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Exercise E1 Magnetic Flux Density
(written test, approx. 6 % of a 120-minute written test, SS2021)
An electric motor is operated for experiments in the laboratory. An alternating current with an amplitude of $\hat{I} = 100~\rm A$ is operated.
You stand next to it and think about whether you have any health problems to worry about. The figure below shows the supply line between $\rm A$ and $\rm B$ and its position.
$\mu_{0} = 4\pi\cdot 10^{-7} {{\rm Vs}\over{\rm Am}}$, $\mu_{r}=1$
a) What is the highest magnetic flux density through the line in your body? (3 points)
The magnetic field strength for a conducting wire is given as:
\begin{align*} H &= {{I}\over{2\pi \cdot r}} \end{align*}
The magnetic flux density $B$ is given as: $B = \mu_0 \mu_r H$
Here, the maximum current is $\hat{I} = 100~\rm A$ and the distance to the cable is $r = \sqrt{(0.1 {~\rm m})^2 + (0.4 {~\rm m})^2}= 0.412... ~\rm m$.
Therefore: \begin{align*} B &= 4\pi\cdot 10^{-7} {{\rm Vs}\over{\rm Am}} \cdot 1 \cdot {{100 ~\rm A}\over{2\pi \cdot 0.412... ~\rm m}} \end{align*}
b) The limit value for the magnetic flux density at the frequency used is $B_0 = 100~\rm \mu T$.
At what distance around the conductor is this value exceeded? (3 points, independent)
The formula for the magnetic field strength can be rearranged: \begin{align*} H &= {{I}\over{2\pi \cdot r}} \\ r &= {{I}\over{2\pi \cdot H}} \\ \end{align*}
Again, the magnetic flux density $B$ is given as: $B = \mu_0 \mu_r H$
Therefore:
\begin{align*}
r &= \mu_0 \mu_r {{ I }\over{2\pi \cdot B}} \\
&= 4\pi\cdot 10^{-7} {{\rm Vs}\over{\rm Am}} {{100 ~\rm A}\over{2\pi \cdot 100\cdot 10^{-6} {~\rm T}}} \\
\end{align*}