Question

A small laser emits light at power 6.88 mW and wavelength 441 nm. The laser beam is focused (narrowed) until its diameter matches the 1255 nm diameter of a sphere placed in its path. The sphere is perfectly absorbing and has density 4.00×10³kg/m³.

What is the beam intensity at the sphere's location?

Calculate the radiation pressure on the sphere.

Calculate the magnitude of the corresponding force.

Calculate the magnitude of the acceleration that force alone would give the sphere?

Answer 1

Given

Power of laser P = 6.88 x 10^-3 W

Wavelength λ = 441 x 10^-9 m

Diameter of the sphere and laser D = 1255 x 10^-9 m

Density of the sphere ρ= 4.00×10³ kg/m³

Solution

Radius of the sphere r = D/2 = 1255 x 10^-9 / 2 = 627.5 x 10^-9 m

Intensity

I = power/ area

I = P/πr²

I = 6.88 x 10^-3 / 3.14 x (627.5 x 10^-9)²

I = 5.56 x 10⁹ W/m²

Radiation pressure

P_rad  =I / c

P_rad = 5.56 x 10⁹ / 3 x 10⁸

P_rad = 18.55 Pa

Force.

F = P_rad x πr²

F = 18.55 x 3.14 x (627.5 x 10^-9)²

F = 2.29 x 10^-11 N

Acceleration

F = ma

m = ρV

m = 4ρπr³/3

a = F/m

a = 3F/4ρπr³

a = 3 x 2.29 x 10^-11/4 x 4 x 10³ x 3.14 x (627.5 x 10^-9)³

a = 5.53 x 10³ m/s²