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Within a uniform spherical body of radius R, density ρ, and mass m, the gravitational force g inside the sphere varies linearly with distance r from the center, giving the gravitational potential inside the sphere, which is [7] [8] = [] = [],, which differentiably connects to the potential function for the outside of the sphere (see the figure ...
A corollary is that inside a solid sphere of constant density, the gravitational force within the object varies linearly with distance from the center, becoming zero by symmetry at the center of mass. This can be seen as follows: take a point within such a sphere, at a distance from the center of the sphere. Then you can ignore all of the ...
For example, a hollow sphere does not produce any net gravity inside. The gravitational field inside is the same as if the hollow sphere were not there (i.e. the resultant field is that of all masses not including the sphere, which can be inside and outside the sphere).
Assuming a constant density , the masses of a shell and the sphere inside it are: = and = The required energy for a shell is the negative of the gravitational potential energy: d U = − G m s h e l l m i n t e r i o r r {\displaystyle dU=-G{\frac {m_{\mathrm {shell} }m_{\mathrm {interior} }}{r}}}
Siméon Denis Poisson. Poisson's equation is an elliptic partial differential equation of broad utility in theoretical physics.For example, the solution to Poisson's equation is the potential field caused by a given electric charge or mass density distribution; with the potential field known, one can then calculate the corresponding electrostatic or gravitational (force) field.
Examples of such potentials are the electric potential, the magnetic potential and the gravitational potential. For clarity, we illustrate the expansion for a point charge , [ 1 ] then generalize to an arbitrary charge density ρ ( r ′ ) . {\displaystyle \rho (\mathbf {r} ').}
ρ g is mass density, with SI unit kg⋅m −3; ρ is charge density; J g is mass current density or mass flux, with SI unit kg⋅m −2 ⋅s −1; J is electric current density; G is the gravitational constant; ε 0 is the vacuum permittivity; c is both the speed of propagation of gravity and the speed of light.
The electric flux is then a simple product of the surface area and the strength of the electric field, and is proportional to the total charge enclosed by the surface. Here, the electric field outside (r > R) and inside (r < R) of a charged sphere is being calculated (see Wikiversity).