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Mathematically, we can state the law of charge conservation as a continuity equation: = ˙ ˙ (). where / is the electric charge accumulation rate in a specific volume at time t, ˙ is the amount of charge flowing into the volume and ˙ is the amount of charge flowing out of the volume; both amounts are regarded as generic functions of time.
All of the conservation laws listed above are local conservation laws. A local conservation law is expressed mathematically by a continuity equation, which states that the change in the quantity in a volume is equal to the total net "flux" of the quantity through the surface of the volume. The following sections discuss continuity equations in ...
The laws of physics are invariant with respect to orientation—for example, floating in outer space, there is no measurement you can do to say "which way is up"; the laws of physics are the same regardless of how you are oriented. This symmetry leads to the continuity equation for conservation of angular momentum.
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.
where: is the rate of change of the energy density in the volume. ∇•S is the energy flow out of the volume, given by the divergence of the Poynting vector S. J•E is the rate at which the fields do work on charges in the volume (J is the current density corresponding to the motion of charge, E is the electric field, and • is the dot product).
The law of conservation of charge always applies, giving the object from which a negative charge is taken a positive charge of the same magnitude, and vice versa. Even when an object's net charge is zero, the charge can be distributed non-uniformly in the object (e.g., due to an external electromagnetic field , or bound polar molecules).
Electric field from positive to negative charges. Gauss's law describes the relationship between an electric field and electric charges: an electric field points away from positive charges and towards negative charges, and the net outflow of the electric field through a closed surface is proportional to the enclosed charge, including bound charge due to polarization of material.
Continuous charge distribution. The volume charge density ρ is the amount of charge per unit volume (cube), surface charge density σ is amount per unit surface area (circle) with outward unit normal n̂, d is the dipole moment between two point charges, the volume density of these is the polarization density P.