Question

Someone plans to float a small, totally absorbing sphere 0.500 m above an isotropic point source of light,so that the upward radiation force from the light matches the downward gravitational force on the sphere. The sphere’s density is 19.0 g/cm3, and its radius is 2.00 mm. (a) What power would be required of the light source?

Answer 1

468449163762.0812 W

Step-by-step explanation:

m = Mass =
`\rhoV`

V = Volume =
`(4)/(3)\pi r^3`

r = Distance of sphere from isotropic point source of light = 0.5 m

R = Radius of sphere = 2 mm

`\rho` = Density = 19 g/cm³

c = Speed of light =
`3* 10^8\ m/s`

A = Area =
`\pi R^2`

I = Intensity =
`(P)/(4\pi r^2)`

g = Acceleration due to gravity = 9.81 m/s²

Force due to radiation is given by

`F=(IA)/(c)\\\Rightarrow F=\frac{(P)/(4\pi r^2){\pi R^2}}{c}\\\Rightarrow F=(PR^2)/(4r^2c)`

According to the question

`F=mg\\\Rightarrow (PR^2)/(4r^2c)=\rho (4)/(3)\pi R^3g\\\Rightarrow P=(16r^2\rho c\pi Rg)/(3)\\\Rightarrow P=(16* 0.002* 19000* \pi* 0.5^2* 9.81* 3* 10^8)/(3)\\\Rightarrow P=468449163762.0812\ W`

The power required of the light source is 468449163762.0812 W