Exercise E6 Lorentz Force
(written test, approx. 8 % of a 120-minute written test, SS2024)
A robotic shuttle system uses magnets to lift the mobile shuttle over the fixed floor. To do so, coils in the floor repel the permanent magnets of the mobile shuttle (see image).
A single coil shall have the following properties:
- $N=500$ windings
- radius $r=40 ~\rm mm$
- current $I = 1.6 ~\rm A$
- magnetic field of the shuttle of is homogeneous with $B=0.5 ~\rm T$
1. Calculate the magnitude of the resulting force on one coil!
The Lorentz force on a conductor the length $l$ and the current $I$ in a $B$-field is
\begin{align*}
|\vec{F}_{L}| &= I \cdot l \cdot B \cdot \cos(\angle \vec{B}, \vec{l}) \\
&= I \cdot (N \cdot 2\pi r) \cdot B \cdot \cos(\angle \vec{B}, \vec{l}) \\
&= \rm 1.6 ~A \cdot (500 \cdot 2\pi 40\cdot 10^{-3}~m) \cdot 0.5 ~T \cdot \cos 90°
\end{align*}
$|\vec{F}_{L}|= 100~\rm N$
2. For one winding of the left coil, the cross-sections are marked in bold in the image. Draw the resulting force vectors into the image for each side of the winding.
The orientations are as following:
- $B$-field downwards
- conductors inwards and outwards
The resulting force has to be perpendicular to $B$-field and conductor.
Since the resulting force has to be perpendicular to $B$-field and conductor, the force has to point to the left or the right.
The right-hand rule leads to forces pointing radially into the coil
The right-hand rule leads to forces pointing radially into the coil
3. Does the Lorentz force lift the shuttle for a homogeneous $B$-field of the shuttle? Explain.
No. For a homogeneous $B$ field („constant magnetic field of the shuttle“), the Lorentz forces cancel each other out.
The Lorentz force can only have a lifting effect in an inhomogeneous field.
In this case, there sum of the forces results in a repulsing force, see image.
Beside boundary effects, The field gets also inhomogeneous, by the additional field of the coils.
The Lorentz force can only have a lifting effect in an inhomogeneous field.
In this case, there sum of the forces results in a repulsing force, see image.
Beside boundary effects, The field gets also inhomogeneous, by the additional field of the coils.