In
physics
Physics is the scientific study of matter, its Elementary particle, fundamental constituents, its motion and behavior through space and time, and the related entities of energy and force. "Physical science is that department of knowledge whi ...
,
chemistry
Chemistry is the scientific study of the properties and behavior of matter. It is a physical science within the natural sciences that studies the chemical elements that make up matter and chemical compound, compounds made of atoms, molecules a ...
and
biology
Biology is the scientific study of life and living organisms. It is a broad natural science that encompasses a wide range of fields and unifying principles that explain the structure, function, growth, History of life, origin, evolution, and ...
, a potential gradient is the local
rate of change of the
potential
Potential generally refers to a currently unrealized ability. The term is used in a wide variety of fields, from physics to the social sciences to indicate things that are in a state where they are able to change in ways ranging from the simple r ...
with respect to
displacement
Displacement may refer to:
Physical sciences
Mathematics and physics
*Displacement (geometry), is the difference between the final and initial position of a point trajectory (for instance, the center of mass of a moving object). The actual path ...
, i.e. spatial
derivative
In mathematics, the derivative is a fundamental tool that quantifies the sensitivity to change of a function's output with respect to its input. The derivative of a function of a single variable at a chosen input value, when it exists, is t ...
, or
gradient
In vector calculus, the gradient of a scalar-valued differentiable function f of several variables is the vector field (or vector-valued function) \nabla f whose value at a point p gives the direction and the rate of fastest increase. The g ...
. This quantity frequently occurs in equations of physical processes because it leads to some form of
flux
Flux describes any effect that appears to pass or travel (whether it actually moves or not) through a surface or substance. Flux is a concept in applied mathematics and vector calculus which has many applications in physics. For transport phe ...
.
Definition
One dimension
The simplest definition for a potential gradient ''F'' in one dimension is the following:
:
where is some type of
scalar potential
In mathematical physics, scalar potential describes the situation where the difference in the potential energies of an object in two different positions depends only on the positions, not upon the path taken by the object in traveling from one p ...
and is
displacement
Displacement may refer to:
Physical sciences
Mathematics and physics
*Displacement (geometry), is the difference between the final and initial position of a point trajectory (for instance, the center of mass of a moving object). The actual path ...
(not
distance
Distance is a numerical or occasionally qualitative measurement of how far apart objects, points, people, or ideas are. In physics or everyday usage, distance may refer to a physical length or an estimation based on other criteria (e.g. "two co ...
) in the direction, the subscripts label two different positions , and potentials at those points, . In the limit of
infinitesimal
In mathematics, an infinitesimal number is a non-zero quantity that is closer to 0 than any non-zero real number is. The word ''infinitesimal'' comes from a 17th-century Modern Latin coinage ''infinitesimus'', which originally referred to the " ...
displacements, the ratio of differences becomes a ratio of
differentials:
:
The direction of the electric potential gradient is from
to
.
Three dimensions
In
three dimensions,
Cartesian coordinates
In geometry, a Cartesian coordinate system (, ) in a plane is a coordinate system that specifies each point uniquely by a pair of real numbers called ''coordinates'', which are the signed distances to the point from two fixed perpendicular o ...
make it clear that the resultant potential gradient is the sum of the potential gradients in each direction:
:
where are
unit vector
In mathematics, a unit vector in a normed vector space is a Vector (mathematics and physics), vector (often a vector (geometry), spatial vector) of Norm (mathematics), length 1. A unit vector is often denoted by a lowercase letter with a circumfle ...
s in the directions. This can be compactly written in terms of the
gradient
In vector calculus, the gradient of a scalar-valued differentiable function f of several variables is the vector field (or vector-valued function) \nabla f whose value at a point p gives the direction and the rate of fastest increase. The g ...
operator ,
:
although this final form holds in any
curvilinear coordinate system, not just Cartesian.
This expression represents a significant feature of any
conservative vector field
In vector calculus, a conservative vector field is a vector field that is the gradient of some function. A conservative vector field has the property that its line integral is path independent; the choice of path between two points does not chan ...
, namely has a corresponding potential .
Using
Stokes' theorem
Stokes' theorem, also known as the Kelvin–Stokes theorem after Lord Kelvin and George Stokes, the fundamental theorem for curls, or simply the curl theorem, is a theorem in vector calculus on \R^3. Given a vector field, the theorem relates th ...
, this is equivalently stated as
:
meaning the
curl
cURL (pronounced like "curl", ) is a free and open source computer program for transferring data to and from Internet servers. It can download a URL from a web server over HTTP, and supports a variety of other network protocols, URI scheme ...
, denoted ∇×, of the vector field vanishes.
Physics
Newtonian gravitation
In the case of the
gravitational field
In physics, a gravitational field or gravitational acceleration field is a vector field used to explain the influences that a body extends into the space around itself. A gravitational field is used to explain gravitational phenomena, such as ...
, which can be shown to be conservative, it is equal to the gradient in
gravitational potential
In classical mechanics, the gravitational potential is a scalar potential associating with each point in space the work (energy transferred) per unit mass that would be needed to move an object to that point from a fixed reference point in the ...
:
:
There are opposite signs between gravitational field and potential, because the potential gradient and field are opposite in direction: as the potential increases, the gravitational field strength decreases and vice versa.
Electromagnetism
In
electrostatics
Electrostatics is a branch of physics that studies slow-moving or stationary electric charges.
Since classical antiquity, classical times, it has been known that some materials, such as amber, attract lightweight particles after triboelectric e ...
, the
electric field
An electric field (sometimes called E-field) is a field (physics), physical field that surrounds electrically charged particles such as electrons. In classical electromagnetism, the electric field of a single charge (or group of charges) descri ...
is independent of time , so there is no induction of a time-dependent
magnetic field
A magnetic field (sometimes called B-field) is a physical field that describes the magnetic influence on moving electric charges, electric currents, and magnetic materials. A moving charge in a magnetic field experiences a force perpendicular ...
by
Faraday's law of induction:
:
which implies is the gradient of the electric potential , identical to the classical gravitational field:
:
In
electrodynamics
In physics, electromagnetism is an interaction that occurs between particles with electric charge via electromagnetic fields. The electromagnetic force is one of the four fundamental forces of nature. It is the dominant force in the interacti ...
, the field is time dependent and induces a time-dependent field also (again by Faraday's law), so the curl of is not zero like before, which implies the electric field is no longer the gradient of electric potential. A time-dependent term must be added:
:
where is the electromagnetic
vector potential. This last potential expression in fact reduces Faraday's law to an identity.
Fluid mechanics
In
fluid mechanics
Fluid mechanics is the branch of physics concerned with the mechanics of fluids (liquids, gases, and plasma (physics), plasmas) and the forces on them.
Originally applied to water (hydromechanics), it found applications in a wide range of discipl ...
, the
velocity field describes the fluid motion. An
irrotational flow means the velocity field is conservative, or equivalently the
vorticity
In continuum mechanics, vorticity is a pseudovector (or axial vector) field that describes the local spinning motion of a continuum near some point (the tendency of something to rotate), as would be seen by an observer located at that point an ...
pseudovector
In physics and mathematics, a pseudovector (or axial vector) is a quantity that transforms like a vector under continuous rigid transformations such as rotations or translations, but which does ''not'' transform like a vector under certain ' ...
field is zero:
:
This allows the
velocity potential
A velocity potential is a scalar potential used in potential flow theory. It was introduced by Joseph-Louis Lagrange in 1788.
It is used in continuum mechanics, when a continuum occupies a simply-connected region and is irrotational. In such a ca ...
to be defined simply as:
:
Chemistry
In an
electrochemical
Electrochemistry is the branch of physical chemistry concerned with the relationship between electrical potential difference and identifiable chemical change. These reactions involve electrons moving via an electronically conducting phase (typi ...
half-cell, at the interface between the
electrolyte
An electrolyte is a substance that conducts electricity through the movement of ions, but not through the movement of electrons. This includes most soluble Salt (chemistry), salts, acids, and Base (chemistry), bases, dissolved in a polar solven ...
(an
ionic
solution) and the
metal
A metal () is a material that, when polished or fractured, shows a lustrous appearance, and conducts electrical resistivity and conductivity, electricity and thermal conductivity, heat relatively well. These properties are all associated wit ...
electrode
An electrode is an electrical conductor used to make contact with a nonmetallic part of a circuit (e.g. a semiconductor, an electrolyte, a vacuum or a gas). In electrochemical cells, electrodes are essential parts that can consist of a varie ...
, the standard
electric potential difference
Voltage, also known as (electrical) potential difference, electric pressure, or electric tension, is the difference in electric potential between two points. In a Electrostatics, static electric field, it corresponds to the Work (electrical), ...
is:
[Physical chemistry, P.W. Atkins, Oxford University Press, 1978, ]
:
where ''R'' =
gas constant
The molar gas constant (also known as the gas constant, universal gas constant, or ideal gas constant) is denoted by the symbol or . It is the molar equivalent to the Boltzmann constant, expressed in units of energy per temperature increment p ...
, ''T'' =
temperature
Temperature is a physical quantity that quantitatively expresses the attribute of hotness or coldness. Temperature is measurement, measured with a thermometer. It reflects the average kinetic energy of the vibrating and colliding atoms making ...
of solution, ''z'' =
valency of the metal, ''e'' =
elementary charge
The elementary charge, usually denoted by , is a fundamental physical constant, defined as the electric charge carried by a single proton (+1 ''e'') or, equivalently, the magnitude of the negative electric charge carried by a single electron, ...
, ''N''
A =
Avogadro constant
The Avogadro constant, commonly denoted or , is an SI defining constant with an exact value of when expressed in reciprocal moles.
It defines the ratio of the number of constituent particles to the amount of substance in a sample, where th ...
, and ''a''
M+z is the
activity of the ions in solution. Quantities with superscript ⊖ denote the measurement is taken under
standard conditions. The potential gradient is relatively abrupt, since there is an almost definite boundary between the metal and solution, hence the interface term.
Biology
In
biology
Biology is the scientific study of life and living organisms. It is a broad natural science that encompasses a wide range of fields and unifying principles that explain the structure, function, growth, History of life, origin, evolution, and ...
, a potential gradient is the net difference in
electric charge
Electric charge (symbol ''q'', sometimes ''Q'') is a physical property of matter that causes it to experience a force when placed in an electromagnetic field. Electric charge can be ''positive'' or ''negative''. Like charges repel each other and ...
across a
cell membrane
The cell membrane (also known as the plasma membrane or cytoplasmic membrane, and historically referred to as the plasmalemma) is a biological membrane that separates and protects the interior of a cell from the outside environment (the extr ...
.
Non-uniqueness of potentials
Since gradients in potentials correspond to
physical field
In science, a field is a physical quantity, represented by a scalar, vector, or tensor, that has a value for each point in space and time. An example of a scalar field is a weather map, with the surface temperature described by assigning a nu ...
s, it makes no difference if a constant is added on (it is erased by the gradient operator which includes
partial differentiation
In mathematics, a partial derivative of a function of several variables is its derivative with respect to one of those variables, with the others held constant (as opposed to the total derivative, in which all variables are allowed to vary). Par ...
). This means there is no way to tell what the "absolute value" of the potential "is" – the zero value of potential is completely arbitrary and can be chosen anywhere by convenience (even "at infinity"). This idea also applies to vector potentials, and is exploited in
classical field theory
A classical field theory is a physical theory that predicts how one or more fields in physics interact with matter through field equations, without considering effects of quantization; theories that incorporate quantum mechanics are called qua ...
and also
gauge field theory.
Absolute values of potentials are not physically observable, only gradients and path-dependent potential differences are. However, the
Aharonov–Bohm effect
The Aharonov–Bohm effect, sometimes called the Ehrenberg–Siday–Aharonov–Bohm effect, is a quantum mechanics, quantum-mechanical phenomenon in which an electric charge, electrically charged point particle, particle is affected by an elect ...
is a
quantum mechanical
Quantum mechanics is the fundamental physical theory that describes the behavior of matter and of light; its unusual characteristics typically occur at and below the scale of atoms. Reprinted, Addison-Wesley, 1989, It is the foundation of a ...
effect which illustrates that non-zero
electromagnetic potentials along a closed loop (even when the and fields are zero everywhere in the region) lead to changes in the phase of the
wave function
In quantum physics, a wave function (or wavefunction) is a mathematical description of the quantum state of an isolated quantum system. The most common symbols for a wave function are the Greek letters and (lower-case and capital psi (letter) ...
of an electrically
charged particle
In physics, a charged particle is a particle with an electric charge. For example, some elementary particles, like the electron or quarks are charged. Some composite particles like protons are charged particles. An ion, such as a molecule or atom ...
in the region, so the potentials appear to have measurable significance.
Potential theory
Field equations, such as Gauss's laws
for electricity,
for magnetism, and
for gravity, can be written in the form:
:
where is the electric
charge density
In electromagnetism, charge density is the amount of electric charge per unit length, surface area, or volume. Volume charge density (symbolized by the Greek letter ρ) is the quantity of charge per unit volume, measured in the SI system in co ...
,
monopole density (should they exist), or
mass density
Density (volumetric mass density or specific mass) is the ratio of a substance's mass to its volume. The symbol most often used for density is ''ρ'' (the lower case Greek language, Greek letter rho), although the Latin letter ''D'' (or ''d'') ...
and is a constant (in terms of
physical constant
A physical constant, sometimes fundamental physical constant or universal constant, is a physical quantity that cannot be explained by a theory and therefore must be measured experimentally. It is distinct from a mathematical constant, which has a ...
s , , and other numerical factors).
Scalar potential gradients lead to
Poisson's equation
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 t ...
:
:
A general
theory of potentials has been developed to solve this equation for the potential. The gradient of that solution gives the physical field, solving the field equation.
See also
*
Tensors in curvilinear coordinates
References
{{reflist
Concepts in physics
Spatial gradient