Question

The figure shows the cross section of a toroid with a rectangular cross-section. The inner and outer radii of the toroid are $R_1$ and $R_2$, and $h$ is the height of the toroid. The coil consist on N turns and the core has a permeability $\mu$. The H-field lines and current $i$ flowing through the toroid are shown. a) Apply Ampère's law to find the magnetic field intensity as a function of the distance $r$. Where $r$ is the distance from the central axis of the toroid: 1) For $r < R_1$ in the air. 2) For $R_1 < r < R_2$ inside the toroid. 3) For $r > R_2$ in the air. b) Use the infinitesimal area element $da = h\,dr$ shown to find the flux $\Phi$ in the core. c) Find the inductance $L$ using the flux linkage.

          The figure shows the cross section of a toroid with a rectangular cross-section. The inner and
outer radii of the toroid are $R_1$ and $R_2$, and $h$ is the height of the toroid. The coil consist on N
turns and the core has a permeability $\mu$. The H-field lines and current $i$ flowing through the
toroid are shown.

a) Apply Ampère's law to find the magnetic field intensity as a function of the distance
$r$. Where $r$ is the distance from the central axis of the toroid:
1) For $r < R_1$ in the air.
2) For $R_1 < r < R_2$ inside the toroid.
3) For $r > R_2$ in the air.
b) Use the infinitesimal area element $da = h\,dr$ shown to find the flux $\Phi$ in the core.
c) Find the inductance $L$ using the flux linkage.

The figure shows the cross section of a toroid with a rectangular cross-section. The inner and
outer radii of the toroid are R1 and R2, and h is the height of the toroid. The coil consist on N
turns and the core has a permeability μ. The H-field lines and current i flowing through the
toroid are shown.

a) Apply Ampère's law to find the magnetic field intensity as a function of the distance
r. Where r is the distance from the central axis of the toroid:
1) For r < R1 in the air.
2) For R1 < r < R2 inside the toroid.
3) For r > R2 in the air.
b) Use the infinitesimal area element da = h dr shown to find the flux Φ in the core.
c) Find the inductance L using the flux linkage.

Added by Alexis A.

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