{{tag>magnetostatic flux_density coil toroid exam_ee2_SS2021}}{{include_n>1180}}
 
#@TaskTitle_HTML@##@Lvl_HTML@#~~#@ee1_taskctr#~~ Toroidal Coil\\
<fs medium>(written test, approx. 5 % of a 120-minute written test, SS2021)</fs> #@TaskText_HTML@#

A magnetic field with a flux density of at least $50 ~\rm mT$ is to be achieved in a ring-shaped coil (toroidal coil). \\
The coil has 60 turns, wound around soft iron with $\mu_{\rm r} = 1200$. \\
The average field line length in the coil should be $l = 12  ~\rm cm$. 

$\mu_{0} = 4\pi\cdot 10^{-7} {{\rm Vs}\over{\rm Am}}$

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What is the minimum current that must flow through a single winding?

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The magnetic field strength of a toroidal coil is given as:

\begin{align*}
 H &= {{N \cdot I}\over{l}}
\end{align*}

Based on the flux density the magnetic field strength can be derived by $B = \mu_0 \mu_{\rm r} \cdot H$. \\
By this, the formula can be rearranged:

\begin{align*}
 H                                &= {{N \cdot I}\over{l}} \\
 {{B}\over{ \mu_0 \mu_{\rm r}}}   &= {{N \cdot I}\over{l}} \\
 I                                &= {{B \cdot l}\over{ \mu_0 \mu_{\rm r} \cdot N}}
\end{align*}

Putting in the numbers: 
\begin{align*}
 I                                &= {{ 0.05 {~\rm T} \cdot 0.12{~\rm m} }\over{ 4\pi\cdot 10^{-7} {{\rm Vs}\over{\rm Am}}  \cdot 1'200 \cdot 60}} \\
                                  &= 0.6631... {{\rm T\cdot m}\over{ {{\rm Vs}\over{\rm Am}} }} 
                                  &= 0.6631... {{\rm {{\rm Vs}\over{\rm m^2}} \cdot m}\over{ {{\rm Vs}\over{\rm Am}} }} 
                                  &= 0.6631... ~\rm A 
\end{align*}
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$I = 66 ~\rm mA$
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