Question

.An industrial carbon dioxide laser produces a beam of radiation with average power of \(6.00 \mathrm{~kW}\) at a wavelength of \(10.6 \mu \mathrm{m}\). Such a laser can be used to cut steel up to \(25 \mathrm{~mm}\) thick. The laser light is polarized in the \(x\) -direction, travels in the positive \(z\) -direction, and is collimated (neither diverging or converging) at a constant diameter of \(100.0 \mu \mathrm{m} .\) Write the equations for the laser light's electric and magnetic fields as a function of time and of position \(z\) along the beam. Recall that \(\vec{E}\) and \(\vec{B}\) are vectors. Leave the overall phase unspecified, but be sure to check the relative phase between \(\vec{E}\) and \(\vec{B}\) .

Short answer

Question: Determine the equations for the electric and magnetic fields of an industrial carbon dioxide laser beam as a function of time and position \(z\) along the beam, given the laser's average power is \(6.00 \mathrm{~kW}\), the beam diameter is \(100.0 \mu \mathrm{m}\), and the wavelength of the light produced by the laser is \(10.6 \mu \mathrm{m}\). The electric field is polarized in the \(x\)-direction, and the magnetic field is polarized in the \(y\)-direction. Answer: The equations for the electric and magnetic fields of the industrial carbon dioxide laser as a function of time and position along the beam are: \(\vec{E}(z, t) = E_0 \cos{(kz - \omega t)} \hat{x}\) \(\vec{B}(z, t) = B_0 \cos{(kz - \omega t)} \hat{y}\) where \(E_0\) represents the electric field amplitude, \(B_0\) represents the magnetic field amplitude, \(k\) is the wave number, \(\omega\) is the angular frequency, and \(\hat{x}\) and \(\hat{y}\) are unit vectors in the \(x\) and \(y\) directions, respectively.

Step-by-step solution

Calculate the laser beam intensity

We can start by calculating the intensity (power per unit area) of the laser beam. The average power of the laser is given by \(P = 6.00 \mathrm{~kW}\) and the diameter of the beam is given by \(d = 100.0 \mu \mathrm{m}\). First, calculate the beam's cross-sectional area, then divide the power by the area to get the intensity. The area of the laser beam: \(A = \dfrac{\pi d^2}{4}\) Intensity: \(I = \dfrac{P}{A}\)

Calculate the amplitude of the electric field

Now we can use the intensity to calculate the amplitude of the electric field. The intensity is related to the electric field amplitude \(E_0\) and the speed of light \(c\) by the following formula: \(I = \dfrac{c \varepsilon_0 E_0^2}{2}\) where \(\varepsilon_0\) is the permittivity of free space. We can rearrange the formula to solve for the electric field amplitude: \(E_0 = \sqrt{\dfrac{2I}{c \varepsilon_0}}\)

Calculate the amplitude of the magnetic field

Next, we can calculate the amplitude of the magnetic field using the relation between electric and magnetic fields in a plane wave: \(B_0 = \dfrac{E_0}{c}\)

Write the equations for the electric and magnetic fields

Now that we have the amplitudes of the electric and magnetic fields, we can write the equations for these fields as a function of time and position \(z\) along the beam. The electric field is polarized in the \(x\)-direction and propagates in the \(z\)-direction. The equation for the electric field: \(\vec{E}(z, t) = E_0 \cos{(kz - \omega t)} \hat{x}\) The magnetic field is perpendicular to both the electric field and propagation direction, so it is polarized in the \(y\)-direction: \(\vec{B}(z, t) = B_0 \cos{(kz - \omega t)} \hat{y}\) Here, \(k\) is the wave number, and \(\omega\) is the angular frequency. They can be written in terms of the given wavelength \(\lambda = 10.6 \mu \mathrm{m}\) and the speed of light: \(k = \dfrac{2\pi}{\lambda}\) \(\omega = kc\) These equations describe the electric and magnetic fields of the industrial carbon dioxide laser as a function of time and position along the beam.

Key concepts

Electric Field Amplitude

Electric field amplitude is an important aspect of understanding electromagnetic waves, especially when analyzing laser beams. For a laser beam, this amplitude, denoted as \(E_0\), represents the peak value of the electric field strength in the wave. To calculate it, we can use the intensity of the beam. The relationship is given by the formula: \[ E_0 = \sqrt{\dfrac{2I}{c \varepsilon_0}} \] where \(I\) is the intensity of the beam, \(c\) is the speed of light, and \(\varepsilon_0\) is the permittivity of free space. This formula shows that the electric field amplitude is directly related to the intensity of the beam. As intensity increases, the electric field amplitude increases, indicating a stronger laser beam. Understanding \(E_0\) helps to predict how the laser will interact with materials, such as cutting through steel in an industrial setting.

Magnetic Field Amplitude

When dealing with electromagnetic waves, not only the electric field amplitude is crucial but also the magnetic field amplitude, represented by \(B_0\). It is related to the electric field amplitude via the speed of light \(c\). The relationship is given by: \[ B_0 = \dfrac{E_0}{c} \] This equation highlights that the magnetic field amplitude is proportional to the electric field amplitude. In electromagnetic waves like a laser beam, these fields are perpendicular to each other and to the direction of wave propagation. - The electric field might be in the \(x\)-direction, while the magnetic field is in the \(y\)-direction. - This perpendicular orientation is essential for maintaining the energy flow in the \(z\)-direction, which is the direction of wave propagation. Knowing both amplitudes helps us understand the beam's full energy dynamics.

Intensity of Laser Beam

Intensity is a key parameter in characterizing laser beams. It refers to the power per unit area carried by the beam and is vital for applications requiring focused energy delivery. The intensity \(I\) is calculated using the formula: \[ I = \dfrac{P}{A} \] where \(P\) is the power of the laser and \(A\) is the cross-sectional area of the beam, given by \(A = \dfrac{\pi d^2}{4}\) for a circular beam of diameter \(d\). This relationship emphasizes the importance of the beam's area in determining how concentrated the laser’s power is. Higher intensity implies more energy concentrated on a smaller surface area, crucial for applications like cutting or welding. Hence, understanding and calculating intensity is fundamental for optimizing laser performance.

Wave Equation

The wave equation describes how the electric and magnetic fields propagate through space and time. For a laser beam, the electric field \(\vec{E}(z, t)\) and the magnetic field \(\vec{B}(z, t)\) can be represented as sinusoidal functions: - Electric Field: \[ \vec{E}(z, t) = E_0 \cos{(kz - \omega t)} \hat{x} \] - Magnetic Field: \[ \vec{B}(z, t) = B_0 \cos{(kz - \omega t)} \hat{y} \] These equations show the dimensionless fields oscillating in their respective directions. The wave number \(k\) and angular frequency \(\omega\) are derived from the wavelength \(\lambda\) and speed of light \(c\): - \(k = \dfrac{2\pi}{\lambda}\) - \(\omega = kc\) These parameters are crucial for calculating how fast and in what manner these fields oscillate as they travel through space. The wave equations illustrate how laser light maintains its coherent state over a long distance.

Polarization

Polarization describes the orientation of the electric field vector in a light wave. For this carbon dioxide laser, it is polarized in the \(x\)-direction. This means the electric field oscillates primarily in the horizontal plane. - Polarization is a key factor in laser applications that require specific field orientations. - It can influence the interaction of light with surfaces, affecting reflection and transmission. Understanding polarization is crucial for tasks that need precise light control, such as in polarized lenses or when using lasers for precise cutting and welding. By controlling the polarization, the behavior of light can be accurately predicted and utilized for industrial, scientific, or even artistic purposes.