Consider a small, spherical particle of radius r locate in...

Consider a small, spherical particle of radius $r$ locate in space a distance $R$ from the Sun. (a) Show that th ratio $F_{\text {rad }} / F_{\text {grav }}$ is proportional to $1 / r$, where $F_{\text {rad }}$ is th force exerted by solar radiation and $F_{\mathrm{grav}}$ is the force of gravitational attraction. (b) The result of part (a) means that, for a sufficiently small value of $r$, the force exerted on the particle by solar radiation exceeds the force of gravitational attraction. Calculate the value of $r$ for which the particle is in equilibrium under the two forces. (Assume that the particle has a perfectly absorbing surface and a mass density of $1.50 \mathrm{~g} / \mathrm{cm}^{3}$. Let the particle be located $3.75 \times 10^{11} \mathrm{~m}$ from the Sun, and use $214 \mathrm{~W} / \mathrm{m}^{2}$ as the value of the solar intensity at that point.)

Key Concepts

Solar Intensity and Its Role in Radiation Pressure

Solar intensity, which is the power per unit area received from the Sun, plays a crucial role in determining the magnitude of radiation pressure. The intensity of solar radiation decreases with the square of the distance from the Sun, and it directly influences the force exerted by solar radiation on any object in space.

Force Equilibrium

Force equilibrium occurs when two opposing forces acting on an object balance each other exactly, resulting in a net force of zero. In this case, equilibrium between radiation pressure and gravitational force determines a specific particle size at which the outward push of solar radiation is exactly countered by the inward pull of gravity.

Radiation Pressure

Radiation pressure is the pressure exerted upon any surface due to the exchange of momentum between the object and electromagnetic radiation. When photons from a light source, such as the Sun, strike or are absorbed by a particle, they impart momentum, resulting in a force that is directly proportional to the intensity of the radiation and the area of the surface exposed.

Gravitational Force

Gravitational force is the attractive interaction between any two masses, described by Newton's law of universal gravitation. It is dependent on the masses involved and inversely proportional to the square of the distance between them. In the context of small particles in space, gravitational force determines the attractive pull from massive bodies like the Sun.

Scaling of Forces with Particle Size

For a spherical particle, the force due to radiation pressure scales with the cross-sectional area (proportional to the square of the particle's radius), while the gravitational force scales with its mass (which is proportional to the volume, or the cube of the radius). This difference in scaling leads to a dependency of the ratio of radiation to gravitational force on the inverse of the particle's radius.

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