Question

The cylindrical beam of a 10.0 -mW laser is \(0.85 \mathrm{cm}\) in diameter. What is the rms value of the electric field?

Short answer

The rms value of the electric field is approximately 290 V/m.

Step-by-step solution

Understanding Laser Power and Area

First, we need to determine the area of the laser beam. The beam is circular and has a diameter of 0.85 cm. The radius of the beam can be calculated as half of the diameter. The area of a circle is given by the formula \( A = \pi r^2 \), where \( r \) is the radius.

Calculate the Radius

Convert the diameter of the beam from centimeters to meters by dividing by 100: 0.85 cm = 0.0085 m. The radius \( r \) is half of the diameter: \( r = \frac{0.0085}{2} = 0.00425 \) m.

Calculate the Area

Now, we calculate the area using the formula \( A = \pi r^2 \): \[ A = \pi (0.00425)^2 \approx 5.67 \times 10^{-5} \, \text{m}^2 \]

Understanding Power and Intensity

The intensity \( I \) of the laser beam is defined as power per unit area, and is given by the formula \( I = \frac{P}{A} \), where \( P = 10.0 \times 10^{-3} \) W and \( A = 5.67 \times 10^{-5} \, \text{m}^2 \).

Calculate Intensity

Calculate the intensity using the power and area:\[ I = \frac{10.0 \times 10^{-3}}{5.67 \times 10^{-5}} \approx 176.31 \, \text{W/m}^2 \]

Relation Between Intensity and RMS Electric Field

The intensity \( I \) is related to the rms value of the electric field \( E_{rms} \) by the formula: \[ I = \frac{1}{2} c \varepsilon_0 E_{rms}^2 \]where \( c \) is the speed of light in a vacuum \( c = 3 \times 10^8 \, \text{m/s} \) and \( \varepsilon_0 \) is the permittivity of free space \( \varepsilon_0 = 8.85 \times 10^{-12} \, \text{F/m} \).

Solve for RMS Electric Field

Rearrange the equation to solve for \( E_{rms} \):\[ E_{rms} = \sqrt{\frac{2I}{c\varepsilon_0}} \]Substitute the values:\[ E_{rms} = \sqrt{\frac{2 \times 176.31}{3 \times 10^8 \times 8.85 \times 10^{-12}}} \approx 289.8 \, \text{V/m} \]

Key concepts

Electric Field

An electric field represents the force per unit charge experienced by a charge in a particular area. It is a vector field, meaning it has both magnitude and direction. In the context of a laser beam, the electric field oscillates as the beam propagates through space.
In laser physics, the root mean square (rms) value of the electric field is often used. This is a statistical measure of the amount of electric field present. Unlike the peak value, the rms value accounts for the variations in the field's intensity over time.
This value is crucial in applications where exact electric field strength influences the interaction of the laser with other materials, like in precision cutting or medical treatments. Understanding how to derive it from laser beam characteristics, such as intensity, ensures we can predict how effective a laser will be in a given task.

Laser Beam Intensity

Laser beam intensity is a measure of the amount of power per unit area that a laser beam carries. It is crucial in determining how much energy the beam is capable of delivering to a target area, whether for cutting, illuminating, or other applications.
Intensity is given by the formula:

• \( I = \frac{P}{A} \)
  where \( I \) is the intensity, \( P \) is the power of the beam, and \( A \) is the area of the beam.

Intensity plays a fundamental role because it directly impacts the effects of the laser beam on different materials.
For example, higher intensity might be needed if you wish to cut through denser materials. Conversely, delicate tasks might require lower intensity to prevent damage.
Understanding how to manipulate intensity through beam area adjustments or power changes is key in many scientific and industrial applications.

Permittivity of Free Space

The permittivity of free space, often denoted as \( \varepsilon_0 \), is a fundamental physical constant that characterizes how electric fields interact with the vacuum of space.
It is a measure of the resistance encountered when forming an electric field in a vacuum. A lower permittivity means that it is easier to form an electric field, while a higher permittivity implies greater difficulty.
The value of permittivity of free space is approximately \( 8.85 \times 10^{-12} \, \text{F/m} \), which is essential knowledge in both electrodynamics and laser physics. This constant appears in the relation between intensity and the rms electric field in our equations, underscoring its importance.
In many laser applications, optimizing the interaction between the laser beam and its environment requires careful consideration of \( \varepsilon_0 \), due to its implication in field formation and propagation.

Speed of Light

The speed of light is one of the most well-known constants in physics, represented by \( c \). It is the speed at which all light waves propagate in a vacuum and is approximately \( 3 \times 10^8 \, \text{m/s} \).
This constant is pivotal in the equations connecting the intensity of a laser to the electric field, where it appears alongside other constants like the permittivity of free space.
Its significance extends beyond just laser physics. The speed of light forms the cornerstone of the theory of relativity, impacting how we understand time, space, and energy relationships.
In everyday life and technology, knowing the speed of light allows us to manage and predict the behavior of light-related technologies and phenomena accurately. It's crucial in designing sophisticated systems like lasers, which depend on precise timing and energy delivery for efficacy.