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 as a „causer field“ (a field caused by magnets) and an „acting field“ (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 at 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. $M = const. \neq f(\varphi)$ : 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 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 $[H] = {{[I]}\over{[l]}}= 1{{A}\over{m}}$

magnetic Field Strength part 2: straight conductor

The previous derivation from the toroidal coil is now to be used to derive the field strength around a long, straight conductor. For a single conductor the parte $N \cdot I$ 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 ={{\theta}\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 the magnetic field is the current in the winding 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 = N \cdot I} \end{align*} The unit of $\theta$ is: $[\theta]= 1A$ (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 potential difference $\theta$. The naming refers to the Electromotive Force, which depicts the ability of a voltage source to be able to drive a current in order to build a a defined voltage. Both „forces“ shall not be confused with the mechanical force $\vec{F}= m \cdot \vec{a}$, but 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 rotationally symmetric problems could be solved. Now this shall be generalized. For this purpose we will have a look back to 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 ifs given by adding up the field strength along the path of a probe from one plate to the other. This was extended to $U = \int_s E ds$. If we pompare this idea to the magnetic field strength $H$ of a toroidal coil with the average field line 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 ($V_m = H \cdot s$) 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 $ds$ along a field line, the resulting infinitesimally small magnetic potential difference $dV_m = H \cdot ds$ 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_{m12} = V_m(\vec{s_1}, \vec{s_2}) = \int_\vec{s_1}^\vec{s_2} H(\vec{s}) ds \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 $d \vec{s}$. This is made possible by scalar multiplication. Thus, it is generally valid:

\begin{align*} \boxed{V_{m12} = \int_\vec{s_1}^\vec{s_2} \vec{H} \cdot 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{\int_{closed \\ path} \vec{H} \cdot d \vec{s} = \iint_{enclosed \\ surface \; A} \vec{S} 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 gib
  • 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: $\theta = \iint_{enclosed \\ surface \; A} \vec{S} d\vec{A}$
• $d\vec{s}$ and $d\vec{A}$ build a right hand system: once the thumb of the right hand is pointing along $d\vec{A}$, the fingers of the right hand show the correct direction for $d\vec{s}$ for positive $\vec{H}$ and $\vec{S}$

Recap: Application of magn. Field Strength

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

• The closed path $C$ is on revolution of a field line in the center of the coil
• the surface $A$ is the enclosed surface

This leads to:

\begin{align*} \int_C \vec{H} \cdot d \vec{s} &= \iint_{A} \vec{S} d\vec{A} = \theta \end{align*}

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

\begin{align*} \int_C H \cdot ds &= \iint_{A} \vec{S} d\vec{A} \end{align*}

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

\begin{align*} \int_C H \cdot ds &= N\cdot I \end{align*}

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

\begin{align*} H \cdot \int_C ds &= 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 toroidel 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 subchapter the field effect on a magnet caused by currents was analyzed. Now, the field acting onto currents will get deeper investigation. In order 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 differenct 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} {{Vs}\over{Am}} = 1.257 \cdot 10^{-7} {{Vs}\over{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 onto 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 $I_1$ is summarized to $B$ - the magnetic flux density. $B$ has the unit: \begin{align*} [B] &= {{[F]}\over{[I]\cdot[l]}} = 1 {{N}\over{Am}} = 1 {{{VAs}\over{m}}\over{Am}} = 1 {{Vs}\over{m^2}} \\ &= 1 T \quad\quad(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 $\vec{F}$ by means of vector mutiplication to:

\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 a (elementary) charge.
In order to find this force the previous force onto a conductor can be used as a start. However, the formula will be investigated infinidesimally for small parts $\vec{dl}$ of the conductor:

\begin{align*} \vec{dF_L} = I \cdot \vec{dl} \times \vec{B} \end{align*}

The current is now substituted by $I = dQ/dt$, where $dQ$ is the small charge packet in the length $\vec{dl}$ of the conductor.

\begin{align*} \vec{dF_L} = {{dQ}\over{dt}} \cdot \vec{dl} \times \vec{B} \end{align*}

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

\begin{align*} \vec{dF_L} &= {{dQ \cdot \vec{dl}}\over{dt}} \times \vec{B} \\ &= dQ \cdot {{\vec{dl}}\over{dt}} \times \vec{B} \\ &= dQ \cdot {{\vec{dl}}\over{dt}} \times \vec{B} \\ \end{align*}

Here, the part ${{\vec{dl}}\over{dt}}$ represents the speed $\vec{v}$ of the small charge packet $dQ$.

\begin{align*} \vec{dF_L} &= dQ \cdot \vec{v} \times \vec{B} \end{align*}

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

\begin{align*} \boxed{\vec{F_L} &= Q \cdot \vec{v} \times \vec{B}} \end{align*}

Please have a look at the contents (text, videos, exercises) on the page of the KIT-Brückenkurs >> 4.2.3 Lorentz-Kraft. Make sure that „Total“ is selected in the selection bar at the top. The last part on „Magnetic field with matter“ can be skipped.

3.5 Matter in the magnetic field

3.6 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

Force effect on dia- and paramagnetic materials in the magnetic field.

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

Explanation of the hysteresis curve

Nice illustration of magnetization and demagnetization of soft magnetic material.

Wandering magnetic domains in a ferromagnetic material (from [email protected]).

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