A laser beam is directed upward to make a tiny blackened glass sphere of mass 0.000000001 kg and diameter 20micro meter afloat in air. calculate the intensity of the laser beam.

To determine the intensity of the laser beam required to make a tiny blackened glass sphere float in air, we first need to understand the forces acting on the sphere. The primary force to consider here is the gravitational force acting on the sphere, which must be balanced by the radiation pressure exerted by the laser beam.

Understanding the Forces

The gravitational force (weight) acting on the sphere can be calculated using the formula:

F_gravity = m × g

Where:

• m = mass of the sphere = 0.000000001 kg
• g = acceleration due to gravity ≈ 9.81 m/s²

Plugging in the values:

F_gravity = 0.000000001 kg × 9.81 m/s² = 0.00000000981 N

Radiation Pressure from the Laser Beam

The radiation pressure (P) exerted by the laser beam can be expressed as:

P = I/c

Where:

• I = intensity of the laser beam (in W/m²)
• c = speed of light ≈ 3 × 10^8 m/s

To keep the sphere afloat, the radiation pressure must equal the gravitational force:

P = F_gravity/A

Here, A is the cross-sectional area of the sphere, which can be calculated using the formula for the area of a circle:

A = π × (r²)

Given that the diameter of the sphere is 20 micrometers, the radius (r) is:

r = 20 μm / 2 = 10 μm = 10 × 10^-6 m

Now, calculating the area:

A = π × (10 × 10^-6 m)² ≈ 3.14 × 10^-10 m²

Setting Up the Equation

Now we can set the radiation pressure equal to the gravitational force divided by the area:

F_gravity/A = I/c

Substituting the values we have:

0.00000000981 N / (3.14 × 10^-10 m²) = I / (3 × 10^8 m/s)

Calculating the Intensity

Rearranging the equation to solve for intensity (I):

I = (0.00000000981 N × 3 × 10^8 m/s) / (3.14 × 10^-10 m²)

Now, performing the calculations:

I ≈ (0.00000000981 × 3 × 10^8) / (3.14 × 10^-10)

I ≈ 9.36 × 10^3 W/m²

Final Result

The intensity of the laser beam required to make the tiny blackened glass sphere float in air is approximately 9,360 W/m². This value indicates the power per unit area that the laser must deliver to counteract the gravitational force acting on the sphere.