Problem 20 - University Physics with Modern Physics

Question

The intensity of a cylindrical laser beam is 0.800 \(\mathrm{W} / \mathrm{m}^{2}\) The cross-sectional area of the beam is \(3.0 \times 10^{-4} \mathrm{m}^{2}\) and the intensity is uniform across the cross section of the beam. (a) What is the average power output of the laser? (b) What is the rms value of the electric field in the beam?

Step-by-Step Solution

Answer

(a) The average power is \(2.4 \times 10^{-4} \, \mathrm{W}\). (b) The rms value of the electric field is \(547 \, \mathrm{V/m}\).

Step 1: Understand the Problem

We need to find two things: (a) the average power output of the laser, given its intensity and cross-sectional area, and (b) the rms value of the electric field, given the intensity of the laser beam.

Step 2: Recall Relevant Formulas

For part (a), we use the formula to calculate power from intensity: \( P = I \times A \), where \( P \) is power, \( I \) is intensity, and \( A \) is area. For part (b), the rms value of the electric field \( E_{rms} \) in a beam related to intensity is given by \( E_{rms} = \sqrt{\frac{2I}{c\epsilon_0}} \), where \( c \) is the speed of light in a vacuum \( (3 \times 10^8 \, \mathrm{m/s}) \) and \( \epsilon_0 \) is the permittivity of free space \( (8.85 \times 10^{-12} \, \mathrm{F/m}) \).

Step 3: Calculate Average Power Output

Using the formula \( P = I \times A \):\[P = 0.800 \, \mathrm{W/m^2} \times 3.0 \times 10^{-4} \, \mathrm{m^2}\]Calculating this gives:\[P = 2.4 \times 10^{-4} \, \mathrm{W}\]Thus, the average power output is \(2.4 \times 10^{-4} \, \mathrm{W}\).

Step 4: Calculate RMS Value of Electric Field

Plug in the values into the formula \( E_{rms} = \sqrt{\frac{2I}{c\epsilon_0}} \):\[E_{rms} = \sqrt{\frac{2 \times 0.800}{3 \times 10^8 \times 8.85 \times 10^{-12}}}\]Simplifying inside the square root first and then calculating gives:\[ E_{rms} = 547 \, \mathrm{V/m} \]Therefore, the rms value of the electric field is approximately \(547 \, \mathrm{V/m}\).

Key Concepts

Power Calculation

Understanding power calculation is a significant aspect of physics, especially when dealing with laser beams. Power refers to the rate at which energy is transferred or converted. In this scenario, the power output of a laser can be calculated using its intensity and the cross-sectional area through which it passes. This is expressed with the formula:

• \( P = I \times A \)

where:

• \( P \) is the power output (in Watts).
• \( I \) is the intensity of the laser (in \( \, \mathrm{W/m^2} \)).
• \( A \) is the cross-sectional area (in square meters).

To find the power, simply multiply the intensity by the area. By applying this formula to the problem at hand with values \( I = 0.800 \, \mathrm{W/m^2} \) and \( A = 3.0 \times 10^{-4} \, \mathrm{m^2} \), we find that the power output is \( P = 2.4 \times 10^{-4} \, \mathrm{W} \). This calculated value shows how much energy the laser is outputting per second, which can help understand its efficiency and capacity.

Electric Field RMS Value

The RMS (root mean square) value of the electric field in a laser beam is crucial for understanding the energy distribution and the strength of the electromagnetic waves it produces. This value is determined using the formula:

• \( E_{rms} = \sqrt{\frac{2I}{c\epsilon_0}} \)

where:

• \( E_{rms} \) is the RMS value of the electric field (in Volts per meter).
• \( I \) is the intensity (in \( \, \mathrm{W/m^2} \)).
• \( c \) is the speed of light in a vacuum (\( 3 \times 10^8 \, \mathrm{m/s} \)).
• \( \epsilon_0 \) is the permittivity of free space (\( 8.85 \times 10^{-12} \, \mathrm{F/m} \)).

For the given problem, substituting the intensity \( I = 0.800 \, \mathrm{W/m^2} \) yields \( E_{rms} = 547 \, \mathrm{V/m} \). This measure tells us the effective value of the electric field, bridging the gap between the peak field values and their utility in practical applications.

Physics Problem Solving

Physics problem-solving is an essential skill that involves understanding the problem, applying relevant formulas, and computing the correct values. A systematic approach aids in solving even complex problems efficiently. Here’s how to effectively tackle physics questions:

• **Understand the problem:** Identify all given information, what needs finding, and how different elements relate.
• **Recall relevant formulas:** Use known physics equations that connect the given information with what is being asked. Familiarity with equations allows quick and accurate solutions.
• **Perform the calculations:** Substituting the known values into the equations will yield the desired solutions. It's crucial to manage units properly and ensure consistency.
• **Review the result:** Check if the calculation is logical and whether the result aligns with physical intuition, confirming that it makes sense contextually.

Applying this process to the laser beam problem, you first identify the data: intensity and area, and then choose the corresponding equations for power and electric field calculations. By breaking down the steps and focusing on each element, errors are minimized, leading to successful problem resolution.