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Consider a long, thin wire of charge and length .To calculate the average linear charge density, ¯, of this one dimensional object, we can simply divide the total charge, , by the total length, : ¯ = If we describe the wire as having a varying charge (one that varies as a function of position along the length of the wire, ), we can write: = Each infinitesimal unit of charge, , is equal to ...
where ρ is the charge density, which can (and often does) depend on time and position, ε 0 is the electric constant, μ 0 is the magnetic constant, and J is the current per unit area, also a function of time and position. The equations take this form with the International System of Quantities.
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 nĚ‚, d is the dipole moment between two point charges, the volume density of these is the polarization density P.
Integrating the definitions gives the total charge Q of a region according to line integral of the linear charge density λ q (r) over a line or 1d curve C, = similarly a surface integral of the surface charge density σ q (r) over a surface S, = and a volume integral of the volume charge density ρ q (r) over a volume V, = where the subscript ...
Heuristically, this can be regarded as nature "attempting" to forecast what the present field would be by linear extrapolation to the present time. [5] The last term, proportional to the second derivative of the unit direction vector e r ′ {\displaystyle e_{r'}} , is sensitive to charge motion perpendicular to the line of sight.
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Where no such symmetry exists, Gauss's law can be used in its differential form, which states that the divergence of the electric field is proportional to the local density of charge. The law was first [1] formulated by Joseph-Louis Lagrange in 1773, [2] followed by Carl Friedrich Gauss in 1835, [3] both
the total electric charge density (total charge per unit volume), ρ, and; the total electric current density (total current per unit area), J. The universal constants appearing in the equations (the first two ones explicitly only in the SI formulation) are: the permittivity of free space, ε 0, and; the permeability of free space, μ 0, and