Electric potential
Electric potential is the difference in electric potential energy per unit of electric charge between two points in a static electric field. More precisely, electric potential is the amount of work needed to move a test charge from a reference point to a specific point in a static electric field, normalized to a unit of charge. The test charge used is small enough that disturbance to the field-producing charges is unnoticeable, and its 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 expressed as the gradient of the electrostatic potential, which is a scalar quantity denoted by or occasionally, equal to the electric potential energy of any charged particle at any location divided by the charge of that particle. 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 or a dynamic electric field at a specific time with the unit joules per coulomb or volt. 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 as a scalar potential. Instead, the electric field can be expressed as 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 a continuous function in all space, because a spatial derivative of a discontinuous electric potential yields an electric field of impossibly infinite magnitude. Notably, the electric potential due to an idealized point charge is continuous in all space except at the location of the point charge. Though electric field is not continuous across an idealized surface charge, it is not infinite at any point. Therefore, the electric potential is continuous across an idealized surface charge. Additionally, an idealized line of charge has electric potential is continuous everywhere except on the line of charge.
Introduction
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 a force acting on 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 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 a gravitational field and an electric field. Such fields affect objects because of the intrinsic properties and positions of the objects.
An object may possess a property known as electric charge. Since an electric field exerts force on a charged object, if the object has a positive charge, the force will be in the direction of the electric field vector at the location of the charge; if the charge is negative, the force will be in the opposite direction.
The magnitude of force is given by the quantity of the charge multiplied by the magnitude of the electric field vector,
Electrostatics
An electric potential at a point in a static electric field is given by the line integralwhere is an arbitrary path from some fixed reference point to ; it is uniquely determined up to a constant that is added or subtracted from the integral. In electrostatics, the Maxwell-Faraday equation reveals that the curl is zero, making the electric field conservative. Thus, the line integral above does not depend on the specific path chosen but only on its endpoints, making well-defined everywhere. The gradient theorem then allows us to write:
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:
where is the total charge density and denotes the divergence.
The concept of electric potential is closely linked with potential energy. A test charge,, has an electric potential energy,, given by
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 i.e., in the case of a non-conservative electric field. The generalization of electric potential to this case is described in the section.
Electric potential due to a point charge
The electric potential arising from a point charge,, at a distance,, from the location of is observed to bewhere is the permittivity of vacuum, is known as the Coulomb potential. Note that, in contrast to the magnitude of an electric field due to a point charge, the electric potential scales respective to the reciprocal of the radius, rather than the radius squared.
The electric potential at any location,, in a system of point charges is equal to the sum of the individual electric potentials due to every point charge in the system. This fact simplifies calculations significantly, because addition of potential fields is much easier than addition of the electric fields. Specifically, the potential of a set of discrete point charges at points becomes
where
- is a point at which the potential is evaluated;
- is a point at which there is a nonzero charge; and
- is the charge at the point.
where
- is a point at which the potential is evaluated;
- is a region containing all the points at which the charge density is nonzero;
- is a point inside ; and
- is the charge density at the point.
Generalization to electrodynamics
When time-varying magnetic fields are present, it is not possible to describe the electric field simply as a scalar potential because the electric field is no longer conservative: is path-dependent because .Instead, one can still define a scalar potential by also including the magnetic vector potential. In particular, is defined to satisfy:
where is the magnetic field. By the fundamental theorem of vector calculus, such an can always be found, since the divergence of the magnetic field is always zero due to the absence of magnetic monopoles. Now, the quantity
is a conservative field, since the curl of is canceled by the curl of according to the Maxwell–Faraday equation. One can therefore write
where is the scalar potential defined by the conservative field.
The electrostatic potential is simply the special case of this definition where is time-invariant. On the other hand, for time-varying fields,
unlike electrostatics.
Gauge freedom
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 scalar field,, we can perform the following gauge transformation to find a new set of potentials that produce exactly the same electric and magnetic fields: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
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: