# Magnetic field strength

Physical size
Surname Magnetic field strength
Formula symbol ${\ displaystyle {\ vec {H}}}$
Size and
unit system
unit dimension
SI A · m -1 I · L −1
Gauss ( cgs ) Oe L -1/2 · M 02/01 · T -1
esE ( cgs ) Stata · cm -1 L 01/02 · M 02/01 · T -2
emE ( cgs ) Oe L -1/2 · M 02/01 · T -1

The magnetic field strength (symbol: ), also called magnetic excitation referred classified as vector size any point in space and direction of a thickness by the magnetomotive force generated magnetic field to. It is related to the magnetic flux density via the material equations of electrodynamics (within linear, homogeneous, isotropic, time-invariant matter to :) . ${\ displaystyle H}$${\ displaystyle {\ vec {B}} = \ mu \ cdot {\ vec {H}}}$ ${\ displaystyle B}$

The international unit of magnetic field strength is the ampere per meter :

${\ displaystyle \ left [H \ right] = \, {\ mathrm {A} \ over \ mathrm {m}}}$

With a straight conductor , the field strength is constant along a circular field line. If the magnetic field strength outside a straight conductor through which current flows, denotes the current strength in the conductor and the radius of the circular field line, then the magnitude of the magnetic field strength in material with homogeneous magnetic permeability is : ${\ displaystyle H}$${\ displaystyle r}$${\ displaystyle I}$${\ displaystyle r}$

${\ displaystyle H = {\ frac {I} {2 \ pi \ cdot r}}}$

Numerical example: At a distance of 5 cm from the axis of a straight conductor carrying a current of 50 A, the magnetic field strength is: ${\ displaystyle r}$${\ displaystyle I}$

${\ displaystyle H = {\ frac {I} {2 \ pi \ cdot r}} = {\ frac {50 \, \ mathrm {A}} {2 \ pi \ cdot 0 {,} 05 \, \ mathrm { m}}} = 159 {,} 15 \, \ mathrm {\ frac {A} {m}}}$

### Current-carrying ring

If the current flows through a single turn with the radius ( conductor loop ), the field strength is measured at a point on the coil axis at a distance from the center of the ring ${\ displaystyle r}$${\ displaystyle I}$${\ displaystyle x}$

${\ displaystyle H = {\ frac {I \ cdot r ^ {2}} {2 (x ^ {2} + r ^ {2}) ^ {3/2}}}}$

For the derivation see: Biot-Savart - Circular conductor loop

### Solenoid

Solenoid
Magnetic field of a solenoid (in cross section). The wire windings are marked by "×" (current flows into the image plane) and "·" (current flows out of the image plane).

If the current flows through a coil of the length , diameter and turns , the field strength is measured in the center${\ displaystyle l}$${\ displaystyle D}$${\ displaystyle N}$ ${\ displaystyle I}$${\ displaystyle H}$

${\ displaystyle H = {\ frac {I \ cdot N} {\ sqrt {l ^ {2} + D ^ {2}}}}}$

If it is an elongated coil (length much larger than diameter, for short coils only approximate formulas exist), the above formula can be simplified and obtained:

${\ displaystyle H = {\ frac {I \ cdot N} {l}} = {\ frac {U_ {m}} {l}} = {\ frac {\ Theta} {l}}}$

The product is also referred to as ampere-turns or as magnetic voltage , the magnetic voltage - due to historical term formation - also as magnetic flux with the symbol . ${\ displaystyle I \ cdot N}$${\ displaystyle U_ {m}}$${\ displaystyle \ Theta}$

Along the coil axis, it is exactly half as large at the ends of the coil as in the middle. The interior of the coil is almost independent of the distance to the coil axis and almost homogeneous. Strong deviations are only measured at the ends of the coil. ${\ displaystyle H}$${\ displaystyle H}$

### Helmholtz coil

Helmholtz coil pair

Two short, round coils which are identical in size and number of turns and flow through in the same direction of rotation at a distance of their radius build a largely homogeneous magnetic field between them. In the middle of this arrangement, known as the Helmholtz coil , the magnetic field has the field strength

${\ displaystyle H = {\ frac {8 \, N \, I} {R {\ sqrt {125}}}}}$.

Here is the number of turns (per coil). ${\ displaystyle N}$

## Relationships with other quantities

From the material equations of electrodynamics, the relationship between the magnetic field strength and the magnetic flux density within linear, homogeneous, isotropic, time-invariant matter results in vector notation: ${\ displaystyle H}$${\ displaystyle B}$

${\ displaystyle {\ vec {H}} = {\ vec {B}} \ cdot {1 \ over \ mu}}$,

where expresses the magnetic conductivity (permeability) of the point in space considered. In general, the following applies: ${\ displaystyle \ mu}$

${\ displaystyle {\ vec {B}} = \ mu _ {0} {\ vec {H}} + {\ vec {J}}}$,

with the magnetic polarization ( not to be confused with the electrical current density, which is traditionally also referred to as). If the magnetic polarization is generated exclusively by the magnetic field strength : ${\ displaystyle {\ vec {J}}}$${\ displaystyle {\ vec {J}}}$${\ displaystyle {\ vec {H}}}$

${\ displaystyle {\ vec {J}} = \ mu _ {0} \ chi _ {m} {\ vec {H}}}$,

with the magnetic susceptibility . ${\ displaystyle \ chi _ {m}}$

Within linear, homogeneous, isotropic, time-invariant matter, the following applies:

${\ displaystyle {\ vec {B}} = \ mu _ {0} {\ vec {H}} + {\ vec {J}} = \ mu _ {0} {\ vec {H}} + \ mu _ {0} \ chi _ {m} {\ vec {H}} = \ mu _ {0} \ left (1+ \ chi _ {m} \ right) {\ vec {H}} = \ mu _ {0 } \ mu _ {m} {\ vec {H}} = \ mu {\ vec {H}}}$,

where describes the magnetic peamability tensor, which in many cases is assumed to be a scalar. ${\ displaystyle \ mu}$

### Relationship to the electric current density ${\ displaystyle {\ vec {J}}}$

The relationship

${\ displaystyle \ operatorname {rot} \ {\ vec {H}} = {\ vec {J}} + {\ frac {\ partial {\ vec {D}}} {\ partial t}}}$

from Maxwell's equations represents the local form of the flow rate theorem. The electrical current density and the second term with the time derivative of the electrical flux density expresses the density of the displacement current . In the simple static case without a change in time, the second summand disappears and the following applies: ${\ displaystyle {\ vec {J}}}$ ${\ displaystyle {\ vec {D}}}$

${\ displaystyle \ operatorname {red} \ {\ vec {H}} = {\ vec {J}}}$

This means that the vortex density of the magnetic field in every point in space is equal to the local conduction current density . The meaning is that the source freedom of the magnetic field is mathematically expressed and the magnetic field lines are always self-contained. ${\ displaystyle {\ vec {H}}}$

In the harmonically steady state (HZE), it is sufficient to consider the Fourier transform of Ampere's law:

${\ displaystyle \ operatorname {red} \ {\ vec {H ^ {*}}} = {\ vec {J ^ {*}}} + j \ omega {\ vec {D ^ {*}}} = \ kappa ^ {*} {\ vec {E ^ {*}}} + j \ omega \ varepsilon ^ {*} {\ vec {E ^ {*}}} = j \ omega \ left (\ varepsilon ^ {*} - {\ frac {\ kappa ^ {*}} {\ omega}} j \ right) {\ vec {E ^ {*}}}}$,

${\ displaystyle \ varepsilon ^ {*}}$is the complex electrical permittivity, which takes into account electrical relaxation processes or dielectric losses in the material. is the complex conductivity that describes ohmic losses as well as a phase shift from to in the material. (The deformation only applies if there is no impressed electric field strength in the material, which is caused, for example, by chemical processes.) ${\ displaystyle \ kappa ^ {*}}$${\ displaystyle {\ vec {E ^ {*}}}}$${\ displaystyle {\ vec {J ^ {*}}}}$

where are complex vector fields. Application of the rotation and other Maxwell equations (Gauss law for magnetic fields, induction law) gives: ${\ displaystyle {\ vec {H ^ {*}}}, {\ vec {J ^ {*}}}, {\ vec {D ^ {*}}}}$

${\ displaystyle \ operatorname {red} \ \ operatorname {red} \ {\ vec {H ^ {*}}} = \ operatorname {grad} \ left (\ operatorname {div} \, {\ vec {H ^ {* }}} \ right) - \ Delta {\ vec {H ^ {*}}} = - \ Delta {\ vec {H ^ {*}}} = j \ omega \ left (\ varepsilon ^ {*} - { \ frac {\ kappa ^ {*}} {\ omega}} j \ right) \ operatorname {red} \ {\ vec {E ^ {*}}} = j \ omega \ left (\ varepsilon ^ {*} - {\ frac {\ kappa ^ {*}} {\ omega}} j \ right) \ left (-j \ omega {\ vec {B ^ {*}}} \ right) = \ omega ^ {2} \ mu ^ {*} \ left (\ varepsilon ^ {*} - {\ frac {\ kappa ^ {*}} {\ omega}} j \ right) {\ vec {H ^ {*}}} = {\ vec { k ^ {*}}} ^ {2} {\ vec {H ^ {*}}}}$

where the complex permittivity constant describes magnetic relaxation processes or losses due to periodic magnetic reversal of polarization (usually only relevant in the terahertz range) and is the complex wave number vector of a corresponding TEM wave. The Helmholz equation for the magnetic field strength obviously results as: ${\ displaystyle \ mu ^ {*}}$${\ displaystyle {\ vec {k ^ {*}}}}$

${\ displaystyle \ Delta {\ vec {H ^ {*}}} + {\ vec {k ^ {*}}} ^ {2} {\ vec {H ^ {*}}} = 0}$

## literature

Karl Küpfmüller, Gerhard Kohn: Theoretical electrical engineering and electronics . 16th edition. Springer Verlag, 2005, ISBN 3-540-20792-9 .