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Scalar potentials play a prominent role in many areas of physics and engineering. The gravity potential is the scalar potential associated with the force of gravity per unit mass, or equivalently, the acceleration due to the field, as a function of position.
In physics, scalar fields often describe the potential energy associated with a particular force. The force is a vector field , which can be obtained as a factor of the gradient of the potential energy scalar field.
Position vectors r and r′ used in the calculation. The starting point is Maxwell's equations in the potential formulation using the Lorenz gauge: =, = where φ(r, t) is the electric potential and A(r, t) is the magnetic vector potential, for an arbitrary source of charge density ρ(r, t) and current density J(r, t), and is the D'Alembert operator. [2]
The scalar potential is given by = † + (†), where >, so that acquires a non-zero Vacuum expectation value, which generates masses for the Electroweak gauge fields (the Higgs mechanism), and >, so that the potential is bounded from below.
A scalar in physics and other areas of science is also a scalar in mathematics, as an element of a mathematical field used to define a vector space.For example, the magnitude (or length) of an electric field vector is calculated as the square root of its absolute square (the inner product of the electric field with itself); so, the inner product's result is an element of the mathematical field ...
In theoretical physics, scalar field theory can refer to a relativistically invariant classical or quantum theory of ... This is known as a double well potential, ...
The gravitational potential (V) at a location is the gravitational potential energy (U) at that location per unit mass: =, where m is the mass of the object. Potential energy is equal (in magnitude, but negative) to the work done by the gravitational field moving a body to its given position in space from infinity.
In mathematics and physics, an equipotential or isopotential refers to a region in space where every point is at the same potential. [1] [2] [3] This usually refers to a scalar potential (in that case it is a level set of the potential), although it can also be applied to vector potentials.