A laser pointer delivers 0.10 -m \(\mathrm{W}\) average power in a beam \(0.90 \mathrm{mm}\) in diameter. Find (a) the average intensity, (b) the peak electric field, and (c) the peak magnetic field.

Understanding laser beam intensity begins with grasping the concept of average intensity calculation. It's a measure of how much energy is being transmitted in the form of light through a given area. To quantify this, we use the formula for intensity, which is denoted as:
\[\begin{equation} I = P / A, \text{where:}\begin{aligned} P & = \text{the power of the laser beam (in watts), and} \ A & = \text{the cross-sectional area through which the beam is distributed (in square meters).} \- \text{For beams with circular cross-sections,} \ A &= \text{use the area formula for a circle:} \ A &= \text{which simplifies to:} \ A &= \text{where:} \ d &= \text{the diameter of the beam (in meters)}. \end{aligned} \text{By determining the area and dividing the laser’s power by this value}, \text{students can calculate the average intensity of the beam}. \text{In practical situations}, \text{this enables us to understand how powerful a laser pointer is based on its stated milliwatt (mW) power specification and the size of its beam}. \text{Knowing the average intensity is crucial}, \text{especially in areas like medical procedures or material processing where precise energy delivery is critical}. \text{To illustrate}, \text{the given exercise involved a laser pointer with a power of 0.10 mW (0.0001 watts) and a diameter of 0.90 mm (0.0009 meters)}, \text{yielding an average intensity after calculation}. \end{equation}\]

The peak electric field represents the maximum strength of the electric field component in an electromagnetic wave, like that produced by a laser. This value tells us about the energy per unit charge at a particular point in the field. In the case of a laser beam, it's linked to the intensity of the beam with the formula:
\[\begin{equation} E_0 = \text{where:}\begin{aligned} \mu_0 & = \text{and} \ c & = \text{the speed of light (approximately } \text{).} \end{aligned} \end{equation}\]The peak electric field is an important quantity in various applications, from the design of optical communication systems, where it can influence the signal strength and integrity, to the realm of safety standards, determining safe exposure levels to electromagnetic radiation. Calculating this allows us to predict how different materials will interact with the laser light, vital when considering its use for cutting, engraving, or medical treatment.

Coupled with the peak electric field, the peak magnetic field characterizes the magnetic component of the electromagnetic wave. For a laser beam, the peak magnetic field is related to the peak electric field by a simple, yet profound relationship:
\[\begin{equation} B_0 = E_0 / c, \text{where:}\begin{aligned} E_0 & = \text{and} \ c & = \text{.} \end{aligned} \end{equation}\]This formula indicates that the magnetic field is directly proportional to the electric field but inversely proportional to the speed of light. Hence, with a known peak electric field, calculating the peak magnetic field becomes straightforward. This concept is essential in both theoretical and practical physics, enabling scientists and engineers to understand and manipulate the magnetic properties of light for advanced technologies like magnetic resonance imaging (MRI) and other forms of electromagnetic analysis. It demystifies how a beam of light can influence magnetic materials and is pivotal in determining the safety of laser devices.