ebab7d0da53c51438720d60c6e2e467c - /page/wikibook/entry.mu

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Scalar potential

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</style><div role="note" class="hatnote navigation-not-searchable">This article is about a general description of a function used in mathematics and physics to describe conservative fields. For the scalar potential of electromagnetism, see <a href="Electric_potential" title="Electric potential">electric potential</a>. For all other uses, see <a href="Potential" title="Potential">potential</a>.</div>

In <a href="Mathematical_physics" title="Mathematical physics">mathematical physics</a>, scalar potential describes the situation where the difference in the <a href="Potential_energy" title="Potential energy">potential energies</a> of an object in two different positions depends only on the positions, not upon the path taken by the object in traveling from one position to the other. It is a <a href="Scalar_field" title="Scalar field">scalar field</a> in <a href="Three-space" class="mw-redirect" title="Three-space">three-space</a>: a directionless value (<a href="Scalar_(mathematics)" title="Scalar (mathematics)">scalar</a>) that depends only on its location. A familiar example is <a href="Gravitational_energy" title="Gravitational energy">potential energy due to gravity</a>.

A scalar <a href="Potential" title="Potential">potential</a> is a fundamental concept in <a href="Vector_analysis" class="mw-redirect" title="Vector analysis">vector analysis</a> and <a href="Physics" title="Physics">physics</a> (the adjective scalar is frequently omitted if there is no danger of confusion with <a href="Vector_potential" title="Vector potential">vector potential</a>). The scalar potential is an example of a <a href="Scalar_field" title="Scalar field">scalar field</a>. Given a <a href="Vector_field" title="Vector field">vector field</a> F, the scalar potential P is defined such that: <a href="#cite_note-1">[1]</a>

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<annotation encoding="application/x-tex">{\displaystyle \mathbf {F} =-\nabla P=-\left({\frac {\partial P}{\partial x}},{\frac {\partial P}{\partial y}},{\frac {\partial P}{\partial z}}\right),}</annotation>

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where ∇P is the <a href="Gradient" title="Gradient">gradient</a> of P and the second part of the equation is minus the gradient for a function of the <a href="Cartesian_coordinate_system" title="Cartesian coordinate system">Cartesian coordinates</a> x, y, z.<a href="#cite_note-2">[a]</a> In some cases, mathematicians may use a positive sign in front of the gradient to define the potential.<a href="#cite_note-3">[2]</a> Because of this definition of P in terms of the gradient, the direction of F at any point is the direction of the steepest decrease of P at that point, its magnitude is the rate of that decrease per unit length.

In order for F to be described in terms of a scalar potential only, any of the following equivalent statements have to be true:

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<annotation encoding="application/x-tex">{\displaystyle -\int _{a}^{b}\mathbf {F} \cdot d\mathbf {l} =P(\mathbf {b} )-P(\mathbf {a} ),}</annotation>

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</math><img src="./4d8d1e2f7ebac8e5ebdaf08d31d74568ccf5535a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:27.784ex; height:6.343ex;" alt="{\displaystyle -\int _{a}^{b}\mathbf {F} \cdot d\mathbf {l} =P(\mathbf {b} )-P(\mathbf {a} ),}" loading="lazy"> where the integration is over a <a href="Jordan_arc" class="mw-redirect" title="Jordan arc">Jordan arc</a> passing from location a to location b and P(b) is P evaluated at location b.</li>

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<annotation encoding="application/x-tex">{\displaystyle \oint \mathbf {F} \cdot d\mathbf {l} =0,}</annotation>

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</math><img src="./f1cd922559916c8909b9c0579ab3b2fc62bc31bb.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:12.808ex; height:5.676ex;" alt="{\displaystyle \oint \mathbf {F} \cdot d\mathbf {l} =0,}" loading="lazy"> where the integral is over any simple closed path, otherwise known as a <a href="Jordan_curve" class="mw-redirect" title="Jordan curve">Jordan curve</a>.</li>

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<annotation encoding="application/x-tex">{\displaystyle {\nabla }\times {\mathbf {F} }=0.}</annotation>

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The first of these conditions represents the <a href="Gradient_theorem" title="Gradient theorem">fundamental theorem of the gradient</a> and is true for any vector field that is a gradient of a <a href="Differentiable_function" title="Differentiable function">differentiable</a> <a href="Single-valued_function" class="mw-redirect" title="Single-valued function">single valued</a> scalar field P. The second condition is a requirement of F so that it can be expressed as the gradient of a scalar function. The third condition re-expresses the second condition in terms of the <a href="Curl_(mathematics)" title="Curl (mathematics)">curl</a> of F using the <a href="Stokes'_theorem" title="Stokes' theorem">fundamental theorem of the curl</a>. A vector field F that satisfies these conditions is said to be <a href="Irrotational_vector_field" class="mw-redirect" title="Irrotational vector field">irrotational</a> (conservative).

Scalar potentials play a prominent role in many areas of physics and engineering. The <a href="Gravity_potential" class="mw-redirect" title="Gravity potential">gravity potential</a> is the scalar potential associated with the force of gravity per unit mass, or equivalently, the <a href="Acceleration" title="Acceleration">acceleration</a> due to the field, as a function of position. The gravity potential is the <a href="Gravitational_energy" title="Gravitational energy">gravitational potential energy</a> per unit mass. In <a href="Electrostatics" title="Electrostatics">electrostatics</a> the <a href="Electric_potential" title="Electric potential">electric potential</a> is the scalar potential associated with the <a href="Electric_field" title="Electric field">electric field</a>, i.e., with the <a href="Electrostatic_force" class="mw-redirect" title="Electrostatic force">electrostatic force</a> per unit <a href="Electric_charge" title="Electric charge">charge</a>. The electric potential is in this case the electrostatic potential energy per unit charge. In <a href="Fluid_dynamics" title="Fluid dynamics">fluid dynamics</a>, irrotational <a href="Lamellar_field" class="mw-redirect" title="Lamellar field">lamellar fields</a> have a scalar potential only in the special case when it is a <a href="Laplacian_field" class="mw-redirect" title="Laplacian field">Laplacian field</a>. Certain aspects of the <a href="Nuclear_force" title="Nuclear force">nuclear force</a> can be described by a <a href="Yukawa_potential" title="Yukawa potential">Yukawa potential</a>. The potential play a prominent role in the <a href="Lagrangian_mechanics" title="Lagrangian mechanics">Lagrangian</a> and <a href="Hamiltonian_mechanics" title="Hamiltonian mechanics">Hamiltonian</a> formulations of <a href="Classical_mechanics" title="Classical mechanics">classical mechanics</a>. Further, the scalar potential is the fundamental quantity in <a href="Quantum_mechanics" title="Quantum mechanics">quantum mechanics</a>.

Not every vector field has a scalar potential. Those that do are called <a href="Conservative_vector_field" title="Conservative vector field">conservative</a>, corresponding to the notion of <a href="Conservative_force" title="Conservative force">conservative force</a> in physics. Examples of non-conservative forces include frictional forces, magnetic forces, and in fluid mechanics a <a href="Solenoidal" class="mw-redirect" title="Solenoidal">solenoidal field</a> velocity field. By the <a href="Helmholtz_decomposition" title="Helmholtz decomposition">Helmholtz decomposition</a> theorem however, all vector fields can be describable in terms of a scalar potential and corresponding <a href="Vector_potential" title="Vector potential">vector potential</a>. In electrodynamics, the electromagnetic scalar and vector potentials are known together as the <a href="Electromagnetic_four-potential" title="Electromagnetic four-potential">electromagnetic four-potential</a>.

<div class="mw-heading mw-heading2"><h2 id="Integrability_conditions">Integrability conditions</h2></div>

If F is a <a href="Conservative_vector_field" title="Conservative vector field">conservative vector field</a> (also called irrotational, <a href="Curl_(mathematics)" title="Curl (mathematics)">curl</a>-free, or potential), and its components have <a href="Continuous_function" title="Continuous function">continuous</a> <a href="Partial_derivative" title="Partial derivative">partial derivatives</a>, the potential of F with respect to a reference point r0 is defined in terms of the <a href="Line_integral" title="Line integral">line integral</a>:

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<annotation encoding="application/x-tex">{\displaystyle V(\mathbf {r} )=-\int _{C}\mathbf {F} (\mathbf {r} )\cdot \,d\mathbf {r} =-\int _{a}^{b}\mathbf {F} (\mathbf {r} (t))\cdot \mathbf {r} '(t)\,dt,}</annotation>

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where C is a <a href="Parametrization_(geometry)" title="Parametrization (geometry)">parametrized</a> path from r0 to r,

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<annotation encoding="application/x-tex">{\displaystyle \mathbf {r} (t),a\leq t\leq b,\mathbf {r} (a)=\mathbf {r_{0}} ,\mathbf {r} (b)=\mathbf {r} .}</annotation>

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The fact that the line integral depends on the path C only through its terminal points r0 and r is, in essence, the path independence property of a conservative vector field. The <a href="Gradient_theorem" title="Gradient theorem">fundamental theorem of line integrals</a> implies that if V is defined in this way, then F = –∇V, so that V is a scalar potential of the conservative vector field F. Scalar potential is not determined by the vector field alone: indeed, the gradient of a function is unaffected if a constant is added to it. If V is defined in terms of the line integral, the ambiguity of V reflects the freedom in the choice of the reference point r0.

<div class="mw-heading mw-heading2"><h2 id="Altitude_as_gravitational_potential_energy">Altitude as gravitational potential energy</h2></div>

<div role="note" class="hatnote navigation-not-searchable">Main article: <a href="Gravitational_potential" title="Gravitational potential">Gravitational potential</a></div>

An example is the (nearly) uniform <a href="Gravitational_field" title="Gravitational field">gravitational field</a> near the Earth's surface. It has a potential energy

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<annotation encoding="application/x-tex">{\displaystyle U=mgh}</annotation>

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where U is the gravitational potential energy and h is the height above the surface. This means that gravitational potential energy on a <a href="Contour_map" class="mw-redirect" title="Contour map">contour map</a> is proportional to altitude. On a contour map, the two-dimensional negative gradient of the altitude is a two-dimensional vector field, whose vectors are always perpendicular to the contours and also perpendicular to the direction of gravity. But on the hilly region represented by the contour map, the three-dimensional negative gradient of U always points straight downwards in the direction of gravity; F. However, a ball rolling down a hill cannot move directly downwards due to the <a href="Normal_force" title="Normal force">normal force</a> of the hill's surface, which cancels out the component of gravity perpendicular to the hill's surface. The component of gravity that remains to move the ball is parallel to the surface:

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<annotation encoding="application/x-tex">{\displaystyle \mathbf {F} _{\mathrm {S} }=-mg\ \sin \theta }</annotation>

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where θ is the angle of inclination, and the component of FS perpendicular to gravity is

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<annotation encoding="application/x-tex">{\displaystyle \mathbf {F} _{\mathrm {P} }=-mg\ \sin \theta \ \cos \theta =-{1 \over 2}mg\sin 2\theta .}</annotation>

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This force FP, parallel to the ground, is greatest when θ is 45 degrees.

Let Δh be the uniform interval of altitude between contours on the contour map, and let Δx be the distance between two contours. Then

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<annotation encoding="application/x-tex">{\displaystyle \theta =\tan ^{-1}{\frac {\Delta h}{\Delta x}}}</annotation>

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so that

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<annotation encoding="application/x-tex">{\displaystyle F_{P}=-mg{\Delta x\,\Delta h \over \Delta x^{2}+\Delta h^{2}}.}</annotation>

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However, on a contour map, the gradient is inversely proportional to Δx, which is not similar to force FP: altitude on a contour map is not exactly a two-dimensional potential field. The magnitudes of forces are different, but the directions of the forces are the same on a contour map as well as on the hilly region of the Earth's surface represented by the contour map.

<div class="mw-heading mw-heading2"><h2 id="Pressure_as_buoyant_potential">Pressure as buoyant potential</h2></div>

In <a href="Fluid_mechanics" title="Fluid mechanics">fluid mechanics</a>, a fluid in equilibrium, but in the presence of a uniform gravitational field is permeated by a uniform buoyant force that cancels out the gravitational force: that is how the fluid maintains its equilibrium. This <a href="Buoyancy" title="Buoyancy">buoyant force</a> is the negative gradient of <a href="Pressure" title="Pressure">pressure</a>:

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<annotation encoding="application/x-tex">{\displaystyle \mathbf {f_{B}} =-\nabla p.}</annotation>

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Since buoyant force points upwards, in the direction opposite to gravity, then pressure in the fluid increases downwards. Pressure in a static body of water increases proportionally to the depth below the surface of the water. The surfaces of constant pressure are planes parallel to the surface, which can be characterized as the plane of zero pressure.

If the liquid has a vertical <a href="Vortex" title="Vortex">vortex</a> (whose axis of rotation is perpendicular to the surface), then the vortex causes a depression in the pressure field. The surface of the liquid inside the vortex is pulled downwards as are any surfaces of equal pressure, which still remain parallel to the liquids surface. The effect is strongest inside the vortex and decreases rapidly with the distance from the vortex axis.

The buoyant force due to a fluid on a solid object immersed and surrounded by that fluid can be obtained by integrating the negative <a href="Pressure_gradient" title="Pressure gradient">pressure gradient</a> along the surface of the object:

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<annotation encoding="application/x-tex">{\displaystyle F_{B}=-\oint _{S}\nabla p\cdot \,d\mathbf {S} .}</annotation>

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<div class="mw-heading mw-heading2"><h2 id="Scalar_potential_in_Euclidean_space">Scalar potential in Euclidean space</h2></div>

In 3-dimensional Euclidean space ⁠<math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {R} ^{3}}">

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<annotation encoding="application/x-tex">{\displaystyle \mathbb {R} ^{3}}</annotation>

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</math><img src="./f936ddf584f8f3dd2a0ed08917001b7a404c10b5.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.732ex; height:2.676ex;" alt="{\displaystyle \mathbb {R} ^{3}}" loading="lazy">⁠, the scalar potential of an <a href="Irrotational_vector_field" class="mw-redirect" title="Irrotational vector field">irrotational vector field</a> E is given by

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<annotation encoding="application/x-tex">{\displaystyle \Phi (\mathbf {r} )={\frac {1}{4\pi }}\int _{\mathbb {R} ^{3}}{\frac {\operatorname {div} \mathbf {E} (\mathbf {r} ')}{\left\|\mathbf {r} -\mathbf {r} '\right\|}}\,dV(\mathbf {r} ')}</annotation>

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where dV(r') is an infinitesimal <a href="Volume_element" title="Volume element">volume element</a> with respect to r'. Then

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<annotation encoding="application/x-tex">{\displaystyle \mathbf {E} =-\mathbf {\nabla } \Phi =-{\frac {1}{4\pi }}\mathbf {\nabla } \int _{\mathbb {R} ^{3}}{\frac {\operatorname {div} \mathbf {E} (\mathbf {r} ')}{\left\|\mathbf {r} -\mathbf {r} '\right\|}}\,dV(\mathbf {r} ')}</annotation>

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This holds provided E is <a href="Continuous_function" title="Continuous function">continuous</a> and vanishes asymptotically to zero towards infinity, decaying faster than 1/r and if the <a href="Divergence" title="Divergence">divergence</a> of E likewise vanishes towards infinity, decaying faster than 1/r 2.

Written another way, let

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<annotation encoding="application/x-tex">{\displaystyle \Gamma (\mathbf {r} )={\frac {1}{4\pi }}{\frac {1}{\|\mathbf {r} \|}}}</annotation>

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be the <a href="Newtonian_potential" title="Newtonian potential">Newtonian potential</a>. This is the <a href="Fundamental_solution" title="Fundamental solution">fundamental solution</a> of the <a href="Laplace_equation" class="mw-redirect" title="Laplace equation">Laplace equation</a>, meaning that the Laplacian of Γ is equal to the negative of the <a href="Dirac_delta_function" title="Dirac delta function">Dirac delta function</a>:

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<annotation encoding="application/x-tex">{\displaystyle \nabla ^{2}\Gamma (\mathbf {r} )+\delta (\mathbf {r} )=0.}</annotation>

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Then the scalar potential is the divergence of the <a href="Convolution" title="Convolution">convolution</a> of E with Γ:

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<annotation encoding="application/x-tex">{\displaystyle \Phi =\operatorname {div} (\mathbf {E} *\Gamma ).}</annotation>

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Indeed, convolution of an irrotational vector field with a rotationally invariant potential is also irrotational. For an irrotational vector field G, it can be shown that

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<annotation encoding="application/x-tex">{\displaystyle \nabla ^{2}\mathbf {G} =\mathbf {\nabla } (\mathbf {\nabla } \cdot {}\mathbf {G} ).}</annotation>

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Hence

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<annotation encoding="application/x-tex">{\displaystyle \nabla \operatorname {div} (\mathbf {E} *\Gamma )=\nabla ^{2}(\mathbf {E} *\Gamma )=\mathbf {E} *\nabla ^{2}\Gamma =-\mathbf {E} *\delta =-\mathbf {E} }</annotation>

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as required.

More generally, the formula

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<annotation encoding="application/x-tex">{\displaystyle \Phi =\operatorname {div} (\mathbf {E} *\Gamma )}</annotation>

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holds in n-dimensional Euclidean space (n > 2) with the Newtonian potential given then by

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<annotation encoding="application/x-tex">{\displaystyle \Gamma (\mathbf {r} )={\frac {1}{n(n-2)\omega _{n}\|\mathbf {r} \|^{n-2}}}}</annotation>

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where ωn is the volume of the unit n-ball. The proof is identical. Alternatively, <a href="Integration_by_parts" title="Integration by parts">integration by parts</a> (or, more rigorously, the <a href="Convolution#Differentiation" title="Convolution">properties of convolution</a>) gives

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<annotation encoding="application/x-tex">{\displaystyle \Phi (\mathbf {r} )=-{\frac {1}{n\omega _{n}}}\int _{\mathbb {R} ^{n}}{\frac {\mathbf {E} (\mathbf {r} ')\cdot (\mathbf {r} -\mathbf {r} ')}{\|\mathbf {r} -\mathbf {r} '\|^{n}}}\,dV(\mathbf {r} ').}</annotation>

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<ul><li><a href="Gradient_theorem" title="Gradient theorem">Gradient theorem</a></li>

<li><a href="Fundamental_theorem_of_vector_analysis" class="mw-redirect" title="Fundamental theorem of vector analysis">Fundamental theorem of vector analysis</a></li>

<li><a href="Equipotential" title="Equipotential">Equipotential</a> (isopotential) lines and surfaces</li></ul>

<div class="mw-heading mw-heading2"><h2 id="Notes">Notes</h2></div>

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<li id="cite_note-2"><a href="#cite_ref-2">^</a> The second part of this equation is only valid for Cartesian coordinates, other coordinate systems such as cylindrical or spherical coordinates will have more complicated representations, derived from the <a href="Gradient_theorem" title="Gradient theorem">fundamental theorem of the gradient</a>.

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<div class="mw-heading mw-heading2"><h2 id="References">References</h2></div>

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</style><cite id="CITEREFGoldstein1980" class="citation book cs1">Goldstein, Herbert (1980). Classical Mechanics (2 ed.). Addison-Wesley Publishing Company. pp. 3–4. <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>978-0-201-02918-5</bdi>.</cite>

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<li id="cite_note-3"><a href="#cite_ref-3">^</a> See <a rel="nofollow" class="external autonumber" href="http://www.math.umn.edu/~nykamp/m2374/readings/findpot/">[1]</a> for an example where the potential is defined without a negative. Other references such as <cite id="CITEREFLouis_Leithold" class="citation cs2">Louis Leithold, The Calculus with Analytic Geometry (5 ed.), p. 1199</cite> avoid using the term potential when solving for a function from its gradient.

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<div class="mw-heading mw-heading2"><h2 id="External_links">External links</h2></div>

<ul><li> Media related to <a href="https://commons.wikimedia.org/wiki/Category:Scalar_potential" class="extiw external" title="commons:Category:Scalar potential">Scalar potential</a> at Wikimedia Commons</li></ul></div><div><div class="zim-footer">

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