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Categories: Multivariate calculus | Classical mechanics | Quantum mechanics | Electromagnetism

Potential

In vector calculus, any vector field of a certain type has an associated scalar field called the potential. Potentials find broad applications in physics. The vector potential is a related construct.

Mathematical definition

If $\vec F$ is an irrotational (aka conservative, curl-free, or potential) vector field with continuous partial derivatives, the potential of $\vec F$ with respect to a reference point $\mathbf r_0$ is defined in terms of a line integral:

$V(\mathbf r) = \int _{\mathbf r_0} ^{\mathbf r} \vec F \cdot d \mathbf r'$ (1).

where $\mathbf r'$ is a dummy variable of integration. It can be shown that such a scalar field exists for any curl-free vector field.

By the Fundamental Theorem of Calculus, we can alternatively define V as the scalar field that satisfies the following condition:

$\vec F = \nabla V$ (2).

This does not give a unique definition of V. In terms of definition (1), the ambiguity lies in the choice of the reference point. In terms of definition (2), V can change by a constant value throughout all space without changing its gradient.

Applications in physics

As entities described as vector fields occur often in physics, it is often useful to work with their potentials.

For instance, some force fields exert forces on a body equal to the product of the field and some invariant scalar property of the body.

As a body moves through such a force field, it rises and falls in the associated potential. The energy gained or lost by the body through mechanical work performed by the force is defined as its potential energy. It is then possible, in the case of a particle subjected to a known field, to speak directly of its change in energy between two points, without resorting to kinematics, which can be computationally difficult.

The gravitational field is a notable example of such a field. The electric field also behaves this way in many cases, though in the general case it does not (see Electric potential and Faraday's Law).

Because the physically observable field is a spatial derivative of its potential, adding an arbitrary constant field to it—a gauge transformation—will not change anything in the physics of a system. This is an example of the general concept of gauge invariance.

In quantum theory, gauge invariance leads to Aharonov-Bohm effects where an effect of a potential is observable even in regions where the corresponding classical field is zero.

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