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:
1. Calculate the magnitude of the resulting force on one coil!
Path
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*}
Result
$|\vec{F}_{L}|= 100~\rm N$
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.
Path
The orientations are as following:
The resulting force has to be perpendicular to $B$-field and conductor.
Result
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
3. Does the Lorentz force lift the shuttle for a homogeneous $B$-field of the shuttle? Explain.
Result
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.