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A velocity potential is not unique. 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 ...
The first term describes near field effects from the charge, and its direction in space is updated with a term that corrects for any constant-velocity motion of the charge on its distant static field, so that the distant static field appears at distance from the charge, with no aberration of light or light-time correction. This term, which ...
Position vectors r and r′ used in the calculation. Retarded time t r or t′ is calculated with a "speed-distance-time" calculation for EM fields.. If the EM field is radiated at position vector r′ (within the source charge distribution), and an observer at position r measures the EM field at time t, the time delay for the field to travel from the charge distribution to the observer is |r ...
The scalar potential is an example of a scalar field. Given a vector field F, the scalar potential P is defined such that: [1] = = (,,), where ∇P is the gradient of P and the second part of the equation is minus the gradient for a function of the Cartesian coordinates x, y, z. [a] In some cases, mathematicians may use a positive sign in front ...
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 metric captures all the geometric and causal structure of spacetime, being used to define notions such as time, distance, volume, curvature, angle, and separation of the future and the past. In general relativity, the metric tensor plays the role of the gravitational potential in the classical theory of gravitation, although the physical ...
Trajectory of a particle with initial position vector r 0 and velocity v 0, subject to constant acceleration a, all three quantities in any direction, and the position r(t) and velocity v(t) after time t. The initial position, initial velocity, and acceleration vectors need not be collinear, and the equations of motion take an almost identical ...
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 ...