Problem 21 - University Physics with Modern Physics

Question

The intensity of a cylindrical laser beam is 0.800 W/m\(^2\). The cross- sectional area of the beam is 3.0 \(\times\) 10\(^{-4}\) 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 output is 2.4 x 10^-4 W. (b) The rms electric field is 548 V/m.

Step 1: Understanding Intensity and Power

Intensity, denoted as \( I \), is the power per unit area. It is given by the formula \( I = \frac{P}{A} \), where \( P \) is the power and \( A \) is the area. We are given the intensity \( I = 0.800 \, \text{W/m}^2 \) and the area \( A = 3.0 \times 10^{-4} \, \text{m}^2 \). Our task is to find the power output \( P \) by rearranging the formula to \( P = I \times A \).

Step 2: Calculate Average Power Output

Using the formula from Step 1, calculate the power: \[ P = I \times A = 0.800 \, \text{W/m}^2 \times 3.0 \times 10^{-4} \, \text{m}^2 = 2.4 \times 10^{-4} \, \text{W} \]. Therefore, the average power output of the laser is \( 2.4 \times 10^{-4} \text{ W} \).

Step 3: Relate Intensity and Electric Field

The intensity \( I \) is also related to the root mean square (rms) value of the electric field \( E_{\text{rms}} \) through the following expression for electromagnetic waves: \[ I = \frac{1}{2} c \varepsilon_0 E_{\text{rms}}^2 \] where \( \varepsilon_0 = 8.85 \times 10^{-12} \, \text{F/m} \) is the permittivity of free space and \( c = 3 \times 10^8 \, \text{m/s} \) is the speed of light.

Step 4: Solve for RMS Electric Field

Rearrange the expression from Step 3 to solve for \( E_{\text{rms}} \): \[ E_{\text{rms}} = \sqrt{\frac{2I}{c\varepsilon_0}} \]. Substitute \( I = 0.800 \, \text{W/m}^2 \), \( c = 3 \times 10^8 \, \text{m/s} \), and \( \varepsilon_0 = 8.85 \times 10^{-12} \, \text{F/m} \): \[ E_{\text{rms}} = \sqrt{\frac{2 \times 0.800}{3 \times 10^8 \times 8.85 \times 10^{-12}}} \approx 548 \, \text{V/m} \]. Thus, the rms value of the electric field is approximately \( 548 \, \text{V/m} \).

Key Concepts

Average Power Output

The average power output of a laser beam is a fundamental concept when discussing laser beams and their efficiency. The intensity of a laser beam is given as power per unit area. Hence, to determine how much power the laser actually outputs, we need to consider its intensity along with the cross-sectional area of the beam.
To calculate the average power output, we use the formula:

• Intensity (\( I \)) = Power (\( P \)) / Area (\( A \))

Given that the intensity (\( I \)) of the laser beam is 0.800 W/m² and the area (\( A \)) of the beam is 3.0 × 10⁻⁴ m², we can rearrange this formula to solve for power:

• Power (\( P \)) = Intensity (\( I \)) × Area (\( A \))

Plugging in the values, we find that the average power output is:\[P = 0.800 \, \text{W/m}^2 \times 3.0 \times 10^{-4} \, \text{m}^2 = 2.4 \times 10^{-4} \, \text{W}\]Therefore, the average power output of the laser is 0.00024 watts, a relatively small but precise measurement significant in applications involving lasers.

Root Mean Square Electric Field

The root mean square (rms) electric field is another critical aspect when discussing electromagnetic waves like laser beams. This value gives us an understanding of the strength of the electric field component of the wave. The rms electric field can be determined when we know the intensity of the electromagnetic wave.
The relationship between the intensity (\( I \)) and the rms electric field (\( E_{\text{rms}} \)) is expressed by:

• \( I = \frac{1}{2} c \varepsilon_0 E_{\text{rms}}^2 \)

Where:

• \( c = 3 \times 10^8 \, \text{m/s} \) is the speed of light.
• \( \varepsilon_0 = 8.85 \times 10^{-12} \, \text{F/m} \) is the permittivity of free space.

To find the rms electric field, rearrange the formula to:\[E_{\text{rms}} = \sqrt{\frac{2I}{c\varepsilon_0}}\]Substituting the given values, we get:\[E_{\text{rms}} = \sqrt{\frac{2 \times 0.800}{3 \times 10^8 \times 8.85 \times 10^{-12}}} \approx 548 \, \text{V/m}\]This shows that the rms value of the electric field in the laser beam is approximately 548 volts per meter, indicating the wave’s capability of exerting electric forces within the medium.

Electromagnetic Waves

Electromagnetic waves, such as those produced by lasers, are waves of electric and magnetic fields that propagate through space. These waves are crucial in various fields, including telecommunications, medicine, and everyday technologies like WiFi and microwaves.
Key characteristics of electromagnetic waves include:

• They travel at the speed of light (\( c = 3 \times 10^8 \, \text{m/s} \)) in a vacuum.
• They have a frequency (\( f \)) and a wavelength (\( \lambda \)) linked by the equation: \( c = f \times \lambda \).
• They encompass a wide spectrum, from radio waves to gamma rays, with varying frequencies and wavelengths.

Electromagnetic waves, such as laser beams, are generated by oscillating charges or by stimulating particular materials to emit coherent light, like stimulated emission in lasers. This coherent behavior means the waves are in phase, producing highly focused and intense beams of light ideal for precise applications. Understanding these waves allows us to better harness their properties for practical purposes, such as data transmission and medical treatment.