Respuesta :
ANSWERS
(a) I = 50.92 W/m²
(b) B₀ = 6.53 x 10⁻⁷ T
(c) E₀ = 195.9 V/m
EXPLANATION
Given:
• the power output of a helium-neon laser, P = 0.250 mW = 2.5 x 10⁻⁴ W
Find:
• (a), The intensity, I in W/m², if a laser beam is projected into a circular spot with a diameter d = 2.50 mm
,• (b), The peak magnetic field strength, B₀ in T
,• (c), The peak electric field strength, E₀ in V/m
(a) The intensity of a beam of light of power P projected in area A is,
[tex]I=\frac{P}{A}[/tex]We know that the circular spot has a diameter of 2.50 mm, so its radius is,
[tex]r=\frac{d}{2}=\frac{2.50mm}{2}=1.25mm=1.25\cdot10^{-3}m[/tex]So, the area of the circular spot is,
[tex]A=\pi r^2=\pi\cdot(1.25\cdot10^{-3})^2m^2\approx4.91\cdot10^{-6}m^2[/tex]And the intensity is,
[tex]I=\frac{P}{A}=\frac{2.5\cdot10^{-4}W}{4.91\cdot10^{-6}m^2}\approx50.92W/m^2[/tex]Hence, the intensity of the laser beam is 50.92 W/m², rounded to two decimal places.
(b) The intensity of the beam is also related to the peak magnetic field as follows,
[tex]I=\frac{c\cdot B_0^2}{2\mu_0}[/tex]Where c is the speed of light, 3 x 10⁸ m/s, and μ₀ is the vacuum magnetic permeability, 1.257 x 10⁻⁶ H/m.
Solving this equation for B₀,
[tex]B_0=\sqrt{\frac{2I\mu_0}{c}}=\sqrt{\frac{2\cdot50.92W/m^2\cdot1.257\cdot10^{-6}H/m}{3\cdot10^8m/s}}\approx6.53\cdot10^{-7}T[/tex]Hence, the peak magnetic field strength is 6.53 x 10⁻⁷ T.
(c) The peak electric field strength is proportional to the peak magnetic field strength. The constant of proportionality is the speed of light,
[tex]E_0=cB_0[/tex]Using the peak magnetic field strength found in part (b),
[tex]E_0=3\cdot10^8m/s\cdot6.53\cdot10^{-7}T\approx195.9V/m[/tex]Hence, the peak electric field strength is 195.9 V/m.
