Electrical Potential

The electrical potential of a given system at a given point determines the potential energy, due to [electric fields]?, of electric charges at that point. The electrical potential is therefore a scalar field with units of energy per unit charge (Volt). The electric field is the gradient of the electrical potential.

A scalar field has a magnitude, but no direction, at every point in space. Thus, electrical potential is analagous to temperature--every point in space has a given temperature, and temperature gradients (analagous 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. (Some would say the force is exerted on the electric charge itself).

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 moves, that potential energy 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 (electromagnetic) force.

The potential of an electric field is called the electrical potential. The commonly used unit of electrical potential is the Volt.

Mathematical Introduction

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

φ_E = U/q

where U is the potential energy of the charge q. The unit of electrical potential is J/C (Joule per Coulomb); more commonly known as the Volt (V).

More Rigorous Definition

The above definition is a bit oversimplified, and it requires the presence of a charge that is not creating the potential. More commonly, electrical potential is calculated using the [Electric Field]?, thus:

 φE = ∫E·ds
      s

where E is a vector function describing the [Electric Field]?, s is the path the integral covers. Note: this equation breaks down whenever ∇×E = 0; see Maxwell's equations for when this is true. When ∇×E ≠ 0, something like the potential can be calculated along a specific path, but it can not be represented as a function of position. Whenever the above caveat holds, the electric potential can take on the form of a scalar function in space. One requirement of a scalar function is that it has only one value for each point. If the above caveat holds, then each point will be uniquely defined because:

∫E·ds = 0†
s

For any closed path.

For a point charge it has the following form:

φ_E = q/(4πε_or)

where r is the distance from the point charge q.

If E is constant, then φ_E looks like this:

φ_E = E·s

where s is any displacement vector.

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† This equation, in a simplified form, is extremely useful in electrical engineering as one of [Kirchoffs Rules]?.

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References [Electric Potential]