Problem 75 - Chemistry: Structure and Properties

Question

An argon ion laser puts out 5.0 W of continuous power at a wavelength of 532 nm. The diameter of the laser beam is 5.5 mm. If the laser is pointed toward a pinhole with a diameter of 1.2 mm, how many photons will travel through the pinhole per second? Assume that the light intensity is equally distributed throughout the entire cross-sectional area of the beam. (1 W = 1 J>s) Hint: Use the formula for the area of a circle (A = pr2) to find the cross-sectional area of the beam and of the pinhole and determine what fraction of the power gets through the pinhole.

Step-by-Step Solution

Answer

Assuming the calculations are made, find the areas, determine the power fraction through the pinhole, the energy per photon, and then calculate the number of photons passing through the pinhole per second.

Step 1: Calculate the area of the laser beam

Determine the area of the circular laser beam using the formula for the area of a circle: \( A = \text{A}_{beam} = \text{pi} \times r^2 \text \therefore A_{beam} = \text{pi} \times (\text \frac{5.5}{2})^2 \)

Step 2: Calculate the area of the pinhole

Determine the area of the circular pinhole using the formula for the area of a circle: \( A = \text{A}_{pinhole} = \text{pi} \times r^2 \text \therefore A_{pinhole} = \text{pi} \times (\text \frac{1.2}{2})^2 \)

Step 3: Calculate the fraction of power that goes through the pinhole

Divide the area of the pinhole by the area of the laser beam to find the fraction of power that goes through the pinhole: \( \text{Fraction of Power} = \text \frac{A_{pinhole}}{A_{beam}} \)

Step 4: Calculate the power through the pinhole

Multiply the total power of the laser by the fraction of power that goes through the pinhole to find the power through the pinhole: \( \text P_{pinhole} = \text{Fraction of Power} \times \text P_{total} \)

Step 5: Calculate the energy per photon

Use the energy relation for a photon \( E = \ frac {hc}{\lambda} \), where \( h \) is Planck's constant, \( c \) is the speed of light, and \( \lambda \) is the wavelength of light. Insert the wavelength \( \lambda = 532 \) nm into the equation.

Step 6: Calculate the number of photons per second passing through the pinhole

Divide the power through the pinhole by the energy per photon to find the number of photons \( N \) passing through the pinhole per second: \( N = \frac{P_{pinhole}}{E} \)

Key Concepts

Photon Energy

Photon energy is a fundamental concept in quantum mechanics and refers to the amount of energy carried by a single photon, which is the smallest quantifiable unit of light. This energy can be calculated using the equation \( E = \frac{hc}{\lambda} \), where \( h \) is Planck's constant (\( 6.626 \times 10^{-34}\) Joule seconds), \( c \) is the speed of light (approximately \( 3 \times 10^8 \) meters per second), and \( \lambda \) is the wavelength of the photon.

To put it simply, shorter wavelengths correlate with higher energy photons, and longer wavelengths correspond to lower energy photons. This principle is crucial when we analyze the power output of a laser, such as in the given exercise, where we are concerned with an argon ion laser emitting photons with a wavelength of 532 nm.

Laser Beam Power

Laser beam power is a measure of the energy output of a laser per unit of time and is typically expressed in Watts (\(W\)), where 1 Watt is equivalent to 1 Joule per second (\(J/s\)). This quantity is indicative of how much work the laser can do over a given time period, and it is directly related to the number of photons emitted by the laser.

In practical terms, a high-power laser can deliver more energy and thus more photons, which can be useful for a variety of applications such as cutting materials, medical surgeries, or scientific experiments. In the context of the exercise, a continuous power output of 5.0 W implies a continuous emission of photons, where the power associated with the photons that pass through a given area, such as a pinhole, can be calculated and related to the number of photons passing through that area every second.

Circular Cross-Sectional Area

The concept of circular cross-sectional area plays a pivotal role in optics and laser physics, particularly when considering the spatial distribution of a laser beam. For a circular cross-section, the area can be calculated using the formula \( A = \pi r^2 \), where \( r \) is the radius of the circle.

Understanding how to calculate this area is important for determining how much of a laser's power is being directed through a particular region, such as the pinhole in the problem statement. The area ratio between the pinhole and the laser beam's cross-section gives us the fraction of the beam's power that will pass through the pinhole. Therefore, calculating the areas correctly is critical for predicting the beam's behavior and its interaction with objects, such as in applications requiring precise energy delivery.