Question

A \(5.00-\mathrm{mW}\) laser pointer has a beam diameter of \(2.00 \mathrm{~mm}\) a) What is the root-mean-square value of the electric field in this laser beam? b) Calculate the total electromagnetic energy in \(1.00 \mathrm{~m}\) of this laser beam.

Short answer

Answer: The root-mean-square value of the electric field in the laser beam is 1371.94 V/m, and the total electromagnetic energy in 1.00 m of the laser beam is approximately 4.02 mJ.

Step-by-step solution

Find the Area of the Laser Beam

Given the diameter of the laser beam to be \(2.00 \mathrm{~mm}\), we can calculate the area, \(A\), of the laser beam using the formula for the area of a circle: \(A = \pi r^2\). The radius, \(r\), can be found by dividing the diameter by 2. \(r = \frac{2.00 \mathrm{~mm}}{2} = 1.00 \mathrm{~mm} = 1.00 \times 10^{-3} \mathrm{~m}\) \(A = \pi (1.00 \times 10^{-3} \mathrm{~m})^2 = 3.14159 \times 10^{-6} \mathrm{~m}^2\)

Calculate the Intensity of the Laser Beam

The intensity, \(I\), of a laser beam can be found by the formula: \(I = \frac{P}{A}\), where \(P\) is the power of the laser beam, and \(A\) is the area of the laser beam. The given power, \(P\), is \(5.00 \mathrm{~mW} = 5.00 \times 10^{-3} \mathrm{~W}\). \(I = \frac{5.00 \times 10^{-3} \mathrm{~W}}{3.14159 \times 10^{-6} \mathrm{~m}^2} = 1.5915 \times 10^{3} \mathrm{\frac{W}{m^2}}\)

Find the Electric Field Amplitude

Using the intensity, \(I\), we can find the electric field amplitude, \(E_0\), by using the formula: \(I = \frac{c \epsilon_0 E_0^2}{2}\), where \(c\) is the speed of light, and \(\epsilon_0\) is the vacuum permittivity. Rearranging the formula to solve for \(E_0\), we get: \(E_0 = \sqrt{\frac{2I}{c \epsilon_0}}\). We know that \(c = 3 \times 10^8 \mathrm{\frac{m}{s}}\), and \(\epsilon_0 = 8.85 \times 10^{-12} \mathrm{\frac{C^2}{N \cdot m^2}}\). \(E_0 = \sqrt{\frac{2 \times 1.5915 \times 10^{3} \mathrm{\frac{W}{m^2}}}{3 \times 10^8 \mathrm{\frac{m}{s}} \times 8.85 \times 10^{-12} \mathrm{\frac{C^2}{N \cdot m^2}}}} = 1939.84 \mathrm{\frac{V}{m}}\)

Find the Root-Mean-Square Value of the Electric Field

To find the root-mean-square value, \(E_\mathrm{rms}\), of the electric field, we can use the formula: \(E_\mathrm{rms} = \frac{E_0}{\sqrt{2}}\). \(E_\mathrm{rms} = \frac{1939.84 \mathrm{\frac{V}{m}}}{\sqrt{2}} = 1371.94 \mathrm{\frac{V}{m}}\) The root-mean-square value of the electric field in the laser beam is 1371.94 V/m. #b) Calculating the total electromagnetic energy in 1.00 m of the laser beam#

Calculate the Total Electromagnetic Energy

To calculate the total electromagnetic energy, \(U\), in 1.00 m of the laser beam, we can use the formula: \(U = I \times c \times \epsilon_0 \times A \times d\), where \(d\) is the given distance (1.00 m). \(U = 1.5915 \times 10^{3} \mathrm{\frac{W}{m^2}} \times 3 \times 10^8 \mathrm{\frac{m}{s}} \times 8.85 \times 10^{-12} \mathrm{\frac{C^2}{N \cdot m^2}} \times 3.14159 \times 10^{-6} \mathrm{~m}^2 \times 1.00 \mathrm{~m}\) \(U = 4.0188 \times 10^{-3} \mathrm{J}\) The total electromagnetic energy in 1.00 m of the laser beam is approximately 4.02 mJ.

Key concepts

Electric Field in Laser Beam

When we talk about a laser beam, we're referring to a highly focused stream of photons, or light particles, which creates a precise and consistent beam of light. This light is not just made up of photons but also harbors an oscillating electric field—a fundamental characteristic of all electromagnetic waves, including laser light. The electric field in a laser beam is vital because it's what exerts a force on charged particles and can induce current if it interacts with a conductive material.

The electric field strength of a laser beam is determined by its amplitude, which corresponds to the maximum field strength during one wave cycle. High electric field amplitudes in laser beams can lead to significant effects, such as nonlinear optical phenomena. These characteristics make laser beams incredibly useful for a variety of applications from cutting and welding to data transfer and corrective eye surgery. In our exercise, we're concerned with a practical aspect of this concept: how to calculate the root-mean-square (RMS) value of the electric field which is representative of the electric field's strength over time, critical in understanding the laser's impact on materials it interacts with.

Intensity of Laser Beam

The intensity of a laser beam is a measure of the power that the beam delivers per unit area. Understanding the intensity is crucial as it tells us how much energy is being transmitted and how concentrated that energy is. This has practical implications, particularly in medical and industrial applications where specific intensities are required for cutting materials or performing delicate surgeries.

The interested student would note that intensity is calculated by dividing the power of the laser beam by the area over which the beam is spread. But it's not just a technical detail—it reflects the fact that, the smaller the area our laser beam's energy is concentrated over, the higher its intensity will be. For instance, a laser pointer may feel harmless on your skin merely because its intensity is spread out over a larger area, but the same laser, when focused to a point, can burn through plastic. Our exercise provided concrete steps to determine the intensity given the power and beam area, which is foundational knowledge for anyone studying laser technology or its applications.

Root-Mean-Square Value of Electric Field

The root-mean-square (RMS) value of an electric field is a statistical measure which represents the 'effective' value of the field strength. It’s particularly useful when dealing with alternating fields like those found in laser beams—where the field strength varies over time. Essentially, RMS provides a way to quantify the average field strength that would produce the same effect as the actual time-varying field over one cycle of oscillation.

For students, understanding the RMS value of an electric field aids in comprehending the continuous effects of the laser's electric field, like heating. If you were to average out the peaks and valleys of the oscillating field strength, you'd get the RMS value. This value is crucial for practical applications, like calibrating instruments or ensuring safety standards are met. The exercise presented shows how to move from peak electric field amplitude to RMS value—halving the peak value and then dividing by the square root of two—illustrating a foundational concept in the study of electromagnetic fields in physics and engineering courses.