91Ó°ÊÓ

The intensity of a laser beam can be determined using the formula for intensity, which is given by \[I = \frac{P}{A}\]where \(I\) is the intensity in watts per square meter (W/m²), \(P\) is the power of the laser in watts, and \(A\) is the area of the beam in square meters. First, convert the power from milliwatts to watts:\[P = 0.500 \text{ mW} = 0.500 \times 10^{-3} \text{ W} = 5.00 \times 10^{-4} \text{ W}\]Next, calculate the area \(A\) of the circular beam using the formula for the area of a circle \(A = \pi \left( \frac{d}{2} \right)^2\):\[A = \pi \left( \frac{1.00 \times 10^{-3}}{2} \right)^2 \text{ m}^2 = \pi \left( 0.5 \times 10^{-3} \right)^2 \text{ m}^2 = \pi \times 0.25 \times 10^{-6} \text{ m}^2 = 0.7854 \times 10^{-6} \text{ m}^2\]Finally, calculate the intensity:\[I = \frac{5.00 \times 10^{-4}}{0.7854 \times 10^{-6}} \text{ W/m}^2 = 637 \text{ W/m}^2\]

The maximum value of the electric field \(E_{max}\) can be calculated using the relationship between intensity \(I\) and the electric field in a wave:\[I = \frac{1}{2} \cdot c \cdot \varepsilon_0 \cdot E_{max}^2\]where \(c\) is the speed of light \((3 \times 10^8 \text{ m/s})\) and \(\varepsilon_0\) is the permittivity of free space \((8.85 \times 10^{-12} \text{ C}^2/\text{N} \cdot \text{m}^2)\).Rearranging to solve for \(E_{max}\):\[E_{max} = \sqrt{\frac{2I}{c \cdot \varepsilon_0}} = \sqrt{\frac{2 \times 637}{3 \times 10^8 \times 8.85 \times 10^{-12}}} = \sqrt{4.807 \times 10^3} \approx 69.3 \text{ V/m}\]

The maximum magnetic field \(B_{max}\) can be determined using the relationship between \(E_{max}\) and \(B_{max}\):\[B_{max} = \frac{E_{max}}{c}\]Substituting the value of \(E_{max}\):\[B_{max} = \frac{69.3}{3 \times 10^8} \approx 2.31 \times 10^{-7} \text{ T}\]

The average energy density \(u\) in the laser beam can be determined using the intensity and the speed of light with the formula:\[u = \frac{I}{c}\]Substituting the known values:\[u = \frac{637}{3 \times 10^8} \text{ J/m}^3 \approx 2.12 \times 10^{-6} \text{ J/m}^3\]