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 $\overrightarrow{\mathrm{E}}$ and $\bar{B}$ are vectors. Leave the overall phase unspecified, but be sure to check the relative phase between $\vec{E}$ and $\vec{B}$.

Solution

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We will write expressions for the electric field $\vec{E}$ and magnetic field $\vec{B}$ of the electromagnetic wave travelling along the $z$ axis. Since we require amplitudes of electric field and magnetic field in this electromagnetic wave to write these expressions, we will first determine expression for the amplitude of the electric field in the electromagnetic wave. We will do this by equating expression for the intensity of the electromagnetic wave emitted from the laser (laser light) in terms of power of the laser and cross-sectional area of the laser beam with the expression for the intensity of the electromagnetic waves (laser light) in terms of amplitude of the electric field in the electromagnetic wave. After expressing amplitude of the electric field from the obtained equation, we will also determine expression for the amplitude of the magnetic field from the equation that relates amplitude of magnetic field with the amplitude of the magnetic field.
The given values are the average power of the laser beam $P=6 \mathrm{~kW}$, the wavelength of the laser beam $\lambda=10.6 \mathrm{~\mu m}$, and the diameter of the laser beam $d=100 \mathrm{~\mu m}$.