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In physics, the electrical potential is a scalar field: it has a magnitude, but no direction, at every point in space. It determines (for a given system at a given point) the ratio of potential energy to electric charges at that point. The electric field is the negative gradient of the electrical potential.

The electrical potential is therefore measured in units of energy per unit of electric charge:

joules/coulomb = volt.

Electrical potential is (mathematically) analogous to temperature: every point in space has a given temperature, and temperature gradients (analogous to the electric field) determine the direction of heat flows.

## Introduction

Objects may possess a property known as electric charge. In the presence of an electric field, a force is exerted on such objects, accelerating them in the direction of the force. This force has the same direction as the electrical 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.

There is a direct relationship between force and potential energy. As an object moves in the direction the force accelerates it, its potential energy decreases. For example, the gravitational potential energy of a cannonball at the top of a tower is greater than at the base of the tower. As the object falls, that potential energy decreases and is translated to motion, or inertial 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. The potential of an electric field is called the electrical potential.

## Mathematical Introduction

The concept of electrical potential (denoted by: φE or V), also called electromotive force (ε), is closely linked with potential energy. Specifically, one definition for electrical potential is:

where U is the potential energy of the charge q. Here, q must be a test charge small enough as to not significantly affect the field. φE will only depend on the position of q but not on its size. The unit of electrical potential is J/C (joule per coulomb); more commonly known as the volt (V). Note that the potential energy and hence also the electrical potential is only defined up to an additive constant: one may arbitrary choose one position where the potential energy and the electrical potential is zero.

The electrical potential can also be calculated using the electric field E, thus:

where s is an arbitrary path connecting the point with zero potential to the point under consideration. Note: this equation cannot be used and the electrical potential is not defined if ×E ≠ 0, i.e., in the case of a changing magnetic field (see Maxwell's equations). When ×E = 0, the above integral does not depend on the specific path s chosen but only on its endpoints because then:

for any closed path s (this equation, in a simplified form, is extremely useful in electrical engineering as one of Kirchoff's Laws).

If E is constant, then φE looks like this:

where s is the displacement vector from the point of zero potential to the point under consideration.

The electrical potential created by a point charge q can be shown to have the following form:

where r is the distance of the point under consideration from the point charge.

The electrical potentials due to a system of point charges may be computed as the sum of the respective potentials, which simplifies calculations significantly since adding scalar fields is very much easier than adding the electrical fields, which are vector fields.  