Magnetostatics

The Magnetostatics is a branch of electrodynamics . They treated DC magnetic fields , ie temporally constant magnetic fields .

Basics

In magnetostatics, the spatial distribution of magnetic fields in the vicinity of permanent magnets and of stationary currents (concept of the current filament ) is examined. A stationary current is, for example, direct current in an electrical conductor . In addition to the individual magnetic properties of substances such as ferromagnetism , diamagnetism, etc., this also includes the earth's magnetic field . In addition, magnetostatics describes the force effect of such fields on magnets and currents. This includes the behavior of a magnetic dipole in a time-constant magnetic field, for example the behavior of a (freely movable) magnetic needle in the earth's magnetic field.

The basic terms are analogous to electrostatics . The positive and negative electrical charge correspond to the north and south poles, quantitatively: positive and negative pole strength . However, in contrast to electrical charges, magnetic poles cannot be isolated, but always appear together in a body.

illustration

Although there are no isolated magnetic charges ( magnetic monopoles ), magnetostatic effects can be illustrated with an analogy to electrostatics. This is particularly used in school physics: one considers a bar magnet of length l as two opposing magnetic charges at a distance l . The analogue of the electric charge is the magnetic pole strength . The pole strength is defined in such a way that the magnetic force law (also: magnetostatic force law ) can be formulated analogously to the Coulomb force : ${\displaystyle p}$

${\displaystyle F={\frac {1}{4\pi \mu _{0}}}{\frac {p_{1}\cdot p_{2}}{r^{2}}}.}$

Here F is the magnetic force that acts between two magnetic poles of the same pole strength and at a distance ; μ ₀ is the magnetic field constant . The pole strength is of the same dimension as the magnetic flux and is therefore given in Weber units . ${\displaystyle p_{1}}$ ${\displaystyle p_{2}}$ ${\displaystyle r}$

From the definition it follows e.g. B. for a homogeneous field with known flux density B and area A for the force:

${\displaystyle F={\frac {1}{4\pi \mu _{0}}}{\frac {\Phi _{1}\cdot \Phi _{2}}{r^{2}}}={\frac {1}{4\pi \mu _{0}}}{\frac {B_{1}A_{1}\cdot B_{2}A_{2}}{r^{2}}}}$

Field theory

For fields that are constant over time, the equations for electric (E) and magnetic (B) fields “decouple”: if all time derivatives in Maxwell's equations are set equal to 0, equations arise that do not contain E and B at the same time . The phenomena of magnetostatics can be described with the following two reduced Maxwell equations:

1. ${\displaystyle \nabla \cdot {\vec {B}}=0}$
2. ${\displaystyle \nabla \times {\vec {B}}=\mu {\vec {j}}}$

The vector potential is introduced as an auxiliary field with the following definition: ${\displaystyle {\vec {A}}}$

${\displaystyle {\vec {B}}=\nabla \times {\vec {A}}}$

This automatically fulfills the equation , since the divergence of a rotational field is identical to 0 . ${\displaystyle \nabla \cdot {\vec {B}}=0}$ ${\displaystyle \nabla \cdot \left({\nabla \times {\vec {A}}}\right)\equiv 0}$

${\displaystyle {\vec {A}}}$however, is not clearly determined because it is invariant under a gauge transformation with . I.e. the B fields defined by A and A ' are identical. This follows from ${\displaystyle {\vec {B}}}$ ${\displaystyle \chi }$ ${\displaystyle {\vec {A}}'={\vec {A}}+\nabla \chi }$

${\displaystyle {\vec {B}}'=\nabla \times {\vec {A}}'=\nabla \times {\vec {A}}+\nabla \times \nabla \chi =\nabla \times {\vec {A}}={\vec {B}}}$,

since the rotation of the gradient of a scalar field vanishes.

If we put into the inhomogeneous Maxwell equation (above equation 2) ${\displaystyle {\vec {B}}=\nabla \times {\vec {A}}}$

${\displaystyle \mu {\vec {j}}=\nabla \times \nabla \times {\vec {A}}=\nabla \left({\nabla \cdot {\vec {A}}}\right)-\Delta {\vec {A}}}$

a, the Coulomb calibration results in the particularly simple form: ${\displaystyle \nabla \cdot {\vec {A}}=0}$

${\displaystyle \Delta {\vec {A}}=-\mu {\vec {j}}}$

This represents a Poisson equation for each component , which is given by

${\displaystyle {\vec {A}}({\vec {r}})={\frac {\mu _{0}}{4\pi }}\int \mathrm {d} ^{3}r'{\frac {{\vec {j}}({\vec {r}}\,')}{|{\vec {r}}-{\vec {r}}\,'|}}}$

is resolved.

If one applies the rotation to A , one obtains the Biot-Savart law for the physically relevant B field

${\displaystyle {\vec {B}}\left({\vec {r}}\right)=\nabla _{\vec {r}}\times {\vec {A}}\left({\vec {r}}\right)={\frac {\mu _{0}}{4\pi }}\int {\nabla _{\vec {r}}\times {\frac {{\vec {j}}\left({{\vec {r}}\,'}\right)}{\left|{{\vec {r}}-{\vec {r}}\,'}\right|}}}\mathrm {d} ^{3}r'={\frac {\mu _{0}}{4\pi }}\int {\left({\nabla _{\vec {r}}{\frac {1}{\left|{{\vec {r}}-{\vec {r}}\,'}\right|}}}\right)\times }{\vec {j}}\left({{\vec {r}}\,'}\right)\mathrm {d} ^{3}r'={\frac {\mu _{0}}{4\pi }}\int {{\vec {j}}\left({{\vec {r}}\,'}\right)\times {\frac {{\vec {r}}-{\vec {r}}\,'}{\left|{{\vec {r}}-{\vec {r}}\,'}\right|^{3}}}\mathrm {d} ^{3}r'}}$

For a stream filament goes to : ${\displaystyle {\vec {j}}\left({{\vec {r}}\,'}\right)\mathrm {d} ^{3}r'}$ ${\displaystyle I\,\mathrm {d} {\vec {s}}\,'}$

${\displaystyle {\vec {B}}\left({\vec {r}}\right)={\frac {\mu _{0}}{4\pi }}I\int {\mathrm {d} {\vec {s}}\,'\times {\frac {{\vec {r}}-{\vec {r}}\,'}{\left|{{\vec {r}}-{\vec {r}}\,'}\right|^{3}}}}}$

Magnetostatic fields

Magnetostatic fields exist within conductors carrying direct current. They are source-free and there are no magnetic charges,

${\displaystyle \mathrm {div} \;B(r)=0}$.

The cause of magnetostatic fields are moving electrical charges or equivalent direct currents with the vortex density:

${\displaystyle \mathrm {rot} \;H(r)=J_{L}(r)}$.

literature

• Wolfgang Demtröder : Experimental Physics. Vol. 2: Electricity and optics . Springer, Berlin 2004, ISBN 3-540-20210-2
• Wolfgang Nolting : Basic Course Theoretical Physics 3: Electrodynamics . Springer, Berlin 2007, ISBN 978-3-540-71251-0
• Adolf J. Schwab: Conceptual World of Field Theory, Springer Verlag, ISBN 3-540-42018-5

Individual evidence

1. ^ You can also find the definition . In this case the pole strength has the dimension “current strength × length” and the unit A · m. ${\displaystyle F={\tfrac {\mu _{0}}{4\pi }}{\tfrac {p_{1}\cdot p_{2}}{r^{2}}}}$