Physics:Magnetic energy

Short description: Energy from the work of a magnetic force

The potential magnetic energy of a magnet or magnetic moment $\mathbf{𝐦}$ in a magnetic field $\mathbf{𝐁}$ is defined as the mechanical work of the magnetic force on the re-alignment of the vector of the magnetic dipole moment and is equal to: $$E_{\text{p,m}}=-\mathbf{𝐦}\cdot \mathbf{𝐁}$$The mechanical work takes the form of a torque $\mathit{N}$:$$\mathbf{𝐍}=\mathbf{𝐦}\times \mathbf{𝐁}=-\mathbf{𝐫}\times \mathrm{\nabla }{E}_{\text{p,m}}$$ which will act to "realign" the magnetic dipole with the magnetic field.

In an electronic circuit the energy stored in an inductor (of inductance $L$) when a current $I$ flows through it is given by:$$E_{\text{p,m}}=\frac{1}{2}L{I}^2.$$ This expression forms the basis for superconducting magnetic energy storage. It can be derived from a time average of the product of current and voltage across an inductor.

Energy is also stored in a magnetic field itself. The energy per unit volume $u$ in a region of free space with vacuum permeability ${\mu }_0$ containing magnetic field $\mathbf{𝐁}$ is: $$u=\frac{1}{2}\frac{B^2}{{\mu }_0}$$More generally, if we assume that the medium is paramagnetic or diamagnetic so that a linear constitutive equation exists that relates $\mathbf{𝐁}$ and the magnetization $\mathbf{𝐇}$ (for example $\mathbf{𝐇}=\mathbf{𝐁}/\mu$ where $\mu$ is the magnetic permeability of the material), then it can be shown that the magnetic field stores an energy of $$E=\frac{1}{2}\int \mathbf{𝐇}\cdot \mathbf{𝐁}\mathrm{d}V$$ 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: $$E=\frac{1}{2}\int \mathbf{𝐉}\cdot \mathbf{𝐀}\mathrm{d}V$$ where $\mathbf{𝐉}$ is the current density field and $\mathbf{𝐀}$ is the magnetic vector potential. This is analogous to the electrostatic energy expression $\frac{1}{2}\int \rho \varphi \mathrm{d}V$; note that neither of these static expressions apply in the case of time-varying charge or current distributions.

External links

• Magnetic Energy, Richard Fitzpatrick Professor of Physics The University of Texas at Austin.