Electrical Potential Introduction Mathematical Introduction

Like any potential function, only the potential difference ( voltage) between two points is physically meaningful (neglecting quantum Aharonov-Bohm effects), since any constant can be added to φ without affecting E.

The electrical potential is therefore measured in units of energy per unit of electric charge. In SI units, this is:

The electric potential provides a simple way to analyze electrical networkAn electrical network or electrical circuit is an interconnection of analog electrical elements such as resistors, inductors, capacitors, diodes, switches and transistors. It can be as small as an integrated circuit on a silicon chip, or as large as an els with the help of Kirchhoff's voltage lawKirchhoff's circuit laws are a pair of laws that deal with the conservation of charge and energy in electrical circuits, and were first described in 1845 by Gustav Kirchhoff. Widely used in electrical engineering, they are also called Kirchhoff's rules or, without solving the detailed Maxwell's equationsMaxwell's equations are the set of four equations, attributed to James Clerk Maxwell, that describe the behavior of both the electric and magnetic fields, as well as their interactions with matter. Introduction Maxwell's four equations express, respective for the fields of the circuit. For this problem, the electric potential can also be generalized to handle sitations with time-varying magnetic fields, in which case the electric field is not conservative and a potential function cannot be defined everywhere in space. There, an effective potential drop is included, associated with the inductanceInductance is a physical characteristic of an inductor, which is an electrical device that produces at any time a voltage proportional to the instantaneous rate of change in current flowing through it. The symbol L is used for inductance in honour of the of the circuit. This generalized potential difference is also called the electromotive forceAn electromotive force ( emf is the "force", measured in volts, that is produced by interaction between a current and a magnetic field, at least one of which is changing. Since the word " force" now has a very specific meaning in physics, and an emf is no (emf).

1 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 mechanicsClassical mechanics is a model of the physics of forces acting upon bodies. It is often referred to as Newtonian mechanics after Newton and his laws of motion. Classical mechanics is subdivided into statics (which models objects at rest), kinematics (whic 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 that 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 in the absence of time-varying magnetic fields. The potential of an electric field is called the electrical potential.

2 Mathematical introduction

The concept of electrical potential (denoted by: φ or V) is closely linked with potential energy. Specifically, one definition for electrical potential is:

where U is the potential energy of a small charge q due to the electric field. Here, q must be a test charge infinitesimally small, so as to not significantly affect the distribution of charges elsewhere. φ_E will only depend on the position of q but not on its size. 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 Kirchhoff's circuit laws).

If E is constant, then φ_E 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, in SI units:

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.

3 See also

4 References

Electric Potential

Electromagnetism

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