Electric potential

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Electric potential

In classical electromagnetism, the electric potential (a scalar quantity denoted by φ, φE or V and also called the electric field potential or the electrostatic potential) at a point within a defined space is equal to the electric potential energy (measured in joules) at that location divided by the charge there (measured in coulombs). The electric potential at a specific location in the electric field is independent of qt. That is to say, it is a characteristic only of the electric field that is present. The electric potential can be calculated at a point in either a static (time-invariant) electric field or in a dynamic (varying with time) electric field at a specific time, and has the units of joules per coulomb, or volts.

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 the ratio of potential energy to charge, however.

Introduction

Objects may possess a property known as an 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 synonymous term "electrostatic potential" is also in common use.

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.

In electrostatics

The electric potential at a point r in a static electric field E is given by the line integral

$\Delta V_\mathbf{E} = - \int_C \mathbf{E} \cdot \mathrm{d} \boldsymbol{\ell} \, ,$

where C is an arbitrary path connecting the point with zero potential to r. When the curl × E is zero, the line integral above does not depend on the specific path C chosen but only on its endpoints. In this case, the electric field is conservative and determined by the gradient of the potential:

$\mathbf{E} = - \mathbf{\nabla} V_\mathbf{E}. \,$

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

$\mathbf{\nabla} \cdot \mathbf{E} = \mathbf{\nabla} \cdot \left (- \mathbf{\nabla} V_\mathbf{E} \right ) = -\nabla^2 V_\mathbf{E} = \rho / \varepsilon_0, \,$

where ρ is the total charge density (including bound charge) and · denotes the divergence.

The concept of electric potential is closely linked with potential energy. A test charge q has an electric potential energy UE given by

$U_ \mathbf{E} = q\,V. \,$

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 are zero.

These equations cannot be used if the curl × E ≠ 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.

Electric potential due to a point charge

The electric potential created by a point charge Q, at a distance r from the charge (relative to the potential at infinity), can be shown to be

$V_\mathbf{E} = \frac{1}{4 \pi \varepsilon_0} \frac{Q}{r}, \,$

where ε0 is the electric constant (permittivity of free space). This is known as the Coulomb Potential.

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 equation given above for the electric potential (and all the equations used here) are in the forms required by SI units. In some other (less common) systems of units, such as CGS-Gaussian, many of these equations would be altered.

Generalization to electrodynamics

When time-varying magnetic fields are present (which is true whenever there are time-varying electric fields and vice versa), it is not possible to describe the electric field simply in terms of a scalar potential V because the electric field is no longer conservative: $\textstyle\int_C \mathbf{E}\cdot \mathrm{d}\boldsymbol{\ell}$ is path-dependent because × E ≠ 0 (Faraday's law of induction).

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

$\mathbf{B} = \mathbf{\nabla} \times \mathbf{A}, \,$

where B is the magnetic field. Because the divergence of the magnetic field is always zero due to the absence of magnetic monopoles, such an A can always be found. Given this, the quantity

$\mathbf{F} = \mathbf{E} + \frac{\partial\mathbf{A}}{\partial t}$

is a conservative field by Faraday's law and one can therefore write

$\mathbf{E} = -\mathbf{\nabla}V - \frac{\partial\mathbf{A}}{\partial t}, \,$

where V is the scalar potential defined by the conservative field F.

The electrostatic potential is simply the special case of this definition where A is time-invariant. On the other hand, for time-varying fields, note that

$\int_a^b \mathbf{E} \cdot \mathrm{d}\boldsymbol{\ell} \neq V_{(b)} - V_{(a)}, \,$

unlike electrostatics.

Note that this definition of V depends on the gauge choice for the vector potential A (the gradient of any scalar field can be added to A without changing B). One choice is the Coulomb gauge, in which we choose · A = 0. In this case, we obtain

$-\nabla^2 V = \rho/\varepsilon_0, \,$

where ρ is the charge density, just as for electrostatics. Another common choice is the Lorenz gauge, in which we choose A to satisfy

$\mathbf{\nabla} \cdot \mathbf{A} = - \frac{1}{c^2} \frac{\partial V}{\partial t}. \,$

Units

The SI unit of electric potential is the volt (in honor of Alessandro Volta), which is why electric potential is also known as voltage. 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.

Galvani potential versus electrochemical potential

Inside metals (and other solids and liquids), the energy of an electron is affected not only by the electric potential, but also by the specific atomic environment that it is in. When a voltmeter is connected between two different types of metal, it measures not the electric potential difference, but instead the potential difference corrected for the different atomic environments.[1] The quantity measured by a voltmeter is called electrochemical potential or fermi level, while the pure unadjusted electric potential is sometimes called Galvani potential. The terms "voltage" and "electric potential" are a bit ambiguous in that, in practice, they can refer to either of these in different contexts.

References

1. ^ Bagotskii, Vladimir Sergeevich (2006). Fundamentals of electrochemistry. p. 22. ISBN 9780471700586.
• Griffiths, David J. (1998). Introduction to Electrodynamics (3rd. ed.). Prentice Hall. ISBN 0-13-805326-X.
• Jackson, John David (1999). Classical Electrodynamics (3rd. ed.). USA: John Wiley & Sons, Inc.. ISBN 978-0-471-30932-1.
• Wangsness, Roald K. (1986). Electromagnetic Fields (2nd., Revised, illustrated ed.). Wiley. ISBN 9780471811862.

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