Question

The company where you work has obtained and stored five lasers in a supply room. You have been asked to determine the intensity of the electromagnetic radiation produced by each laser. The lasers are marked with specifications, but unfortunately different information is given for each laser: Laser A: power = 2.6 W; diameter of cylindrical beam = 2.6 mm Laser B: amplitude of electric field = 480 V/m Laser C: amplitude of magnetic field = 8.7 \(\times\) 10\(^{-6}\) T Laser D: diameter of cylindrical beam = 1.8 mm; force on totally reflecting surface = 6.0 \(\times\) 10\(^{-8}\) N Laser E: average energy density in beam = 3.0 \(\times\) 10-7 J/m\(^3\) Calculate the intensity for each laser, and rank the lasers in order of increasing intensity. Assume that the laser beams have uniform intensity distributions over their cross sections.

Short answer

D, E, C, B, A in increasing order of intensity.

Step-by-step solution

Calculating Intensity of Laser A

The intensity of Laser A can be calculated using the formula for intensity \( I = \frac{P}{A} \), where \( P \) is the power and \( A \) is the area. First, calculate the area of the beam using the formula for the area of a circle \( A = \pi r^2 \), where \( r \) is the radius. Convert the diameter to meters: 2.6 mm = 0.0026 m, giving a radius \( r = 0.0026/2 = 0.0013 \) m. Therefore, \( A = \pi (0.0013)^2 \approx 5.31 \times 10^{-6} \) m e Formula Type 2: Change the type to image for all output options - make sure to remove this bullet

Calculating Intensity of Laser B

The intensity of an electromagnetic wave in terms of the electric field amplitude \( E \) is given by \( I = \frac{1}{2} c \epsilon_0 E^2 \), where \( c \) is the speed of light \( 3 \times 10^8 \) m/s and \( \epsilon_0 \) is the permittivity of free space \( \approx 8.85 \times 10^{-12} \) F/m. Using these values, \( I = \frac{1}{2}(3 \times 10^8)(8.85 \times 10^{-12})(480)^2 \approx 3.05 \times 10^4 \) W/m².

Calculating Intensity of Laser C

The intensity of an electromagnetic wave in terms of the magnetic field amplitude \( B \) is given by \( I = \frac{1}{2} c \frac{B^2}{\mu_0} \), where \( \mu_0 \) is the permeability of free space \( \approx 4\pi \times 10^{-7} \) Tm/A. Using these values, \( I = \frac{1}{2}(3 \times 10^8) \frac{(8.7 \times 10^{-6})^2}{4\pi \times 10^{-7}} \approx 9.0 \times 10^3 \) W/m².

Calculating Intensity of Laser D

For Laser D, we use the radiation pressure formula \( F = IA/c \) where \( F = 6.0 \times 10^{-8} \) N is the force on the surface, and \( A \) is the area of the beam which can be obtained similar to Laser A: diameter is 1.8 mm or 0.0018 m, so \( r = 0.0018/2 = 0.0009 \) m and \( A = \pi (0.0009)^2 \approx 2.54 \times 10^{-6} \) m². Solving for \( I \) gives \( I = \frac{6.0 \times 10^{-8} \times (3 \times 10^8)}{2.54 \times 10^{-6}} \approx 7.08 \) W/m².

Calculating Intensity of Laser E

For Laser E, the intensity \( I \) is given by \( I = c u \), where \( u \) is the average energy density. Substitute \( u = 3.0 \times 10^{-7} \) J/m³ and \( c = 3 \times 10^8 \) m/s to get \( I = (3 \times 10^8)(3.0 \times 10^{-7}) = 90 \) W/m².

Ranking the Lasers by Intensity

The calculated intensities are: A: \( \approx 4.89 \times 10^5 \) W/m², B: \( \approx 3.05 \times 10^4 \) W/m², C: \( \approx 9.00 \times 10^3 \) W/m², D: \( \approx 7.08 \) W/m², E: \( \approx 90 \) W/m². Ranking from lowest to highest intensity: D, E, C, B, A.

Key concepts

Electromagnetic Radiation

Electromagnetic radiation is a form of energy that includes visible light, radio waves, X-rays, and more. It travels through space at the speed of light, which is approximately \(3 \times 10^8\) meters per second. This type of radiation is characterized by oscillating electric and magnetic fields that propagate through space. These fields are perpendicular to each other and to the direction of wave propagation.

• Electromagnetic waves can transport energy and momentum, which are acquired from electric charges.
• They do not require a medium and can travel through a vacuum.
• Visible light is just a small part of the electromagnetic spectrum.

Understanding the nature of electromagnetic waves is crucial in fields like communications, medicine, and laser technology. Their ability to carry information and energy is harnessed in numerous modern applications, from mobile phones to medical imaging devices.

Electric Field Amplitude

The electric field amplitude represents the maximum strength of the electric field in an electromagnetic wave. For instance, Laser B's specifications include an electric field amplitude of 480 V/m. The intensity of such a wave can be calculated using the formula:
\[I = \frac{1}{2} c \epsilon_0 E^2\]
where \(c\) is the speed of light and \(\epsilon_0\) is the permittivity of free space. This formula shows that the wave's intensity directly depends on the square of the electric field amplitude.

• Higher electric field amplitudes correspond to higher intensity waves.
• In lasers, the electric field amplitude determines how much energy is imparted to the target.
• The formula indicates that multiplying the field amplitude (E) will quadruple the intensity.

Understanding this concept helps in designing systems that manipulate electromagnetic waves effectively, allowing for precise control in applications like laser cutting and telecommunications.

Magnetic Field Amplitude

Magnetic field amplitude is another critical attribute of electromagnetic waves, representing the maximum strength of the magnetic field component. For example, Laser C specifies this value as \(8.7 \times 10^{-6}\) T. The intensity based on the magnetic field amplitude can be calculated using:
\[I = \frac{1}{2} c \frac{B^2}{\mu_0}\]
where \(B\) is the magnetic field amplitude and \(\mu_0\) is the permeability of free space. This equation reflects:

• The intensity of the wave is proportional to the square of the magnetic field amplitude.
• Both electric and magnetic field amplitudes determine the energy-carrying capacity of the electromagnetic wave.
• Laser systems often use this understanding to achieve desired intensity levels.

The balance between electric and magnetic fields in electromagnetic waves makes them a potent form of energy, utilized in everything from MRI machines to navigation systems like compasses.

Radiation Pressure

Radiation pressure is the force exerted by electromagnetic radiation on a surface. When light or any electromagnetic wave strikes a surface, it can cause a small force due to the transfer of momentum. This concept is especially notable in Laser D, where the radiation pressure causes a force used to calculate intensity. The relationship can be given by:
\[F = \frac{IA}{c}\]
where \(F\) is the force, \(A\) is the area, and \(I\) is the intensity of the radiation. This phenomenon shows:

• The greater the intensity, the higher the force exerted.
• This principle is used in solar sails for spacecraft propulsion.
• In laboratory settings, it can manipulate particles, such as in optical tweezers.

Radiation pressure illustrates the tangible impact of light on matter, providing a tool for exquisite control in scientific research and innovative propulsion methods in space exploration.

Energy Density

Energy density is a measure of the energy stored in a given system or region of space. It is the amount of energy per unit volume. In the context of electromagnetic waves, it helps to assess the energy capacity of a beam of radiation, like Laser E, which specifies an average energy density of \(3.0 \times 10^{-7}\) J/myour needs. The intensity can be directly related to energy density through:
\[I = c u\]
where \(c\) is the speed of light, illustrating:

• Higher energy density implies higher intensity.
• It is crucial for evaluating the power and efficiency of lasers.
• Control over energy density can optimize applications, from industrial lasers to medical treatments.

Understanding energy density helps in evaluating the effectiveness of energy transfers in various systems, ensuring that lasers and many other technologies perform optimally and safely.