Dies ist eine alte Version des Dokuments!

Block 17 — Magnetic Flux Density and Forces

Learning objectives

Preparation at Home

Well, again

• read through the present chapter and write down anything you did not understand.
• Also here, there are some clips for more clarification under 'Embedded resources' (check the text above/below, sometimes only part of the clip is interesting).

For checking your understanding please do the following exercises:

• …

90-minute plan

1. Warm-up (x min):
   1. ….
2. Core concepts & derivations (x min):
   1. …
3. Practice (x min): …
4. Wrap-up (x min): Summary box; common pitfalls checklist.

Conceptual overview

Core content

We know from block11 that a static charge $Q_1$ generate a static electric field $D$.
Before in block09, we developed that a static electric field $E= {{1}\over{\varepsilon}} D$ effects a force $F_C$ on a static charge $Q_2$

From the last chapter (block16) we got, that moving charges ${{d}\over{dt}}Q_1 = I_1$ generate a static magnetic field $H$.
So, how does an acting magnetic field effects a force on a moving charge ${{d}\over{dt}}Q_2 = I_2$?

Definition of the Magnetic Flux Density

To derive the forces, we do a step back to the images of field lines.
In Abbildung 1 a) the field lines of a single current-carrying wire is shown.
Abbildung 1 b) depicts the homogenous field of a coil.

When a current-carrying wire is within the homogenous field, we get the superimposed picture of both fields.
This leads to an enrichment of magnetic field on the left and an depletion on the right.
With the knowledge, that the field lines usually do not like to stay next to each other, one can conclude that there will be a force to the right.

When no current is flowing through the conductor the force is equal to zero.
The following is detectable:

1. $|\vec{F}| \sim I$ : The stronger the current, the stronger the force $F$.
2. $|\vec{F}| \sim l$ : As longer the conductor length $l$, as stronger the force $F$ gets.
3. $|\vec{F}| \sim {H}$ : As more current through the coil, as stronger the $H$-field. And a stronger the $H$-field leads to stronger force $F$.

To summarize: \begin{align*} F \sim H \cdot I \cdot l \end{align*}

The proportionality factor is $\mu_0$, the magnetic field constant or permeability : $\mu_0 = 4\pi \cdot 10^{-7} {\rm {Vs}\over{Am}}$.

\begin{align*} F = \mu_0 \cdot H \cdot I \cdot l \end{align*}

When adding an iron core into the coil the force $F$ gets stronger. Therefore, we include a material-dependent constant $\mu_r$, the so-called relative permeability

\begin{align*} F = \mu_0 \mu_r \cdot H \cdot I \cdot l \\ \boxed{F_L = \quad \: B \quad \: \cdot I \cdot l} \end{align*}

The new field quantity is $B$ the magnetic flux density

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 2).

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*} \boxed{ |\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 >> Lorentz-Kraft. The last part „Magnetic field within matter“ can be skipped.

Common pitfalls

• …

Exercises

Embedded resources