Question

A ruby laser produces radiations of wavelength \(662.6 \mathrm{~nm}\) in pulses whose duration are \(1.0 \times 10^{-9} \mathrm{~s}\). If the laser produces \(0.36 \mathrm{~J}\) of energy per pulse, how many photons are produced in each pulse? (a) \(1.2 \times 10^{9}\) (b) \(1.2 \times 10^{27}\) (c) \(1.2 \times 10^{18}\) (d) \(1.2 \times 10^{15}\)

Short answer

The number of photons produced in each pulse is approximately 1.2 × 10^{18}.

Step-by-step solution

Calculate the energy of a single photon

Firstly, determine the energy of a single photon by using the Planck's equation, which relates the energy of a photon (E) to its frequency (ν): E = hν. The frequency can be found from the wavelength (λ) using the speed of light equation c = λν, where c is the speed of light. Rearrange the equation to solve for ν: ν = c/λ. Then, express h (Planck's constant) in joules·seconds and λ in meters to ensure unit consistency. Calculate the energy of one photon with the given wavelength.

Convert the wavelength from nanometers to meters

To convert the wavelength given in nanometers to meters, use the conversion factor 1 nm = 1e-9 m. Perform the conversion to use the value in meters for subsequent calculations.

Calculate the energy of one photon

Substitute the Planck's constant value (h = 6.62607015 × 10^-34 J·s), the speed of light (c = 3.00 × 10^8 m/s), and the wavelength converted to meters into the equation E = hν = hc / λ to find the energy of one photon in joules.

Calculate the number of photons per pulse

Divide the total energy per pulse by the energy of one photon to determine the number of photons produced in each pulse. The total energy per pulse is given as 0.36 J. Use the calculation from the previous steps to find the energy of a single photon.

Select the correct answer

Compare the result with the given options and choose the one that matches the calculation result.

Key concepts

Planck's equation

Understanding Planck's equation is the foundation for analyzing the energy of photons, which are the particles that make up light. Planck’s equation, represented as (E = h \( u \), is a formula that establishes the link between the energy (E) of a photon and its frequency (\( u \)). The constant h, known as Planck's constant, is a fundamental value in quantum mechanics, quantifying the action of photons. It’s determined to be approximately 6.62607015 × 10^-34 Joule-seconds (J·s). This equation implies that the energy of a photon is directly proportional to its frequency: as the frequency goes up, so does the energy.

When considering the energy of light emitted by a laser, as with the ruby laser in our example, Planck's equation becomes pivotal to calculate the energy of each individual photon. It establishes an essential relationship used in a multitude of applications, such as determining the power of laser pulses in modern technology and communications.

Wavelength to frequency conversion

Transitioning from wavelength to frequency is necessary to use Planck's equation effectively. The speed of light, denoted by c, which is about 3.00 × 10⁸ meters per second (m/s), serves as a bridge between the two concepts. This relationship is expressed by the equation c = \( \lambdau \), where \( \lambda \) represents wavelength and \( u \) represents frequency.

To find a photon's frequency (\( u \)), just rearrange the equation to \( u = \frac{c}{\lambda} \). In our ruby laser problem, this conversion allows us to relate the known wavelength of the emitted light to the photon frequency needed for Planck's equation. This step is fundamental because Planck's equation takes frequency into account, not wavelength. Thus, understanding this conversion is not just theoretical – it is practically applied when calculating photon energies.

Speed of light

The speed of light (c) is not just a key aspect of wavelength to frequency conversion; it's a universal constant crucial in both classical and quantum physics. Measured as roughly 3.00 × 10⁸ m/s, it stands as the maximum speed at which all conventional matter and information in the universe can travel. Not surprisingly, the constancy of the speed of light sits at the core of Einstein’s theory of relativity, and it allows us to convert between wavelength and frequency. Additionally, the speed of light is used extensively when we communicate using light waves, calculate distances in astronomy, and understand the behavior of particles at high speeds. For example, in laser technology, which is based on the manipulation of light, this constant is essential to ensure that energy calculations for photons are precise and applicable to practical uses like medical treatments, manufacturing, or computer networking.

Energy per pulse

The concept of 'energy per pulse' comes into play when using devices that emit energy in quick, short bursts, such as lasers. With energy per pulse, we describe the total energy emitted in a single burst or pulse of light. This measure is particularly relevant in various laser applications, from cutting materials to medical surgeries. To obtain the number of photons in a laser pulse, as in our given problem, we must divide the total energy per pulse by the energy of a single photon calculated using Planck's equation.

For example, with our ruby laser emitting 0.36 J energy per pulse, we can calculate how many packets of light – photons – this energy corresponds to. By doing so, we can understand the laser's strength and its effects at the photonic level. This kind of calculation is not only essential for theoretical physics but also for practical applications in industry and technology where precise energy measurements are paramount.