3 The magnetostatic Field

So far the magnetic field was defined quite pragmatically by the effect on a compass. For a deeper analysis of the magnetic field, the field is now to be considered again - as with the electric field - from two directions. The magnetic field will also be considered a „causer field“ (a field caused by magnets) and an „acting field“ (a field acting on a magnet). This chapter will first discuss the acting magnetic field. For this, it is convenient to consider the effects inside a toroidal coil (= donut-like setup). This can be seen in Abbildung 6. For reasons of symmetry, it is also clear here that the field lines form concentric circles.

In an experiment, a magnetic needle inside the toroidal coil is now to be aligned perpendicular to the field lines. Then, the magnetic field will generate a torque $M$ which tries to align the magnetic needle in the field direction.

It now follows:

1. $M = {\rm const.} \neq f(\varphi)$: For the same distance from the axis of symmetry, the torque $M$ is independent of the angle $\phi$.
2. $M \sim I$: The stronger the current flowing through a winding, the stronger the effect, i.e. the stronger the torque.
3. $M \sim N$: The greater the number $N$ of windings, the stronger the torque $M$.
4. $M \sim {1 \over l}$ : The smaller the average coil circumference $l$ the greater the torque. The average coil circumference $l$ is equal to the mean magnetic path length (=average field line length).

To summarize: \begin{align*} M \sim {{I \cdot N}\over{l}} \end{align*}

The magnetic field strength $H$ inside the toroidal coil is given as: \begin{align*} \boxed{H ={{I \cdot N}\over{l}}} \quad \quad | \quad \text{applies to toroidal coil only} \end{align*}

For the unit of the magnetic field strength $H$ we get $[H] = {{[I]}\over{[l]}}= \rm 1~{{A}\over{m}}$

Magnetic Field Strength part 2: straight conductor

The previous derivation from the toroidal coil is now used to derive the field strength around a long, straight conductor. For a single conductor the part $N \cdot I$ of the formula can be reduced to $ N \cdot I = 1 \cdot I = I$ since there is only one conductor. For the toroidal coil, the magnetic field strength was given by this current(s) divided by the (average) field line length. Because of the (same rotational) symmetry, this is also true for the single conductor. Also here the field line length has to be taken into account.

The length of a field line around the conductor is given by the distance $r$ of the field line from the conductor: $l = l(r) = 2 \cdot \pi \cdot r$.
For the magnetic field strength of the single conductor we then get: \begin{align*} \boxed{H ={I\over{l}} = {{I}\over{2 \cdot \pi \cdot r}}} \quad \quad | \quad \text{applies only to the long, straight conductor} \end{align*}

In the electric field, the field line density was a measure of the strength of the field. This is also used for the magnetic field. Looking at the simulations in Falstad (e.g. Abbildung 8) with this understanding, one notices an inconsistency: contrary to the relationship just given, the field line density in the Falstad simulation not indicates the strength of the field. A realistic simulation is shown in Abbildung 9 for comparison, which makes the difference clear: the field is stronger near the conductor. Thus the field line density must also be stronger there.

Magnetic Voltage

The cause of a magnetic field is a current. As seen for the coil, sometimes the current has to be counted up ($N \cdot I$), e.g. by the number of windings of the coil. If this current $I$ and/or the number $N$ of windings is increased, the effect is amplified. To make this easier to handle, we introduce the Magnetic Voltage. The magnetic voltage $\theta$ is defined as

\begin{align*} \boxed{\theta = \sum I = N \cdot I} \end{align*} The unit of $\theta$ is: $[\theta]= 1~ \rm A$ (obsoletely called ampere-turn). For the magnetic voltage the currents which flow through the surface enclosed by the closed path have to be considered. A detailed definition will be given below after more analysis.

Thus, the magnetic field strength $H$ of the toroidal coil is then given by: $H= {{\theta}\over{l}}$

In the English literature often the name Magnetomotive Force $\mathcal{F}$ is used instead of magnetic voltage $\theta$. The naming refers to the Electromotive Force. The electromotive force describes the root cause of a (voltage) source to be able to drive a current and therefore generate a defined voltage. Both „forces“ shall not be confused with the mechanical force $\vec{F}= m \cdot \vec{a}$. They only describe the driving cause behind the electric or magnetic fields. The German courses in higher semesters use the term Magnetische Spannung - therefore, the English equivalent is introduced here.

Magnetic Field Strength part 3: Generalization

So far, only rotational symmetric problems could be solved. Now, this shall be generalized. For this purpose, we will have a look back at the electric field. For the electric field strength $E$ of a capacitor with two plates at a distance of $s$ and the potential difference $U$ holds:

\begin{align*} U = E \cdot s \quad \quad | \quad \text{applies to capacitor only} \end{align*}

In words: The potential difference is given by adding up the field strength along the path of a probe from one plate to the other. This was extended to $U = \int_s E {\rm d}s$. If we compare this idea to the magnetic field strength $H$ of a toroidal coil with the mean magnetic path length $l$, we had

\begin{align*} \theta = H \cdot l \quad \quad | \quad \text{applies to toroidal coil only} \end{align*}

Can you see the similarities? Again, the magnitude of the field strength is summed up along a path to arrive at another field-describing quantity (here, the magnetic voltage $\theta$). Because of the similarity - which continues below - the so-called magnetic potential difference $V_m$ is introduced:

\begin{align*} V_m = H \cdot s \quad \quad | \quad \text{applies to toroidal coil only} \end{align*}

Now, what is the difference between the magnetic potential difference $V_m$ and the magnetic voltage $\theta$?

1. The first equation of the toroidal coil ($\theta = H \cdot l$) is valid for exactly one turn along a field line with the length $l$. In addition, the magnetic voltage was equal to the current times the number of windings: $\theta = N \cdot I$.
2. The second equation ($V_{\rm m} = H \cdot s$) is independent of the length of the field line $l$. Only if $s = l$ is chosen, the magnetic voltage equals the magnetic potential difference. The path length $s$ can be a fraction or multiple of a single revolution $l$ for the magnetic potential difference.

Thus, for each infinitesimally small path ${\rm d}s$ along a field line, the resulting infinitesimally small magnetic potential difference ${\rm d}V_{\rm m} = H \cdot {\rm d}s$ can be determined. If now along the field line the magnetic field strength $H = H(\vec{s})$ changes, then the magnetic potential difference from point $\vec{s_1}$ to point $\vec{s_2}$ results to:

\begin{align*} V_{\rm m12} = V_{\rm m}(\vec{s_1}, \vec{s_2}) = \int_\vec{s_1}^\vec{s_2} H(\vec{s}) {\rm d}s \end{align*}

Up to now, only the situation was considered that one always walks along one single field line. $\vec{s}$ therefore always arrived at the same spot of the field line. If one wants to extend this to arbitrary directions (also perpendicular to field lines), then only that part of the magnetic field strength $\vec{H}$ may be used in the formula, which is parallel to the path ${\rm d} \vec{s}$. This is made possible by scalar multiplication. Thus, it is generally valid:

\begin{align*} \boxed{V_{\rm m12} = \int_\vec{s_1}^\vec{s_2} \vec{H} \cdot {\rm d} \vec{s}} \end{align*}

The magnetic voltage $\theta$ (and therefore the current) is the cause of the magnetic field strength. From the chapter The stationary Electric Flow the general representation of the current through a surface is known. This leads to the Ampere's Circuital Law

\begin{align*} \boxed{\oint_{s} \vec{H} \cdot {\rm d} \vec{s} = \iint_A \; \vec{S} {\rm d}\vec{A} = \theta} \end{align*}

• The path integral of the magnetic field strength along an arbitrary closed path is equal to the free currents (= current density) through the surface enclosed by the path.
• The magnetic voltage $\theta$ can be given as
  • for a single conductor: $\theta = I$
  • for a coil: $\theta = N \cdot I$
  • for multiple conductors: $\theta = \sum_n \cdot I_n$
  • for spatial distribution: $\theta = \iint_A \; \vec{S} {\rm d}\vec{A}$
• ${\rm d}\vec{s}$ and ${\rm d}\vec{A}$ build a right-hand system: once the thumb of the right hand is pointing along ${\rm d}\vec{A}$, the fingers of the right hand show the correct direction for ${\rm d}\vec{s}$ for positive $\vec{H}$ and $\vec{S}$

Recap: Application of magnetic Field Strength

Ampere's Circuital Law shall be applied to find the magnetic field strength $H$ inside the toroidal coil (Abbildung 13).

• The closed path ${\rm s}$ is on a revolution of a field line in the center of the coil
• The surface $A$ is the enclosed surface

This leads to:

\begin{align*} \oint_s \vec{H} \cdot {\rm d} \vec{s} &= \iint_A \vec{S} {\rm d}\vec{A} = \theta \end{align*}

Since $\vec{H} \uparrow \uparrow {\rm d} \vec{s}$ the term $\vec{H} \cdot {\rm d} \vec{s}$ can be substituted by $H {\rm d}s$:

\begin{align*} \oint_s H \cdot {\rm d}s &= \iint_A \vec{S} {\rm d}\vec{A} \end{align*}

The magnetic voltage is the current through the surface and is given as $N\cdot I$:

\begin{align*} \oint_{\rm s} H \cdot {\rm d}s &= N\cdot I \end{align*}

The magnetic field strength $H$ can be considered constant:

\begin{align*} H \cdot \int_C {\rm d}s &= N\cdot I \end{align*}

The average coil circumference is $2\pi \cdot {{d}\over{2}}$:

\begin{align*} H \cdot 2\pi \cdot {{d}\over{2}} &= N\cdot I \end{align*}

Therefore, the magnetic field strength in the toroidal coil is \begin{align*} H &= {{N\cdot I}\over{ \pi \cdot d }} \end{align*}

Similarly, the magnetic field strength $H$ in the distance $r$ to a single conductor with the current $I$ can be derived. In this situation, the result is:

\begin{align*} H &= {{I}\over{ 2\pi \cdot r }} \end{align*}

3.3 Magnetic Flux Density and Lorentz Law

Definition of the Magnetic Flux Density

In the last sub-chapter, the field effect on a magnet caused by currents was analyzed. Now, the field acting onto currents will get deeper investigation. To do so, the effect between two parallel conductors has to be examined closer. The experiment consists of a part $l$ of two very long¹⁾ conductors with the different currents $I_1, I_2$ in the distance $r$ (see Abbildung 14).

When no current is flowing through the conductors the forces er equal to zero. Once the currents flow in the same direction (e.g. $I_1>0, I_2>0$) attracting forces $\vec{F}_{12} = - \vec{F}_{21}$ appear. The force $\vec{F}_{xy}$ shall be the force on the conductor $x$ caused by conductor $y$. In the following the force $\vec{F}_{12}$ on the conductor $1$ will be examined.

The following is detectable:

1. $|\vec{F}_{12}| \sim I_1$, $|\vec{F}_{12}| \sim I_2$ : The stronger each current, the stronger the force $F_{12}$.
2. $|\vec{F}_{12}| \sim l$ : As longer the conductor length $l$, as stronger the force $F_{12}$ gets.
3. $|\vec{F}_{12}| \sim {{1}\over{r}}$ : A smaller distance $r$ leads to stronger force $F_{12}$.

To summarize: \begin{align*} {F}_{12} \sim {I_1 \cdot I_2 \cdot {{l}\over{r}}} \end{align*}

The proportionality factor is arbitrarily chosen as: \begin{align*} {{{F}_{12} \cdot r} \over {I_1 \cdot I_2 \cdot {l}}} = {{\mu}\over{2\pi}} \end{align*}

Here $\mu$ is the magnetic permeability and for vacuum (vacuum permeability): \begin{align*} \mu = \mu_0 = 4\pi \cdot 10^{-7} {{\rm Vs}\over{\rm Am}} = 1.257 \cdot 10^{-7} {{\rm Vs}\over{\rm Am}} \end{align*}

This leads to the Ampere's Force Law: \begin{align*} |\vec{F}_{12}| = {{\mu}\over{2 \pi}} \cdot {{I_1 \cdot I_2 }\over{r}} \cdot l \end{align*}

Since we want to investigate the effect on the current $I_1$, the following rearrangement can be done: \begin{align*} |\vec{F}_{12}| &= {{\mu}\over{2 \pi}} \cdot {{I_2 }\over{r}} &\cdot I_1 \cdot l \\ &= B &\cdot I_1 \cdot l \\ \end{align*}

The properties of the field from $I_2$ acting on $I_1$ are summarized to $B$ - the magnetic flux density.
$B$ has the unit: \begin{align*} [B] &= {{[F]}\over{[I]\cdot[l]}} = 1 \rm {{N}\over{Am}} = 1 {{{VAs}\over{m}}\over{Am}} = 1 {{Vs}\over{m^2}} \\ &= 1 {\rm T} \quad\quad({\rm Tesla}) \end{align*}

This formula can be generalized with the knowledge of the directions of the conducting wire $\vec{l}$, the magnetic field strength $\vec{B}$ and the force $\vec{F}$ using vector multiplication too:

\begin{align*} \boxed{\vec{F_L} = I \cdot \vec{l} \times \vec{B}} \end{align*}

The absolute value can be calculated by

\begin{align*} \boxed{|\vec{F_L}| = I \cdot |l| \cdot |B| \cdot \sin(\angle \vec{l},\vec{B} )} \end{align*}

The force is often called Lorentz Force $F_L$. For the orientation, another right-hand rule can be applied.

Lorentz Law and Lorentz Force

The true Lorentz force is not the force on the whole conductor but the single force onto an (elementary) charge.
To find this force the previous force onto a conductor can be used as a start. However, the formula will be investigated infinitesimally for small parts ${\rm d} \vec{l}$ of the conductor:

\begin{align*} \vec{{\rm d}F}_{\rm L} = I \cdot {\rm d}\vec{l} \times \vec{B} \end{align*}

The current is now substituted by $I = {\rm d}Q/{\rm d}t$, where ${\rm d}Q$ is the small charge packet in the length $\vec{{\rm d}l}$ of the conductor.

\begin{align*} \vec{{\rm d}F}_{\rm L} = {{{\rm d}Q}\over{{\rm d}t}} \cdot {\rm d}\vec{l} \times \vec{B} \end{align*}

Mathematically not quite correct, but in a physical way true the following rearrangement can be done:

\begin{align*} \vec{{\rm d}F}_{\rm L} &= {{{\rm d}Q \cdot {\rm d}\vec{l}}\over{{\rm d}t}} \times \vec{B} \\ &= {\rm d}Q \cdot {{{\rm d}\vec{l}}\over{{\rm d}t}} \times \vec{B} \\ &= {\rm d}Q \cdot {{{\rm d}\vec{l}}\over{{\rm d}t}} \times \vec{B} \\ \end{align*}

Here, the part ${{{\rm d}\vec{l}}\over{{\rm d}t}}$ represents the speed $\vec{v}$ of the small charge packet ${\rm d}Q$.

\begin{align*} \vec{{\rm d}F}_{\rm L} &= {\rm d}Q \cdot \vec{v} \times \vec{B} \end{align*}

The Lorenz Force on a finite charge packet is the integration:

\begin{align*} \boxed{\vec{F}_{\rm L} = Q \cdot \vec{v} \times \vec{B}} \end{align*}

Please have a look at the German contents (text, videos, exercises) on the page of the KIT-Brückenkurs >> 5.2.3 Lorentz-Kraft. Make sure that Gesamt is selected in the selection bar at the top. The last part „Magnetic field within matter“ can be skipped.

3.4 Matter in the Magnetic Field

The magnetic interaction is also represented via two fields in the previous subchapters similar to the electric field. Since the magnetic field strength $\vec{H}$ is only dependent on the field causing current $I$, this field is independent of surrounding matter.
The magnetic flux density $\vec{B}$ on the other hand, showed with the magnetic permeability a factor that can contain the material effects.

Both fields are connected. The Ampere's Force Law gave the formula

\begin{align*} |\vec{F}_{12}| &= {{\mu}\over{2 \pi}} \cdot {{I_2 }\over{r}} &\cdot I_1 \cdot l \\ &= B &\cdot I_1 \cdot l \\ \end{align*}

Therefore, the magnetic flux density $\vec{B}$ is equal to:

\begin{align*} B &= {{\mu}\over{2 \pi}} \cdot {{I_2 }\over{r}} \end{align*}

$I_2$ was here the field-causing current.
For the field strength of the straight conductor, we had:

\begin{align*} H &= {{I}\over{2 \pi \cdot r }} \end{align*}

The connection between the two fields is also $B = \mu \cdot H$. Since the field lines of both fields are always parallel to each other it results to

\begin{align*} \boxed{ \vec{B} = \mu \cdot \vec{H} } \end{align*}

This is similar to the $\vec{D} = \varepsilon \cdot \vec{E}$. Similarly, the permeability is separated into two parts:

\begin{align*} \mu &= \mu_0 \cdot \mu_{\rm r} \end{align*}

where

• $\mu_0 $ is the permeability of the vacuum
• $\mu_{\rm r}$ is the relative permeability, for the case that there is material in the magnetic field

The material can be divided into different types by looking at its relative permeability. Abbildung 16 shows the relative permeability in the magnetization curve (also called $B$-$H$-curve). In this diagram, the different effect ($B$-field on $y$-axis) based on the causing external $H$-field (on $x$-axis) for different materials is shown. The three most important material types shall be discussed shortly.

Diamagnetic Materials

• Diamagnetic materials weaken the magnetic field, compared to the vacuum.
• The weakening is very low (see Tabelle 1).
• For diamagnetic materials applies $0<\mu_{\rm r}<1$
• The principle behind the effect is based on quantum mechanics (see Abbildung 17):
  • Without the external field no counteracting field is generated by the matter.
  • With an external magnetic field an antiparallel-orientated magnet is induced.
  • The reaction weakens the external field. This is similar to the weakening of the electric field by the dipoles of materials.
• Due to the repulsion of the outer magnetic field the material tends to move out of a magnetic field.
  For very strong magnetic fields small objects can be levitated (see clip).

A living insect („diamagnet“) floats in a very strong magnetic field

Paramagnetic Materials

• Paramagnetic materials strengthen the magnetic field, compared to the vacuum.
• The strengthening is very low (see Tabelle 2).
• For paramagnetic materials applies $\mu_{\rm r}>1$
• The principle behind the effect is again based on quantum mechanics (see Abbildung 18):
  • Without the external field no counteracting field is generated by the matter.
  • With an external magnetic field internal „tiny magnets“ based on the electrons in their orbitals are orientated similarly.
  • This reaction strengthens the external field.

Explanation of diamagnetism and paramagnetism

Ferromagnetic Materials

• Ferromagnetic materials strengthen the magnetic field strongly, compared to the vacuum.
• The strengthening can create a field multiple times stronger than in a vacuum.
• For ferromagnetic materials applies $\mu_{\rm r} \gg 1$
• The principle behind the effect is based on quantum mechanics:
  • Also here internal „tiny magnets“ based on the electrons in their orbitals are oriented based on the external field.
  • However, these „tiny magnets“ interact with each other in such a way that the field becomes stable even without the external field.
  • Ferromagnetic materials can be permanently magnetized. They build up permanent magnets.
  • The magnetic flux density of the ferromagnetic matter is depending on the external field strength and the history of the material.
  • The dependency between $\mu_{\rm r}$ and the external field strength $H$ is nonlinear.
• Ferromagnetic materials are characterized by the magnetization curve (see Abbildung 19)
  • Non-magnetized ferromagnets are located in the origin.
  • With an external field $H$ the initial magnetization curve (in German: Neukurve, dashed in Abbildung 19) is passed.
  • Even without an external field ($H=0$) and the internal field is stable.
    The stored field without external field is called remanence $B(H=0) = B_{\rm R}$ (or remanent magnetization).
  • In order to eliminate the stored field the counteracting coercive field strength $H_{\rm C}$ (also called coercivity) has to be applied.
  • The saturation flux density $B_{\rm sat}$ is the maximum possible magnetic flux density (at the maximum possible field strength $H_{\rm sat}$)

Explanation of the hysteresis curve

The ferromagnetic materials can again be subdivided into two groups: magnetically soft and magnetically hard materials.

magnetically soft ferromagnetic Materials

• high permeability
• high saturation flux density $B_{\rm sat}$ (= good magnetic conductor)
• small coercive field strength $H_{\rm C} < 1 ~\rm kA/m$ (= easy to reverse the magnetization)
• low magnetic losses for reversing the magnetization
• Important materials: ferrites

Applications:

• mainly for coil core material
• rotor and stator material for electric motors
• relays, contactors, transformers, compact antennas

Magnetically hard ferromagnetic Materials

• low permeability
• low saturation flux density $B_{\rm sat}$ (= good magnetic conductor)
• high coercive field strength $H_{\rm C}$ up to $2'700~\rm kA/m$ (= easy to reverse the magnetization)
• high magnetic losses for reversing the magnetization
• Important materials: ferrites, alloys of iron and cobalt: $\rm AlNiCo$²⁾, $\rm SmCo$³⁾, $\rm NdFeB$⁴⁾

Applications:

• mainly for permanent magnets
• permanent electric motors
• loudspeakers and microphones
• mechanical actuators

Wandering magnetic domains in a ferromagnetic material when the external field is reversed (from [email protected]).

3.5 Poynting Vector (not part of the curriculum)

• Clear picture of the Poynting vector along an electric circuit: https://de.cleanpng.com/png-jyy1vj/
• Good explanation of the Energy flow via a current model: http://amasci.com/elect/poynt/poynt.html
• Very detailed view of the energy flow in an electric circuit: http://sharif.edu/~aborji/25733/files/Energy%20transfer%20in%20electrical%20circuits.pdf

Tasks

Task 1

References to the media used

Element	License	Link
Abbildung 21	CC-BY-SA 3.0	https://en.wikipedia.org/wiki/Magnetite?oldformat=true
Abbildung 2	Public Domain	https://commons.wikimedia.org/wiki/File:Magnetic_field_of_bar_magnets_attracting.png
Abbildung 10	CC-BY-SA 3.0	https://commons.wikimedia.org/wiki/File:VFPt_Solenoid_correct.svg