Magnetic energy

Magnetic energy and electrostatic potential energy are related by Maxwell's equations. The potential energy of a magnet or magnetic moment




m



{\displaystyle \mathbf {m} }
in a magnetic field




B



{\displaystyle \mathbf {B} }
is defined as the mechanical work of the magnetic force (actually magnetic torque) on the re-alignment of the vector of the magnetic dipole moment and is equal to:





E


p
,
m



=
−

m

⋅

B



{\displaystyle E_{\rm {p,m}}=-\mathbf {m} \cdot \mathbf {B} }
while the energy stored in an inductor (of inductance



L


{\displaystyle L}
) when a current



I


{\displaystyle I}
flows through it is given by:





E


p
,
m



=


1
2


L

I

2


.


{\displaystyle E_{\rm {p,m}}={\frac {1}{2}}LI^{2}.}
This second expression forms the basis for superconducting magnetic energy storage.
Energy is also stored in a magnetic field. The energy per unit volume in a region of space of permeability




μ

0




{\displaystyle \mu _{0}}
containing magnetic field




B



{\displaystyle \mathbf {B} }
is:




u
=


1
2





B

2



μ

0






{\displaystyle u={\frac {1}{2}}{\frac {B^{2}}{\mu _{0}}}}
More generally, if we assume that the medium is paramagnetic or diamagnetic so that a linear constitutive equation exists that relates




B



{\displaystyle \mathbf {B} }
and




H



{\displaystyle \mathbf {H} }
, then it can be shown that the magnetic field stores an energy of




E
=


1
2


∫

H

⋅

B



d

V


{\displaystyle E={\frac {1}{2}}\int \mathbf {H} \cdot \mathbf {B} \ \mathrm {d} V}
where the integral is evaluated over the entire region where the magnetic field exists.For a magnetostatic system of currents in free space, the stored energy can be found by imagining the process of linearly turning on the currents and their generated magnetic field, arriving at a total energy of:




E
=


1
2


∫

J

⋅

A



d

V


{\displaystyle E={\frac {1}{2}}\int \mathbf {J} \cdot \mathbf {A} \ \mathrm {d} V}
where




J



{\displaystyle \mathbf {J} }
is the current density field and




A



{\displaystyle \mathbf {A} }
is the magnetic vector potential. This is analogous to the electrostatic energy expression





1
2


∫
ρ
ϕ


d

V


{\textstyle {\frac {1}{2}}\int \rho \phi \ \mathrm {d} V}
; note that neither of these static expressions do apply in the case of time-varying charge or current distributions.

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