electric potential gradient

Electric potential

At a point in space, the electric potential is the potential energy per unit of charge that is associated with a static (time-invariant) electric field. It is typically measured in volts, and is a Lorentz scalar quantity. The difference in electrical potential between two points is known as voltage.

There is also a generalized electric scalar potential that is used in electrodynamics when time-varying electromagnetic fields are present. This generalized electric potential cannot be simply interpreted as potential energy per unit charge, however.

Explanation

Electric potential may be conceived of as "electric pressure". Where this "pressure" is uniform, no current flows and nothing happens. This is similar to why people do not feel normal atmospheric air pressure: there is no difference between the pressure inside the body and outside, so nothing is felt. However, where this electrical pressure varies, an electric field exists, which will create a force on charged particles.

Mathematically, it is the potential φ (a scalar field) associated with the conservative electric field mathbf{E} (mathbf{E}=-mathbf{nabla}varphi) that occurs when the magnetic field is time invariant (so that mathbf{nabla} times mathbf{E}=0 from Faraday's law of induction).

Like any potential function, only the potential difference (voltage) between two points is physically meaningful (neglecting quantum Aharonov-Bohm effects), since any constant can be added to φ without affecting mathbf{E} (gauge invariance).

The electric potential φ is therefore measured in units of energy per unit of electric charge. In SI units, this is:

joule/coulomb = volt.

The electric potential can also be generalized to handle situations with time-varying potential fields, in which case the electric field is not conservative and a potential function cannot be defined everywhere in space. There, an effective potential drop is included, associated with the inductance of the circuit. This generalized potential difference is also called the electromotive force (emf).

Introduction

Objects may possess a property known as electric charge. An electric field exerts a force on charged objects, accelerating them in the direction of the force, in either the same or the opposite direction of the electric field. If the charged object has a positive charge, the force and acceleration will be in the direction of the field. This force has the same direction as the electric field vector, and its magnitude is given by the size of the charge multiplied with the magnitude of the electric field.

Classical mechanics explores the concepts such as force, energy, potential etc. in more detail.

Force and potential energy are directly related. As an object moves in the direction that the force accelerates it, its potential energy decreases. For example, the gravitational potential energy of a cannonball at the top of a hill is greater than at the base of the hill. As the object falls, that potential energy decreases and is translated to motion, or inertial (kinetic) energy.

For certain forces, it is possible to define the "potential" of a field such that the potential energy of an object due to a field is dependent only on the position of the object with respect to the field. Those forces must affect objects depending only on the intrinsic properties of the object and the position of the object, and obey certain other mathematical rules.

Two such forces are the gravitational force (gravity) and the electric force in the absence of time-varying magnetic fields. The potential of an electric field is called the electric potential.

The electric potential and the magnetic vector potential together form a four vector, so that the two kinds of potential are mixed under Lorentz transformations.

Mathematical introduction

The concept of electric potential (denoted by: phi, phi_mathrm{E} or V) is closely linked with potential energy, thus:

U_ mathrm{E} = qphi

where U_mathrm{E} is the electric potential energy of a test charge q due to the electric field. Note that the potential energy and hence also the electric potential is only defined up to an additive constant: one must arbitrarily choose a position where the potential energy and the electric potential is zero.

The proper definition of the electric potential uses the electric field mathbf{E}:

phi_ mathrm{E} = - int_C mathbf{E} cdot mathrm{d} mathbf{ell}

where C is an arbitrary path connecting the point with zero potential to the point under consideration. When mathbf{nabla} times mathbf{E} = 0, the line integral above does not depend on the specific path C chosen but only on its endpoints. Equivalently, the electric potential determines the electric field via its gradient:

mathbf{E} = - mathbf{nabla} phi_mathrm{E}

and therefore, by Gauss's law, the potential satisfies Poisson's equation:

mathbf{nabla} cdot mathbf{E} = mathbf{nabla} cdot left (- mathbf{nabla} phi_mathrm{E} right ) = -nabla^2 phi_mathrm{E} = rho / varepsilon_0

where ρ is the total charge density (including bound charge).

Note: these equations cannot be used if mathbf{nabla}timesmathbf{E} ne 0, i.e., in the case of a nonconservative electric field (caused by a changing magnetic field; see Maxwell's equations). The generalization of electric potential to this case is described below.

Generalization to electrodynamics

When time-varying magnetic fields are present (which is true whenever there are time-varying electric fields and vice versa), one cannot describe the electric field simply in terms of a scalar potential phi; because the electric field is no longer conservative: int mathbf{E}cdot mathrm{d}mathbf{S} is path-dependent because mathbf{nabla} times mathbf{E}neq 0.

Instead, one can still define a scalar potential by also including the magnetic vector potential mathbf{A}. In particular, mathbf{A} is defined by:

mathbf{B} = mathbf{nabla} times mathbf{A}

where mathbf{B} is the magnetic flux density. One can always find such an mathbf{A} because mathbf{nabla} cdot mathbf{B} = 0 (the absence of magnetic monopoles). Given this, the quantity mathbf{F} = mathbf{E} + partialmathbf{A}/partial t is a conservative field by Faraday's law and one can therefore write:

mathbf{E} = -mathbf{nabla}phi - frac{partialmathbf{A}}{partial t}

where φ is the scalar potential defined by the conservative field mathbf{F}.

The electrostatic potential is simply the special case of this definition where mathbf{A} is time-invariant. On the other hand, for time-varying fields, note that int_a^b mathbf{E} cdot mathrm{d}mathbf{S} neq phi(b) - phi(a), unlike electrostatics.

Note that this definition of φ depends on the gauge choice for the vector potential mathbf{A} (the gradient of any scalar field can be added to mathbf{A} without changing mathbf{B}). One choice is the Coulomb gauge, in which we choose mathbf{nabla} cdot mathbf{A} = 0. In this case, we obtain -nabla^2 phi = rho/varepsilon_0, where ρ is the charge density, just as for electrostatics. Another common choice is the Lorenz gauge, in which we choose mathbf{A} to satisfy mathbf{nabla} cdot mathbf{A} = - frac{1}{c^2} frac{partialphi}{partial t}.

Special cases and computational devices

The electric potential at a point mathbf{l} due to a constant electric field mathbf{E} can be shown to be:

phi_mathrm{E} = - int mathbf{E} cdot mathrm{d}mathbf{l}.

The electric potential created by a point charge q, at a distance r from the charge, can be shown to be, in SI units:

phi_mathbf{E} = frac{q} {4 pi epsilon_o r}.

The electric potential due to a system of point charges is equal to the sum of the point charges' individual potentials. This fact simplifies calculations significantly, since addition of potential (scalar) fields is much easier than addition of the electric (vector) fields.

The electric potential created by a tridimensional spherically symmetric gaussian charge density rho(r) given by:

rho(r) = frac{q}{sigma^3sqrt{2pi}^3},e^{-frac{r^2}{2sigma^2}},

where q is the total charge, is obtained by solving the Poisson's equation (in cgs units):

nabla^2 phi_mathbf{E} = - 4 pi rho.

The solution is given by:

phi_mathbf{E}(r) = frac{q}{r},mbox{erf}left(frac{r}{sqrt{2}sigma}right)

where erf(x) is the error function. This solution can be checked explicitly by a careful manual evaluation of nabla^2 phi_mathbf{E}. Note that, for r much greater than σ, erf(x) approaches unity and the potential phi_mathbf{E} approaches the point charge potential frac{q}{r} seen above, as expected.

Applications in electronics

This electric potential, typically measured in volts, provides a simple way to analyze electric circuits without requiring detailed knowledge of the circuit shape or the fields within it.

The electric potential provides a simple way to analyze electrical networks with the help of Kirchhoff's voltage law, without solving the detailed Maxwell's equations for the fields of the circuit.

Units

The SI unit of electric potential is the volt (in honour of Alessandro Volta), which is so widely used that the terms voltage and electric potential are almost synonymous. Older units are rarely used nowadays. Variants of the centimeter gram second system of units included a number of different units for electric potential, including the abvolt and the statvolt.

See also

References

  • Griffiths, David J. (1999). Introduction to Electrodynamics (3rd ed.). Prentice Hall. ISBN 0-13-805326-X.
  • Jackson, John David (1999). Classical Electrodynamics, 3rd ed., New York: Wiley. ISBN 0-471-30932-X.
  • Electromagnetic Fields (2nd Edition), Roald K. Wangsness, Wiley, 1986. ISBN 0-471-81186-6.

External links

Introduction to Electric Potential A conceptual introduction

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