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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.
These equations taken together are as powerful and complete as Maxwell's equations. Moreover, the problem has been reduced somewhat, as the electric and magnetic fields together had six components to solve for. [1] In the potential formulation, there are only four components: the electric potential and the three components of the vector potential.
Assuming the external magnetic field is uniform and shares a common axis with the paramagnet, the extensive parameter characterizing the magnetic state is , the magnetic dipole moment of the system. The fundamental thermodynamic relation describing the system will then be of the form U = U ( S , V , I , N ) {\displaystyle U=U(S,V,I,N)} .
Accordingly, physicists and engineers usually define magnetization as the quantity of magnetic moment per unit volume. [1] It is represented by a pseudovector M . Magnetization can be compared to electric polarization , which is the measure of the corresponding response of a material to an electric field in electrostatics .
Since this equation is primarily used to solve linear problems, the contributions can be added. For a very difficult geometry, numerical integration may be used. For problems where the dominant magnetic material is a highly permeable magnetic core with relatively small air gaps, a magnetic circuit approach is useful.
If the coordinate system is shifted to center it on m 1 and rotated such that the x-axis points in the direction of m 1 then the previous equation simplifies to [9] = () = ( ), where the variables r and θ are measured in a frame of reference with origin in m 1 and oriented such that m 1 is at the origin pointing in the x-direction.
However, due to other nonlinearities, is usually between 1 and 2, and is between 2 and 3. The equation is a simplified form that only applies when the magnetic field has a sinusoidal waveform and does not take into account factors such as DC offset. However, because most electronics expose materials to non-sinusoidal flux waveforms, various ...
For zero net magnetic charge density (ρ m = 0), the original form of Gauss's magnetism law is the result. The modified formula for use with the SI is not standard and depends on the choice of defining equation for the magnetic charge and current; in one variation, magnetic charge has units of webers, in another it has units of ampere-meters.
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