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Concept# Gauss's law for gravity

Summary

In physics, Gauss's law for gravity, also known as Gauss's flux theorem for gravity, is a law of physics that is equivalent to Newton's law of universal gravitation. It is named after Carl Friedrich Gauss. It states that the flux (surface integral) of the gravitational field over any closed surface is proportional to the mass enclosed. Gauss's law for gravity is often more convenient to work from than Newton's law.
The form of Gauss's law for gravity is mathematically similar to Gauss's law for electrostatics, one of Maxwell's equations. Gauss's law for gravity has the same mathematical relation to Newton's law that Gauss's law for electrostatics bears to Coulomb's law. This is because both Newton's law and Coulomb's law describe inverse-square interaction in a 3-dimensional space.
Gravitational field
The gravitational field g (also called gravitational acceleration) is a vector field – a vector at each point of space (and time). It is defined so that the gravitational force experienced by a particle is equal to the mass of the particle multiplied by the gravitational field at that point.
Gravitational flux is a surface integral of the gravitational field over a closed surface, analogous to how magnetic flux is a surface integral of the magnetic field.
Gauss's law for gravity states:
The gravitational flux through any closed surface is proportional to the enclosed mass.
The integral form of Gauss's law for gravity states:
where
(also written ) denotes a surface integral over a closed surface,
∂V is any closed surface (the boundary of an arbitrary volume V),
dA is a vector, whose magnitude is the area of an infinitesimal piece of the surface ∂V, and whose direction is the outward-pointing surface normal (see surface integral for more details),
g is the gravitational field,
G is the universal gravitational constant, and
M is the total mass enclosed within the surface ∂V.
The left-hand side of this equation is called the flux of the gravitational field. Note that according to the law it is always negative (or zero), and never positive.

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