Exercise:
Because of the formal analogy between the gravitational and the electrostatic force Gauss's law can also be used for gravitational fields. We know that the gravitational field outside a spherical mass e.g. a star or a planet is given by gr G fracMr^ where G is the gravitational constant M the object's total mass and r the distance of a po from the object's centre of mass. abcliste abc Show that Gauss's law for gravitational fields can be written as Phi_G pi G sscMenclosed where Phi_G is the flux of the gravitational field through a closed surface and sscMenclosed the mass contained in that surface. abc Derive an expression for the gravitational field on the inside of a sphere with uniform mass distribution. Graph the gravitational field vs. the distance from a sphere with radius R for r R. abcliste

Solution:
abcliste abc For a spherical mass distribution we choose a concentric spherical shell as the Gaussian surface. The gravitational flux through this surface is Phi_G gr Ar G fracMr^ pi r^ pi G M quad square This result is true for any mass distribution and any closed surface as it is the case for the electric field. abc In order to find the gravitational field on the inside of a spherical mass we choose a concentric spherical shell with radius rR as the Gaussian surface. For a uniform mass distribution i.e. a constant density the enclosed mass we find fracsscMenclosedM fracsscVenclosedsscVtot fracr^R^ left fracrR right^ It follows for the gravitational flux Phi_G gr Ar gr pi r^ pi G sscMenclosed pi G M left fracrR right^ Solving for gr yields gr fracG MR^ r On the outside of the sphere we have the standard expression for the gravitational field gr fracG Mr^ On the surface of the sphere the two expresions yield the same value. The figure below displays the gravitational field for rR. The field on the surface is g_gR. center includegraphicswidthtextwidth#image_path:gauss-gravitation-# center abcliste