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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.
A conductive object will experience a repulsion force. This can lift objects against gravity, though with continual power input to replace the energy dissipated by the eddy currents. An example application is separation of aluminum cans from other metals in an eddy current separator. Ferrous metals cling to the magnet, and aluminum (and other ...
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 ...
P o is the initial power; There are often other contributions to power loss for electromagnetic waves that are not included in this expression, such as due to the wall currents of the conductors of a transmission line or waveguide. Also, a similar analysis could be applied to the magnetic permeability where
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.
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: = /
A portative electromagnet is one designed to just hold material in place; an example is a lifting magnet. A tractive electromagnet applies a force and moves something. [8] Electromagnets are very widely used in electric and electromechanical devices, including: Motors and generators; Transformers; Relays; Electric bells and buzzers ...