Electric potential

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https://en.wikipedia.org/wiki/Electric_potential

 

Electric potential
Electric potential around two oppositely charged conducting spheres. Purple represents the highest potential, yellow zero, and cyan the lowest potential. The electric field lines are shown leaving perpendicularly to the surface of each sphere.
Common symbols	V, φ
SI unit	volt
Other units	statvolt
In SI base units	V = kg⋅m2⋅s−3⋅A−1
Extensive?	yes
Dimension	M L2 T−3 I−1

Electromagnetism
Electricity Magnetism History Textbooks

Electrostatics Electric charge Coulomb's law Conductor Charge density Permittivity Electric dipole moment Electric field Electric potential Electric flux / potential energy Electrostatic discharge Gauss's law Induction Insulator Polarization density Static electricity Triboelectricity

Magnetostatics

Electrodynamics

Electrical network

Covariant formulation

The electric potential (also called the electric field potential, potential drop, the electrostatic potential) is defined as the amount of work energy needed to move a unit of electric charge from a reference point to the specific point in an electric field. More precisely, it is the energy per unit charge for a test charge that is so small that the disturbance of the field under consideration is negligible. Furthermore, the motion across the field is supposed to proceed with negligible acceleration, so as to avoid the test charge acquiring kinetic energy or producing radiation. By definition, the electric potential at the reference point is zero units. Typically, the reference point is earth or a point at infinity, although any point can be used.

In classical electrostatics, the electrostatic field is a vector quantity that is expressed as the gradient of the electrostatic potential, which is a scalar quantity denoted by V or occasionally φ, equal to the electric potential energy of any charged particle at any location (measured in joules) divided by the charge of that particle (measured in coulombs). By dividing out the charge on the particle a quotient is obtained that is a property of the electric field itself. In short, an electric potential is the electric potential energy per unit charge.

This value can be calculated in either a static (time-invariant) or a dynamic (varying with time) electric field at a specific time in units of joules per coulomb (J⋅C⁻¹), or volts (V). The electric potential at infinity is assumed to be zero.

In electrodynamics, when time-varying fields are present, the electric field cannot be expressed only in terms of a scalar potential. Instead, the electric field can be expressed in terms of both the scalar electric potential and the magnetic vector 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.

Practically, the electric potential is always a continuous function in space. Otherwise, the spatial derivative of it will yield a field with infinite magnitude, which is practically impossible. Even an idealized point charge has 1 ⁄ r potential, which is continuous everywhere except the origin. The electric field is not continuous across an idealized surface charge, but it is not infinite at any point. Therefore, the electric potential is continuous across an idealized surface charge. An idealized linear charge has ln(r) potential, which is continuous everywhere except on the linear charge.

Introduction

Classical mechanics explores concepts such as force, energy, and potential. Force and potential energy are directly related. A net force acting on any object will cause it to accelerate. As an object moves in the direction of the force that is acting, 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 it rolls downhill, its potential energy decreases and is being translated to motion - kinetic energy.

It is possible to define the potential of certain force fields so that the potential energy of an object in that field depends only on the position of the object with respect to the field. Two such force fields are the gravitational field and an electric field (in the absence of time-varying magnetic fields). Such fields must affect objects due to the intrinsic properties of the object (e.g., mass or charge) and the position of the object.

Objects may possess a property known as electric charge. Since an electric field exerts a force on charged objects, if the charged object has a positive charge, the force will be in the direction of the electric field vector at that point; if the charge is negative, the force will be in the opposite direction.

The magnitude of the force is given by the quantity of the charge multiplied by the magnitude of the electric field vector:

${\displaystyle F=qE}$.

Electrostatics

Main article: Electrostatics

 

Electric potential of separate positive and negative point charges shown as color range from magenta (+), through yellow (0), to cyan (−). Circular contours are equipotential lines. Electric field lines leave the positive charge and enter the negative charge.

 

Electric potential in the vicinity of two opposite point charges.

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

${\displaystyle V_{\mathbf {E} }=-\int _{\mathcal {C}}\mathbf {E} \cdot \mathrm {d} {\boldsymbol {\ell }}\,}$

where C is an arbitrary path from some fixed reference point to ${\textstyle \mathbf {r} }$. In electrostatics, the Maxwell-Faraday equation reveals that the curl ${\textstyle \nabla \times \mathbf {E} }$ is zero, making the electric field conservative. Thus, the line integral above does not depend on the specific path C chosen but only on its endpoints, making ${\textstyle V_{\mathbf {E} }}$ well-defined everywhere. The gradient theorem then allows us to write:

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

This states that the electric field points "downhill" towards lower voltages. By Gauss's law, the potential can also be found to satisfy Poisson's equation:

${\displaystyle \mathbf {\nabla } \cdot \mathbf {E} =\mathbf {\nabla } \cdot \left(-\mathbf {\nabla } V_{\mathbf {E} }\right)=-\nabla ^{2}V_{\mathbf {E} }=\rho /\varepsilon _{0}}$

where ρ is the total charge density and ∇· denotes the divergence.

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

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

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 ${\textstyle \nabla \times \mathbf {E} \neq \mathbf {0} }$, i.e., in the case of a non-conservative electric field (caused by a changing magnetic field; see Maxwell's equations). The generalization of electric potential to this case is described in the section § Generalization to electrodynamics.

Electric potential due to a point charge

See also: Coulomb's law

 

The electric potential created by a charge Q is V = Q/(4πε₀r). Different values of Q will make different values of electric potential V (shown in the image).

The electric potential arising from a point charge Q, at a distance r from the charge is observed to be

$${\displaystyle V_{\mathbf {E} }={\frac {1}{4\pi \varepsilon _{0}}}{\frac {Q}{r}},}$$

where ε₀ is the permittivity of vacuum. V_E is known as the Coulomb potential.

The electric potential for a system of point charges is equal to the sum of the point charges' individual potentials. This fact simplifies calculations significantly, because addition of potential (scalar) fields is much easier than addition of the electric (vector) fields. Specifically, the potential of a set of discrete point charges q_i at points r_i becomes

$${\displaystyle V_{\mathbf {E} }(\mathbf {r} )={\frac {1}{4\pi \varepsilon _{0}}}\sum _{i}{\frac {q_{i}}{|\mathbf {r} -\mathbf {r} _{i}|}},}$$

where

• ${\displaystyle \mathbf {r} }$ is a point at which the potential is evaluated.
• ${\displaystyle \mathbf {r} _{i}}$ is a point at which there is a nonzero charge.
• ${\displaystyle q_{i}}$ is the charge at the point ${\displaystyle \mathbf {r} _{i}}$.

and the potential of a continuous charge distribution ρ(r) becomes

$${\displaystyle V_{\mathbf {E} }(\mathbf {r} )={\frac {1}{4\pi \varepsilon _{0}}}\int _{R}{\frac {\rho (\mathbf {r} ')}{|\mathbf {r} -\mathbf {r} '|}}d^{3}r'.}$$

Where

• ${\displaystyle \mathbf {r} }$ is a point at which the potential is evaluated.
• ${\displaystyle R}$ is a region containing all the points at which the charge density is nonzero.
• ${\displaystyle \mathbf {r} '}$ is a point inside ${\displaystyle R}$.
• ${\displaystyle \rho (\mathbf {r} ')}$ is the charge density at the point ${\displaystyle \mathbf {r} '}$.

The equations 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: ${\displaystyle \textstyle \int _{C}\mathbf {E} \cdot \mathrm {d} {\boldsymbol {\ell }}}$ is path-dependent because ${\displaystyle \mathbf {\nabla } \times \mathbf {E} \neq \mathbf {0} }$ (due to the Maxwell-Faraday equation).

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

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

where B is the magnetic field. By the fundamental theorem of vector calculus, such an A can always be found, since the divergence of the magnetic field is always zero due to the absence of magnetic monopoles. Now, the quantity

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

is a conservative field, since the curl of ${\displaystyle \mathbf {E} }$ is canceled by the curl of ${\displaystyle {\frac {\partial \mathbf {A} }{\partial t}}}$ according to the Maxwell-Faraday equation. One can therefore write

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

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,

${\displaystyle -\int _{a}^{b}\mathbf {E} \cdot \mathrm {d} {\boldsymbol {\ell }}\neq V_{(b)}-V_{(a)}}$

unlike electrostatics.

Gauge freedom

Main article: Gauge fixing

The electrostatic potential could have any constant added to it without affecting the electric field. In electrodynamics, the electric potential has infinitely many degrees of freedom. For any (possibly time-varying or space-varying) scalar field ${\displaystyle \psi }$, we can perform the following gauge transformation to find a new set of potentials that produce exactly the same electric and magnetic fields:

${\displaystyle V^{\prime }=V-{\frac {\partial \psi }{\partial t}}}$

${\displaystyle \mathbf {A} ^{\prime }=\mathbf {A} +\nabla \psi }$

Given different choices of gauge, the electric potential could have quite different properties. In the Coulomb gauge, the electric potential is given by Poisson's equation

${\displaystyle \nabla ^{2}V=-{\frac {\rho }{\varepsilon _{0}}}}$

just like in electrostatics. However, in the Lorenz gauge, the electric potential is a retarded potential that propagates at the speed of light, and is the solution to an inhomogeneous wave equation:

${\displaystyle \nabla ^{2}V-{\frac {1}{c^{2}}}{\frac {\partial ^{2}V}{\partial t^{2}}}=-{\frac {\rho }{\varepsilon _{0}}}}$

Units

The SI derived unit of electric potential is the volt (in honor of Alessandro Volta), which is why a difference in electric potential between two points is known as voltage. Older units are rarely used today. Variants of the centimetre–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

Main articles: Galvani potential, Electrochemical potential, and Fermi level

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 the potential difference corrected for the different atomic environments. The quantity measured by a voltmeter is called electrochemical potential or fermi level, while the pure unadjusted electric potential V is sometimes called Galvani potential ${\displaystyle \phi }$. The terms "voltage" and "electric potential" are a bit ambiguous, however in practice, they can refer to either of these in different contexts.