observed through the photoelectric effect experiment. In this experiment, it was discovered that when light shines on a metal surface, it can eject electrons from that surface. Notably, only light above a certain frequency, called the threshold frequency, was capable of dislodging electrons, regardless of the light’s intensity.

## Mathematical Representation

The energy of a photon (E) is directly proportional to its frequency (f), as given by the Planck-Einstein relation:

\[ E = h f \]

where \( h \) is Planck’s constant (\(6.62607015 \times 10^{-34}\) joule·seconds), and \( f \) is the frequency of the light in hertz (Hz).

A related concept is the wavelength (\( \lambda \)) of light, which is inversely proportional to frequency:

\[ f = \frac{c}{\lambda} \]

where \( c \) represents the speed of light in a vacuum (\(3 \times 10^8\) m/s). Combining these two equations gives the energy of a photon in terms of its wavelength:

\[ E = \frac{hc}{\lambda} \]

Understanding this relationship is crucial in many applications, including spectroscopy, laser technology, and even in the development of renewable energy technologies such as solar cells.

## 20 Problems and Solutions About Relationship Between Energy and Light Frequency

Below are a series of problems that focus on the relationship between energy and light frequency. These will include the use of the formula \( E = h f \), along with conversions between wavelength and frequency.

### Problem 1:
What is the energy of a photon with a frequency of \(5 \times 10^{14}\) Hz?

**Solution 1:**
\[ E = h f \]
\[ E = (6.626 \times 10^{-34} \, \text{J·s})(5 \times 10^{14} \, \text{Hz}) \]
\[ E = 3.313 \times 10^{-19} \, \text{J} \]

### Problem 2:
Calculate the frequency of a photon that has an energy of \(2 \times 10^{-19}\) J.

**Solution 2:**
\[ f = \frac{E}{h} \]
\[ f = \frac{2 \times 10^{-19} \, \text{J}}{6.626 \times 10^{-34} \, \text{J·s}} \]
\[ f \approx 3.017 \times 10^{14} \, \text{Hz} \]

### Problem 3:
What is the wavelength of a photon that has an energy of \(4.5 \times 10^{-19}\) J?

**Solution 3:**
\[ \lambda = \frac{hc}{E} \]
\[ \lambda = \frac{(6.626 \times 10^{-34} \, \text{J·s})(3 \times 10^8 \, \text{m/s})}{4.5 \times 10^{-19} \, \text{J}} \]
\[ \lambda \approx 4.415 \times 10^{-7} \, \text{m} \]

### Problem 4:
Find the energy of a photon with a wavelength of 600 nm in joules.

**Solution 4:**
\[ E = \frac{hc}{\lambda} \]
\[ E = \frac{(6.626 \times 10^{-34} \, \text{J·s})(3 \times 10^8 \, \text{m/s})}{600 \times 10^{-9} \, \text{m}} \]
\[ E \approx 3.31 \times 10^{-19} \, \text{J} \]

### Problem 5:
If the threshold frequency for a metal is \(1 \times 10^{15}\) Hz, what is the minimum energy needed to eject electrons from the metal?

**Solution 5:**
\[ E = h f \]
\[ E = (6.626 \times 10^{-34} \, \text{J·s})(1 \times 10^{15} \, \text{Hz}) \]
\[ E = 6.626 \times 10^{-19} \, \text{J} \]

### Problem 6:
A photon has an energy of \(3.5 \times 10^{-19}\) J. What is its frequency?

**Solution 6:**
\[ f = \frac{E}{h} \]
\[ f = \frac{3.5 \times 10^{-19} \, \text{J}}{6.626 \times 10^{-34} \, \text{J·s}} \]
\[ f \approx 5.28 \times 10^{14} \, \text{Hz} \]

### Problem 7:
Convert a photon wavelength of 500 nm to energy.

**Solution 7:**
\[ E = \frac{hc}{\lambda} \]
\[ E = \frac{(6.626 \times 10^{-34} \, \text{J·s})(3 \times 10^8 \, \text{m/s})}{500 \times 10^{-9} \, \text{m}} \]
\[ E \approx 3.97 \times 10^{-19} \, \text{J} \]

### Problem 8:
What is the energy of light with a wavelength of 450 nm?

**Solution 8:**
\[ E = \frac{hc}{\lambda} \]
\[ E = \frac{(6.626 \times 10^{-34} \, \text{J·s})(3 \times 10^8 \, \text{m/s})}{450 \times 10^{-9} \, \text{m}} \]
\[ E \approx 4.41 \times 10^{-19} \, \text{J} \]

### Problem 9:
Find the frequency of a photon whose energy is \(1.99 \times 10^{-19}\) J.

**Solution 9:**
\[ f = \frac{E}{h} \]
\[ f = \frac{1.99 \times 10^{-19} \, \text{J}}{6.626 \times 10^{-34} \, \text{J·s}} \]
\[ f \approx 3.00 \times 10^{14} \, \text{Hz} \]

### Problem 10:
A green light has a wavelength of 550 nm. What is its frequency?

**Solution 10:**
\[ f = \frac{c}{\lambda} \]
\[ f = \frac{3 \times 10^8 \, \text{m/s}}{550 \times 10^{-9} \, \text{m}} \]
\[ f \approx 5.45 \times 10^{14} \, \text{Hz} \]

### Problem 11:
Calculate the energy of a photon with a wavelength of 700 nm.

**Solution 11:**
\[ E = \frac{hc}{\lambda} \]
\[ E = \frac{(6.626 \times 10^{-34} \, \text{J·s})(3 \times 10^8 \, \text{m/s})}{700 \times 10^{-9} \, \text{m}} \]
\[ E \approx 2.84 \times 10^{-19} \, \text{J} \]

### Problem 12:
What is the wavelength of a photon with an energy of \(5.4 \times 10^{-19}\) J?

**Solution 12:**
\[ \lambda = \frac{hc}{E} \]
\[ \lambda = \frac{(6.626 \times 10^{-34} \, \text{J·s})(3 \times 10^8 \, \text{m/s})}{5.4 \times 10^{-19} \, \text{J}} \]
\[ \lambda \approx 3.67 \times 10^{-7} \, \text{m} \]

### Problem 13:
Determine the energy of a photon with a frequency of \(4.2 \times 10^{14}\) Hz.

**Solution 13:**
\[ E = h f \]
\[ E = (6.626 \times 10^{-34} \, \text{J·s})(4.2 \times 10^{14} \, \text{Hz}) \]
\[ E \approx 2.78 \times 10^{-19} \, \text{J} \]

### Problem 14:
If light has a wavelength of 400 nm, what is its frequency?

**Solution 14:**
\[ f = \frac{c}{\lambda} \]
\[ f = \frac{3 \times 10^8 \, \text{m/s}}{400 \times 10^{-9} \, \text{m}} \]
\[ f = 7.5 \times 10^{14} \, \text{Hz} \]

### Problem 15:
A red light photon has a wavelength of 650 nm. What is its energy?

**Solution 15:**
\[ E = \frac{hc}{\lambda} \]
\[ E = \frac{(6.626 \times 10^{-34} \, \text{J·s})(3 \times 10^8 \, \text{m/s})}{650 \times 10^{-9} \, \text{m}} \]
\[ E \approx 3.06 \times 10^{-19} \, \text{J} \]

### Problem 16:
Find the energy of an ultraviolet photon with a frequency of \(1.2 \times 10^{15}\) Hz.

**Solution 16:**
\[ E = h f \]
\[ E = (6.626 \times 10^{-34} \, \text{J·s})(1.2 \times 10^{15} \, \text{Hz}) \]
\[ E = 7.95 \times 10^{-19} \, \text{J} \]

### Problem 17:
Calculate the frequency for a photon that has an energy of \(6.5 \times 10^{-19}\) J.

**Solution 17:**
\[ f = \frac{E}{h} \]
\[ f = \frac{6.5 \times 10^{-19} \, \text{J}}{6.626 \times 10^{-34} \, \text{J·s}} \]
\[ f \approx 9.80 \times 10^{14} \, \text{Hz} \]

### Problem 18:
What is the wavelength of light with a frequency of \(6 \times 10^{14}\) Hz?

**Solution 18:**
\[ \lambda = \frac{c}{f} \]
\[ \lambda = \frac{3 \times 10^8 \, \text{m/s}}{6 \times 10^{14} \, \text{Hz}} \]
\[ \lambda = 5 \times 10^{-7} \, \text{m} \]

### Problem 19:
A photon has a frequency of \(7.7 \times 10^{14}\) Hz. What is its energy?

**Solution 19:**
\[ E = h f \]
\[ E = (6.626 \times 10^{-34} \, \text{J·s})(7.7 \times 10^{14} \, \text{Hz}) \]
\[ E \approx 5.10 \times 10^{-19} \, \text{J} \]

### Problem 20:
If a photon’s wavelength is 550 nm, find its corresponding energy.

**Solution 20:**
\[ E = \frac{hc}{\lambda} \]
\[ E = \frac{(6.626 \times 10^{-34} \, \text{J·s})(3 \times 10^8 \, \text{m/s})}{550 \times 10^{-9} \, \text{m}} \]
\[ E \approx 3.61 \times 10^{-19} \, \text{J} \]

These problems and solutions provide practical applications of the fundamental relationship between energy and light frequency, which is crucial to developing a deep understanding of electromagnetic radiation and its interactions with matter.

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