# What is Magnetic energy: Definition and 50 Discussions

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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