# 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:

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

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