Someone plans to float a small, totally absorbing sphere 0.500 m above an isotropic point sourc

step 1 In this question, we want to suspend a sphere or make the sphere float in the air at 0 .5 meters

Someone plans to float a small, totally absorbing sphere...

Someone plans to float a small, totally absorbing sphere $0.500 \mathrm{~m}$ 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 $19.0 \mathrm{~g} / \mathrm{cm}^{3},$ and its radius is $2.00 \mathrm{~mm}$. (a) What power would be required of the light source? (b) Even if such a source were made, why would the support of the sphere be unstable?

Key Concepts

Gravitational Force

Gravitational force is the attractive force exerted by a mass due to gravity, calculated as the product of the mass and the acceleration due to gravity. In problems involving equilibrium between light pressure and weight, the gravitational force represents the downward force that must be countered by the upward force from radiation pressure, thereby governing the necessary conditions for balance.

Equilibrium and Stability

Achieving equilibrium means that the upward radiation force equals the downward gravitational force, but stability requires that small perturbations do not lead to unbalanced forces. In systems where forces decay with distance (as with the inverse square law), the equilibrium can be inherently unstable; any slight displacement may alter the balance between forces such that the restoring forces are insufficient to return the object to its original position, leading to instability.

Radiation Pressure

Radiation pressure is the force exerted per unit area by electromagnetic radiation on a surface. When photons impact a surface, they impart momentum, and if the surface is absorbing, all of the photon momentum is transferred, creating a force. This principle is key in calculations where light is used to exert mechanical forces, such as in optical trapping or in balancing gravitational forces with radiation forces.

Inverse Square Law

The inverse square law describes how the intensity of light, or any point-source radiation, decreases with the square of the distance from the source. This law is critical for determining the distribution of radiation pressure around a point source, as the force due to radiation will become significantly weaker as the distance increases, influencing the required power and spatial stability of systems based on electromagnetic forces.

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