Flux

From Wikipedia, the free encyclopedia

The field lines of a vector field F through surfaces with unit normal n, the angle from n to F is θ. Flux is a measure of how much of the field passes through a given surface. F is decomposed into components perpendicular (⊥) and parallel ( ‖ ) to n. Only the parallel component contributes to flux because it is the maximum extent of the field passing through the surface at a point, the perpendicular component does not contribute. Top: Three field lines through a plane surface, one normal to the surface, one parallel, and one intermediate. Bottom: Field line through a curved surface, showing the setup of the unit normal and surface element to calculate flux.

Flux describes the quantity which passes through a surface or substance. A flux is either a concept based in physics or used with applied mathematics. Both concepts have mathematical rigor, enabling comparison of the underlying math when the terminology is unclear. For transport phenomena, flux is a vector quantity, describing the magnitude and direction of the flow of a substance or property. In electromagnetism, flux is a scalar quantity, defined as the surface integral of the component of a vector field perpendicular to the surface at each point.^[1]

Terminology

The word flux comes from Latin: fluxus means "flow", and fluere is "to flow".^[2] As fluxion, this term was introduced into differential calculus by Isaac Newton.

One could argue, based on the work of James Clerk Maxwell,^[3] that the transport definition precedes the way the term is used in electromagnetism. The specific quote from Maxwell is:

In the case of fluxes, we have to take the integral, over a surface, of the flux through every element of the surface. The result of this operation is called the surface integral of the flux. It represents the quantity which passes through the surface.

— James Clerk Maxwell

According to the first definition, flux may be a single vector, or flux may be a vector field / function of position. In the latter case flux can readily be integrated over a surface. By contrast, according to the second definition, flux is the integral over a surface; it makes no sense to integrate a second-definition flux for one would be integrating over a surface twice. Thus, Maxwell's quote only makes sense if "flux" is being used according to the first definition (and furthermore is a vector field rather than single vector). This is ironic because Maxwell was one of the major developers of what we now call "electric flux" and "magnetic flux" according to the second definition. Their names in accordance with the quote (and first definition) would be "surface integral of electric flux" and "surface integral of magnetic flux", in which case "electric flux" would instead be defined as "electric field" and "magnetic flux" defined as "magnetic field". This implies that Maxwell conceived of these fields as flows/fluxes of some sort.

Given a flux according to the second definition, the corresponding flux density, if that term is used, refers to its derivative along the surface that was integrated. By the Fundamental theorem of calculus, the corresponding flux density is a flux according to the first definition. Given a current such as electric current—charge per time, current density would also be a flux according to the first definition—charge per time per area. Due to the conflicting definitions of flux, and the interchangeability of flux, flow, and current in nontechnical English, all of the terms used in this paragraph are sometimes used interchangeably and ambiguously. Concrete fluxes in the rest of this article will be used in accordance to their broad acceptance in the literature, regardless of which definition of flux the term corresponds to.

Flux as flow rate per unit area

In transport phenomena (heat transfer, mass transfer and fluid dynamics), flux is defined as the rate of flow of a property per unit area, which has the dimensions [quantity]·[time]⁻¹·[area]⁻¹.^[4] The area is of the surface the property is flowing "through" or "across". For example, the magnitude of a river's current, i.e. the amount of water that flows through a cross-section of the river each second, or the amount of sunlight energy that lands on a patch of ground each second, are kinds of flux.

General mathematical definition (transport)

Here are 3 definitions in increasing order of complexity. Each is a special case of the following. In all cases the frequent symbol j, (or J) is used for flux, q for the physical quantity that flows, t for time, and A for area. These identifiers will be written in bold when and only when they are vectors.
First, flux as a (single) scalar:

${\displaystyle j={\frac {I}{A}}}$

where:

${\displaystyle I=\lim \limits _{\Delta t\rightarrow 0}{\frac {\Delta q}{\Delta t}}={\frac {\mathrm {d} q}{\mathrm {d} t}}}$

In this case the surface in which flux is being measured is fixed, and has area A. The surface is assumed to be flat, and the flow is assumed to be everywhere constant with respect to position, and perpendicular to the surface.

Second, flux as a scalar field defined along a surface, i.e. a function of points on the surface:

${\displaystyle j(\mathbf {p} )={\frac {\partial I}{\partial A}}(\mathbf {p} )}$

${\displaystyle I(A,\mathbf {p} )={\frac {\mathrm {d} q}{\mathrm {d} t}}(A,\mathbf {p} )}$

As before, the surface is assumed to be flat, and the flow is assumed to be everywhere perpendicular to it. However the flow need not be constant. q is now a function of p, a point on the surface, and A, an area. Rather than measure the total flow through the surface, q measures the flow through the disk with area A centered at p along the surface.

Finally, flux as a vector field:

${\displaystyle \mathbf {j} (\mathbf {p} )={\frac {\partial \mathbf {I} }{\partial A}}(\mathbf {p} )}$

${\displaystyle \mathbf {I} (A,\mathbf {p} )={\underset {\mathbf {\hat {n}} }{\operatorname {arg\,max} }}\,\mathbf {\hat {n}} _{\mathbf {p} }{\frac {\mathrm {d} q}{\mathrm {d} t}}(A,\mathbf {p} ,\mathbf {\hat {n}} )}$

In this case, there is no fixed surface we are measuring over. q is a function of a point, an area, and a direction (given by a unit vector, ${\displaystyle \mathbf {\hat {n}} }$), and measures the flow through the disk of area A perpendicular to that unit vector. I is defined picking the unit vector that maximizes the flow around the point, because the true flow is maximized across the disk that is perpendicular to it. The unit vector thus uniquely maximizes the function when it points in the "true direction" of the flow. [Strictly speaking, this is an abuse of notation because the "arg max" cannot directly compare vectors; we take the vector with the biggest norm instead.]

Properties

These direct definition, especially the last, are rather unwieldy. For example, the argmax construction is artificial from the perspective of empirical measurements, when with a Weathervane or similar one can easily deduce the direction of flux at a point. Rather than defining the vector flux directly, it is often more intuitive to state some properties about it. Furthermore, from these properties the flux can uniquely be determined anyway.

If the flux j passes through the area at an angle θ to the area normal ${\displaystyle \mathbf {\hat {n}} }$, then

${\displaystyle \mathbf {j} \cdot \mathbf {\hat {n}} =j\cos \theta }$

where · is the dot product of the unit vectors. This is, the component of flux passing through the surface (i.e. normal to it) is j cos θ, while the component of flux passing tangential to the area is j sin θ, but there is no flux actually passing through the area in the tangential direction. The only component of flux passing normal to the area is the cosine component.

For vector flux, the surface integral of j over a surface S, gives the proper flowing per unit of time through the surface.

${\displaystyle {\frac {\mathrm {d} q}{\mathrm {d} t}}=\iint _{S}\mathbf {j} \cdot \mathbf {\hat {n}} \,{\rm {d}}A\ =\iint _{S}\mathbf {j} \cdot {\rm {d}}\mathbf {A} }$

A (and its infinitesimal) is the vector area, combination of the magnitude of the area through which the property passes, A, and a unit vector normal to the area, ${\displaystyle \mathbf {\hat {n}} }$. The relation is ${\displaystyle \mathbf {A} =A\mathbf {\hat {n}} }$. Unlike in the second set of equations, the surface here need not be flat.

Finally, we can integrate again over the time duration t₁ to t₂, getting the total amount of the property flowing through the surface in that time (t₂ − t₁):

${\displaystyle q=\int _{t_{1}}^{t_{2}}\iint _{S}\mathbf {j} \cdot {\rm {d}}{\mathbf {A} }\,{\rm {d}}t}$

Transport fluxes

Eight of the most common forms of flux from the transport phenomena literature are defined as follows:

1. Momentum flux, the rate of transfer of momentum across a unit area (N·s·m⁻²·s⁻¹). (Newton's law of viscosity)^[5]
2. Heat flux, the rate of heat flow across a unit area (J·m⁻²·s⁻¹). (Fourier's law of conduction)^[6] (This definition of heat flux fits Maxwell's original definition.)^[3]
3. Diffusion flux, the rate of movement of molecules across a unit area (mol·m⁻²·s⁻¹). (Fick's law of diffusion)^[5]
4. Volumetric flux, the rate of volume flow across a unit area (m³·m⁻²·s⁻¹). (Darcy's law of groundwater flow)
5. Mass flux, the rate of mass flow across a unit area (kg·m⁻²·s⁻¹). (Either an alternate form of Fick's law that includes the molecular mass, or an alternate form of Darcy's law that includes the density.)
6. Radiative flux, the amount of energy transferred in the form of photons at a certain distance from the source per unit area per second (J·m⁻²·s⁻¹). Used in astronomy to determine the magnitude and spectral class of a star. Also acts as a generalization of heat flux, which is equal to the radiative flux when restricted to the electromagnetic spectrum.
7. Energy flux, the rate of transfer of energy through a unit area (J·m⁻²·s⁻¹). The radiative flux and heat flux are specific cases of energy flux.
8. Particle flux, the rate of transfer of particles through a unit area ([number of particles] m⁻²·s⁻¹)

These fluxes are vectors at each point in space, and have a definite magnitude and direction. Also, one can take the divergence of any of these fluxes to determine the accumulation rate of the quantity in a control volume around a given point in space. For incompressible flow, the divergence of the volume flux is zero.

Chemical diffusion

As mentioned above, chemical molar flux of a component A in an isothermal, isobaric system is defined in Fick's law of diffusion as:

${\displaystyle \mathbf {J} _{A}=-D_{AB}\nabla c_{A}}$

where the nabla symbol ∇ denotes the gradient operator, D_AB is the diffusion coefficient (m²·s⁻¹) of component A diffusing through component B, c_A is the concentration (mol/m³) of component A.^[7]

This flux has units of mol·m⁻²·s⁻¹, and fits Maxwell's original definition of flux.^[3]

For dilute gases, kinetic molecular theory relates the diffusion coefficient D to the particle density n = N/V, the molecular mass m, the collision cross section ${\displaystyle \sigma }$, and the absolute temperature T by

${\displaystyle D={\frac {2}{3n\sigma }}{\sqrt {\frac {kT}{\pi m}}}}$

where the second factor is the mean free path and the square root (with Boltzmann's constant k) is the mean velocity of the particles.

In turbulent flows, the transport by eddy motion can be expressed as a grossly increased diffusion coefficient.

Quantum mechanics

In quantum mechanics, particles of mass m in the quantum state ψ(r, t) have a probability density defined as

${\displaystyle \rho =\psi ^{*}\psi =|\psi |^{2}.\,}$

So the probability of finding a particle in a differential volume element d³r is

${\displaystyle {\rm {d}}P=|\psi |^{2}{\rm {d}}^{3}\mathbf {r} .\,}$

Then the number of particles passing perpendicularly through unit area of a cross-section per unit time is the probability flux;

${\displaystyle \mathbf {J} ={\frac {i\hbar }{2m}}\left(\psi \nabla \psi ^{*}-\psi ^{*}\nabla \psi \right).\,}$

This is sometimes referred to as the probability current or current density,^[8] or probability flux density.^[9]

Flux as a surface integral

The flux visualized. The rings show the surface boundaries. The red arrows stand for the flow of charges, fluid particles, subatomic particles, photons, etc. The number of arrows that pass through each ring is the flux.

General mathematical definition (surface integral)

As a mathematical concept, flux is represented by the surface integral of a vector field,^[10]

${\displaystyle \Phi _{F}=\iint _{A}\mathbf {F} \cdot \mathrm {d} \mathbf {A} }$

${\displaystyle \Phi _{F}=\iint _{A}\mathbf {F} \cdot \mathbf {n} \mathrm {d} \mathbf {S} }$

where F is a vector field, and dA is the vector area of the surface A, directed as the surface normal.For the second,n is the outward pointed unit normal vector to the surface.

The surface has to be orientable, i.e. two sides can be distinguished: the surface does not fold back onto itself. Also, the surface has to be actually oriented, i.e. we use a convention as to flowing which way is counted positive; flowing backward is then counted negative.

The surface normal is usually directed by the right-hand rule.

Conversely, one can consider the flux the more fundamental quantity and call the vector field the flux density.

Often a vector field is drawn by curves (field lines) following the "flow"; the magnitude of the vector field is then the line density, and the flux through a surface is the number of lines. Lines originate from areas of positive divergence (sources) and end at areas of negative divergence (sinks).

See also the image at right: the number of red arrows passing through a unit area is the flux density, the curve encircling the red arrows denotes the boundary of the surface, and the orientation of the arrows with respect to the surface denotes the sign of the inner product of the vector field with the surface normals.

If the surface encloses a 3D region, usually the surface is oriented such that the influx is counted positive; the opposite is the outflux.

The divergence theorem states that the net outflux through a closed surface, in other words the net outflux from a 3D region, is found by adding the local net outflow from each point in the region (which is expressed by the divergence).

If the surface is not closed, it has an oriented curve as boundary. Stokes' theorem states that the flux of the curl of a vector field is the line integral of the vector field over this boundary. This path integral is also called circulation, especially in fluid dynamics. Thus the curl is the circulation density.

We can apply the flux and these theorems to many disciplines in which we see currents, forces, etc., applied through areas.

Electromagnetism

One way to better understand the concept of flux in electromagnetism is by comparing it to a butterfly net. The amount of air moving through the net at any given instant in time is the flux. If the wind speed is high, then the flux through the net is large. If the net is made bigger, then the flux is larger even though the wind speed is the same. For the most air to move through the net, the opening of the net must be facing the direction the wind is blowing. If the net is parallel to the wind, then no wind will be moving through the net. The simplest way to think of flux is "how much air goes through the net", where the air is a velocity field and the net is the boundary of an imaginary surface.

Electric flux

An electric "charge," such as a single electron in space, has a magnitude defined in coulombs. Such a charge has an electric field surrounding it. In pictorial form, an electric field is shown as a dot radiating "lines of flux" called Gauss lines.^[11] Electric Flux Density is the amount of electric flux, the number of "lines," passing through a given area. Units are Gauss/square meter.^[12]

Two forms of electric flux are used, one for the E-field:^[13][14]

${\displaystyle \Phi _{E}=}$ ${\displaystyle {\scriptstyle A}}$ ${\displaystyle \mathbf {E} \cdot {\rm {d}}\mathbf {A} }$

and one for the D-field (called the electric displacement):

${\displaystyle \Phi _{D}=}$ ${\displaystyle {\scriptstyle A}}$ ${\displaystyle \mathbf {D} \cdot {\rm {d}}\mathbf {A} }$

This quantity arises in Gauss's law – which states that the flux of the electric field E out of a closed surface is proportional to the electric charge Q_A enclosed in the surface (independent of how that charge is distributed), the integral form is:

${\displaystyle {\scriptstyle A}}$ ${\displaystyle \mathbf {E} \cdot {\rm {d}}\mathbf {A} ={\frac {Q_{A}}{\varepsilon _{0}}}}$

where ε₀ is the permittivity of free space.

If one considers the flux of the electric field vector, E, for a tube near a point charge in the field the charge but not containing it with sides formed by lines tangent to the field, the flux for the sides is zero and there is an equal and opposite flux at both ends of the tube. This is a consequence of Gauss's Law applied to an inverse square field. The flux for any cross-sectional surface of the tube will be the same. The total flux for any surface surrounding a charge q is q/ε₀.^[15]

In free space the electric displacement is given by the constitutive relation D = ε₀ E, so for any bounding surface the D-field flux equals the charge Q_A within it. Here the expression "flux of" indicates a mathematical operation and, as can be seen, the result is not necessarily a "flow", since nothing actually flows along electric field lines.

Magnetic flux

The magnetic flux density (magnetic field) having the unit Wb/m² (Tesla) is denoted by B, and magnetic flux is defined analogously:^[13][14]

${\displaystyle \Phi _{B}=}$ ${\displaystyle {\scriptstyle A}}$ ${\displaystyle \mathbf {B} \cdot {\rm {d}}\mathbf {A} }$

with the same notation above. The quantity arises in Faraday's law of induction, in integral form:

${\displaystyle \oint _{C}\mathbf {E} \cdot d{\boldsymbol {\ell }}=-\int _{\partial C}{\partial \mathbf {B} \over \partial t}\cdot {\rm {d}}\mathbf {s} =-{\frac {{\rm {d}}\Phi _{B}}{{\rm {d}}t}}}$

where dℓ is an infinitesimal vector line element of the closed curve C, with magnitude equal to the length of the infinitesimal line element, and direction given by the tangent to the curve C, with the sign determined by the integration direction.

The time-rate of change of the magnetic flux through a loop of wire is minus the electromotive force created in that wire. The direction is such that if current is allowed to pass through the wire, the electromotive force will cause a current which "opposes" the change in magnetic field by itself producing a magnetic field opposite to the change. This is the basis for inductors and many electric generators.

Poynting flux

Using this definition, the flux of the Poynting vector S over a specified surface is the rate at which electromagnetic energy flows through that surface, defined like before:^[14]

${\displaystyle \Phi _{S}=}$ ${\displaystyle {\scriptstyle A}}$ ${\displaystyle \mathbf {S} \cdot {\rm {d}}\mathbf {A} }$

The flux of the Poynting vector through a surface is the electromagnetic power, or energy per unit time, passing through that surface. This is commonly used in analysis of electromagnetic radiation, but has application to other electromagnetic systems as well.

Confusingly, the Poynting vector is sometimes called the power flux, which is an example of the first usage of flux, above.^[16] It has units of watts per square metre (W/m²).

Gauss's law for gravity

From Wikipedia, the free encyclopedia

This article is about Gauss's law concerning the gravitational field. For analogous laws concerning different fields, see Gauss's law and Gauss's law for magnetism. For Gauss's theorem, a mathematical theorem relevant to all of these laws, see Divergence theorem.

In physics, Gauss's law for gravity, also known as Gauss's flux theorem for gravity, is a law of physics that is essentially equivalent to Newton's law of universal gravitation. It is named after Carl Friedrich Gauss. Although Gauss's law for gravity is equivalent to Newton's law, there are many situations where Gauss's law for gravity offers a more convenient and simple way to do a calculation 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 electricity 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.

Qualitative statement of the law

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.

Integral form

The integral form of Gauss's law for gravity states:

${\displaystyle \scriptstyle \partial V}$ ${\displaystyle \mathbf {g} \cdot d\mathbf {A} =-4\pi GM}$

where

${\displaystyle \scriptstyle \partial V}$ (also written ${\displaystyle \oint _{\partial V}}$) 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. This can be contrasted with Gauss's law for electricity, where the flux can be either positive or negative. The difference is because charge can be either positive or negative, while mass can only be positive.

Differential form

The differential form of Gauss's law for gravity states

${\displaystyle \nabla \cdot \mathbf {g} =-4\pi G\rho ,}$

where ${\displaystyle \nabla \cdot }$ denotes divergence, G is the universal gravitational constant, and ρ is the mass density at each point.

Relation to the integral form

The two forms of Gauss's law for gravity are mathematically equivalent. The divergence theorem states:

${\displaystyle \oint _{\partial V}\mathbf {g} \cdot d\mathbf {A} =\int _{V}\nabla \cdot \mathbf {g} \ dV}$

where V is a closed region bounded by a simple closed oriented surface ∂V and dV is an infinitesimal piece of the volume V (see volume integral for more details). The gravitational field g must be a continuously differentiable vector field defined on a neighborhood of V.

Given also that

${\displaystyle M=\int _{V}\rho \ dV}$

we can apply the divergence theorem to the integral form of Gauss's law for gravity, which becomes:

${\displaystyle \int _{V}\nabla \cdot \mathbf {g} \ dV=-4\pi G\int _{V}\rho \ dV}$

which can be rewritten:

${\displaystyle \int _{V}(\nabla \cdot \mathbf {g} )\ dV=\int _{V}(-4\pi G\rho )\ dV.}$

This has to hold simultaneously for every possible volume V; the only way this can happen is if the integrands are equal. Hence we arrive at

${\displaystyle \nabla \cdot \mathbf {g} =-4\pi G\rho ,}$

which is the differential form of Gauss's law for gravity.

It is possible to derive the integral form from the differential form using the reverse of this method.

Although the two forms are equivalent, one or the other might be more convenient to use in a particular computation.

Relation to Newton's law

Deriving Gauss's law from Newton's law

Gauss's law for gravity can be derived from Newton's law of universal gravitation, which states that the gravitational field due to a point mass is:

${\displaystyle \mathbf {g} (\mathbf {r} )=-GM{\frac {\mathbf {e_{r}} }{r^{2}}}}$

where

e_r is the radial unit vector,

r is the radius, |r|.

M is the mass of the particle, which is assumed to be a point mass located at the origin.

Deriving Newton's law from Gauss's law and irrotationality

It is impossible to mathematically prove Newton's law from Gauss's law alone, because Gauss's law specifies the divergence of g but does not contain any information regarding the curl of g (see Helmholtz decomposition). In addition to Gauss's law, the assumption is used that g is irrotational (has zero curl), as gravity is a conservative force:

${\displaystyle \nabla \times \mathbf {g} =0}$

Even these are not enough: Boundary conditions on g are also necessary to prove Newton's law, such as the assumption that the field is zero infinitely far from a mass.

Poisson's equation and gravitational potential

Since the gravitational field has zero curl (equivalently, gravity is a conservative force) as mentioned above, it can be written as the gradient of a scalar potential, called the gravitational potential:

${\displaystyle \mathbf {g} =-\nabla \phi .}$

Then the differential form of Gauss's law for gravity becomes Poisson's equation:

${\displaystyle \nabla ^{2}\phi =4\pi G\rho .}$

This provides an alternate means of calculating the gravitational potential and gravitational field.

Although computing g via Poisson's equation is mathematically equivalent to computing g directly from Gauss's law, one or the other approach may be an easier computation in a given situation.
In radially symmetric systems, the gravitational potential is a function of only one variable (namely, ${\displaystyle r=|\mathbf {r} |}$), and Poisson's equation becomes (see Del in cylindrical and spherical coordinates):

${\displaystyle {\frac {1}{r^{2}}}{\frac {\partial }{\partial r}}\left(r^{2}\,{\frac {\partial \phi }{\partial r}}\right)=4\pi G\rho (r)}$

while the gravitational field is:

${\displaystyle \mathbf {g} (\mathbf {r} )=-\mathbf {e_{r}} {\frac {\partial \phi }{\partial r}}.}$

When solving the equation it should be taken into account that in the case of finite densities ∂ϕ/∂r has to be continuous at boundaries (discontinuities of the density), and zero for r = 0.

Applications

Gauss's law can be used to easily derive the gravitational field in certain cases where a direct application of Newton's law would be more difficult (but not impossible). See the article Gaussian surface for more details on how these derivations are done. Three such applications are as follows:

Bouguer plate

We can conclude (by using a "Gaussian pillbox") that for an infinite, flat plate (Bouguer plate) of any finite thickness, the gravitational field outside the plate is perpendicular to the plate, towards it, with magnitude 2πG times the mass per unit area, independent of the distance to the plate^[2].

More generally, for a mass distribution with the density depending on one Cartesian coordinate z only, gravity for any z is 2πG times (the mass per unit area above z, minus the mass per unit area below z).

In particular, a combination of two equal parallel infinite plates does not produce any gravity inside.

Cylindrically symmetric mass distribution

In the case of an infinite uniform (in z) cylindrically symmetric mass distribution we can conclude (by using a cylindrical Gaussian surface) that the field strength at a distance r from the center is inward with a magnitude of 2G/r times the total mass per unit length at a smaller distance (from the axis), regardless of any masses at a larger distance.

For example, inside an infinite uniform hollow cylinder, the field is zero.

Spherically symmetric mass distribution

In the case of a spherically symmetric mass distribution we can conclude (by using a spherical Gaussian surface) that the field strength at a distance r from the center is inward with a magnitude of G/r² times only the total mass within a smaller distance than r. All the mass at a greater distance than r from the center can be ignored.
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 any masses inside and outside the sphere only).

Although this follows in one or two lines of algebra from Gauss's law for gravity, it took Isaac Newton several pages of cumbersome calculus to derive it directly using his law of gravity; see the article shell theorem for this direct derivation.

Derivation from Lagrangian

The Lagrangian density for Newtonian gravity is

${\displaystyle {\mathcal {L}}({\vec {x}},t)=-\rho ({\vec {x}},t)\phi ({\vec {x}},t)-{1 \over 8\pi G}(\nabla \phi ({\vec {x}},t))^{2}}$

Applying Hamilton's principle to this Lagrangian, the result is Gauss's law for gravity:

${\displaystyle 4\pi G\rho ({\vec {x}},t)=\nabla ^{2}\phi ({\vec {x}},t).}$

In fiction

In Arthur C. Clarke's science fiction novel, 2010: Odyssey Two, while investigating the alien Monolith orbiting Jupiter, the Leonov 's chief scientist, Vasili Orlov, has engineer Curnow park one of the revived Discovery 's space pods a short distance from the Monolith's two-kilometer-long surface, recalling Bouguer's Anomaly, derived from Gauss's law. He remarks, "I've just remembered an exercise from one of my college astronomy courses - the gravitational attraction of an infinite flat plate. I never thought I'd have a chance of using it in real life." ^[3]