[Home]Electrical Potential

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Changed: 1c1
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.
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.

Changed: 6c6
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).
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.

Changed: 10c10
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.
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.

Changed: 14,16c14,15
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.
Two such forces are the gravitational force (gravity) and the electric force.
The potential of an electric field is called the electrical potential.

Changed: 23c22
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).
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.

Changed: 25,26c24

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:
The electrical potential can also be calculated using the [Electric Field]? E, thus:

Changed: 31c29
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:
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; see Maxwell's equations for when this is true. When ∇×E = 0, the above integral does not depend on the specific path s chosen but only on its endpoints because then:

Changed: 38c36
For a point charge it has the following form:
If E is constant, then φE looks like this:

Changed: 40c38
φE = q/(4πεor)
φE = E·s

Changed: 42c40
where r is the distance from the point charge q.
where s is the displacement vector from the point of zero potential to the point under consideration.

Changed: 44c42
If E is constant, then φE looks like this:
The electrical potential created by a point charge q can be shown to have the following form:

Changed: 46c44,46
φE = E·s
φE = q/(4πεor)

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

Changed: 48c48
where s is any displacement vector.
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.

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. 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:

φE = U/q

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:

 φE = ∫E·ds
      s

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; see Maxwell's equations for when this is true. When ∇×E = 0, the above integral does not depend on the specific path s chosen but only on its endpoints because then:

E·ds = 0
s

for any closed path s (this equation, in a simplified form, is extremely useful in electrical engineering as one of [Kirchoffs Rules]?).

If E is constant, then φE looks like this:

φE = E·s

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:

φE = q/(4πεor)

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.


References:


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Last edited September 23, 2001 12:34 pm by AxelBoldt (diff)
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