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If ϕ is a velocity potential, then ϕ + f(t) is also a velocity potential for u, where f(t) is a scalar function of time and can be constant. Velocity potentials are unique up to a constant, or a function solely of the temporal variable. The Laplacian of a velocity potential is equal to the divergence of the corresponding flow.
The potential equations can be simplified using a procedure called gauge fixing. Since the potentials are only defined up to gauge equivalence, we are free to impose additional equations on the potentials, as long as for every pair of potentials there is a gauge equivalent pair that satisfies the additional equations (i.e. if the gauge fixing ...
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 ...
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 picture of a classical gauge theory developed in the previous section is almost complete, except for the fact that to define the covariant derivatives D, one needs to know the value of the gauge field () at all spacetime points. Instead of manually specifying the values of this field, it can be given as the solution to a field equation.
In electromagnetism, the Lorenz condition is generally used in calculations of time-dependent electromagnetic fields through retarded potentials. [2] The condition is , =, where is the four-potential, the comma denotes a partial differentiation and the repeated index indicates that the Einstein summation convention is being used.
Lorentz force on a charged particle (of charge q) in motion (velocity v), used as the definition of the E field and B field. Here subscripts e and m are used to differ between electric and magnetic charges .
In mathematics and mathematical physics, potential theory is the study of harmonic functions.. The term "potential theory" was coined in 19th-century physics when it was realized that two fundamental forces of nature known at the time, namely gravity and the electrostatic force, could be modeled using functions called the gravitational potential and electrostatic potential, both of which ...