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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).
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.
If electromagnetic energy is not gained from or lost to other forms of energy within some region (e.g., mechanical energy, or heat), then electromagnetic energy is locally conserved within that region, yielding a continuity equation as a special case of Poynting's theorem: = where is the energy density of the electromagnetic field. This ...
In three dimensions, the derivative has a special structure allowing the introduction of a cross product: = + = + from which it is easily seen that Gauss's law is the scalar part, the Ampère–Maxwell law is the vector part, Faraday's law is the pseudovector part, and Gauss's law for magnetism is the pseudoscalar part of the equation.
The electromagnetic wave equation is a second-order partial differential equation that describes the propagation of electromagnetic waves through a medium or in a vacuum. It is a three-dimensional form of the wave equation. The homogeneous form of the equation, written in terms of either the electric field E or the magnetic field B, takes the form:
Magnetic field (green) induced by a current-carrying wire winding (red) in a magnetic circuit consisting of an iron core C forming a closed loop with two air gaps G in it. In an analogy to an electric circuit, the winding acts analogously to an electric battery, providing the magnetizing field , the core pieces act like wires, and the gaps G act like resistors.
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.
In such circuits, simple circuit laws can be used instead of deriving all the behaviour of the circuits directly from electromagnetic laws. Ohm's law states the relationship between the current I and the voltage V of a circuit by introducing the quantity known as resistance R [35] Ohm's law: = /