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A continuity equation is the mathematical way to express this kind of statement. For example, the continuity equation for electric charge states that the amount of electric charge in any volume of space can only change by the amount of electric current flowing into or out of that volume through its boundaries.
In physics a conserved current is a current, , that satisfies the continuity equation =.The continuity equation represents a conservation law, hence the name. Indeed, integrating the continuity equation over a volume , large enough to have no net currents through its surface, leads to the conservation law =, where = is the conserved quantity.
This is called the charge density continuity equation + = The term on the left is the rate of change of the charge density ρ at a point. The term on the right is the divergence of the current density J at the same point. The equation equates these two factors, which says that the only way for the charge density at a point to change is for a ...
The concept of probability current is also used outside of quantum mechanics, when dealing with probability density functions that change over time, for instance in Brownian motion and the Fokker–Planck equation. [1] The relativistic equivalent of the probability current is known as the probability four-current.
With respect to symmetries and invariance principles, three special conservation laws have been described, associated with inversion or reversal of space, time, and charge. Conservation laws are considered to be fundamental laws of nature, with broad application in physics, as well as in other fields such as chemistry, biology, geology, and ...
Since d 2 = 0, the 3-form J satisfies the conservation of current (continuity equation): = = The current 3-form can be integrated over a 3-dimensional space-time region. The physical interpretation of this integral is the charge in that region if it is spacelike, or the amount of charge that flows through a surface in a certain amount of time ...
In electromagnetism, an eddy current (also called Foucault's current) is a loop of electric current induced within conductors by a changing magnetic field in the conductor according to Faraday's law of induction or by the relative motion of a conductor in a magnetic field. Eddy currents flow in closed loops within conductors, in planes ...
In modern terminology, the conserved quantity is called the Noether charge, while the flow carrying that charge is called the Noether current. The Noether current is defined up to a solenoidal (divergenceless) vector field. In the context of gravitation, Felix Klein's statement of Noether's theorem for action I stipulates for the invariants: [7]