A $20 \mathrm{~W}$ laser beam, with an initial diameter of $2.0 \mathrm{~mm}$, spreads out to a diameter of $2.0 \mathrm{~m}$ after traveling $10,000 \mathrm{~m}$. What is the initial intensity of the beam?
A) $6.4 \times 10^{6} \mathrm{~W} / \mathrm{m}^{2}$
B) $1.0 \times 105 \mathrm{~W} / \mathrm{m}^{2}$
C) $2.0 \times 10^{5} \mathrm{~W} / \mathrm{m}^{2}$
D) $3.2 \times 10^{6} \mathrm{~W} / \mathrm{m}^{2}$
E) $20 \times 106 \mathrm{~W} / \mathrm{m}^{2}$

Question

Quizplus · Accepted Answer

The initial intensity of the beam can be calculated using the formula $I = \frac{P}{A}$, where $I$ is the intensity, $P$ is the power, and $A$ is the area of the beam. The initial area $A$ can be found using the formula for the area of a circle, $A = \pi r^2$, where $r$ is the radius of the beam. The initial diameter is $2.0 \mathrm{~mm}$, so the radius is $1.0 \mathrm{~mm}$ or $0.001 \mathrm{~m}$. Thus, the initial area $A = \pi (0.001)^2 = 3.14 \times 10^{-6} \mathrm{~m}^2$. Substituting the values into the intensity formula gives $I = \frac{20}{3.14 \times 10^{-6}} = 6.4 \times 10^{6} \mathrm{~W} / \mathrm{m}^{2}$.

A 20 W20 \mathrm{~W}20 W Laser Beam, with an Initial Diameter Of 2.0 mm2.0 \mathrm{~mm}2.0 mm

A $20 \mathrm{~W}$ Laser Beam, with an Initial Diameter Of $2.0 \mathrm{~mm}$