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A familiar example is potential energy due to gravity. Vector field (right) and corresponding scalar potential (left). A scalar potential is a fundamental concept in vector analysis and physics (the adjective scalar is frequently omitted if there is no danger of confusion with vector potential). The scalar potential is an example of a scalar field.
The vector potential admitted by a solenoidal field is not unique. If is a vector potential for , then so is +, where is any continuously differentiable scalar function. This follows from the fact that the curl of the gradient is zero.
It combines both an electric scalar potential and a magnetic vector potential into a single four-vector. [ 1 ] As measured in a given frame of reference , and for a given gauge , the first component of the electromagnetic four-potential is conventionally taken to be the electric scalar potential, and the other three components make up the ...
The magnetic vector potential, , is a vector field, and the electric potential, , is a scalar field such that: [5] = , =, where is the magnetic field and is the electric field. In magnetostatics where there is no time-varying current or charge distribution , only the first equation is needed.
In advanced classical mechanics it is often useful, and in quantum mechanics frequently essential, to express Maxwell's equations in a potential formulation involving the electric potential (also called scalar potential) φ, and the magnetic potential (a vector potential) A. For example, the analysis of radio antennas makes full use of Maxwell ...
The force is a vector field, which can be obtained as a factor of the gradient of the potential energy scalar field. Examples include: Potential fields, such as the Newtonian gravitational potential, or the electric potential in electrostatics, are scalar fields which describe the more familiar forces.
The Liénard–Wiechert potentials describe the classical electromagnetic effect of a moving electric point charge in terms of a vector potential and a scalar potential in the Lorenz gauge. Stemming directly from Maxwell's equations , these describe the complete, relativistically correct, time-varying electromagnetic field for a point charge in ...
If the force acting on a body varies over space, then one has a force field; such a field is described by vectors at every point in space, which is in-turn called a vector field. A conservative vector field can be simply expressed as the gradient of a certain scalar function, called a scalar potential. The potential energy is related to, and ...