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The magnetic field energy density is [itex]U=\frac{B^2}{2\mu_0}[/itex].

http://scienceworld.wolfram.com/physics/MagneticField.html

The derivative of the magnetic field is equal to [itex]d\mathbf{B}=\frac{\mu_0}{4\pi} \frac{I d\mathbf{l} \times \mathbf{\hat r}}{r^2}[/itex].

Therefore, the magnetic field is proportional to [itex]\frac{Il}{r^2}[/itex], and the magnetic field density is proportional to [itex](\frac{Il}{r^2})^2[/itex]. Electrical power is equal to [itex]P=IV=RI^2[/itex]. Resistivity is equal to [itex]\rho=R\frac{A}{l}[/itex]. Therefore, the central magnetic field energy density produced for a given electrical power is proportional to:

[itex]\frac{(\frac{Il}{r^2})^2}{RI^2}

=\frac{(\frac{l}{r^2})^2}{R}

=\frac{(\frac{l}{r^2})^2}{\rho\frac{l}{A}}

=\frac{l}{\frac{\rho}{A}r^4}

=\frac{l*A}{\rho*r^4}[/itex]

.... The volume of the conductor divided by the product of resistivity and radius to the fourth power.

Is this valid?

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# Magnetic field energy density per electrical power

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