A toroid of rectangular cross-section is wound with turns on an air former whose…

PHYSICS · Multi-part · Fermi University Physics · Electricity, Magnetism & Electromagnetism · Magnetostatics · Magnetostatics

Electricity, Magnetism and Electromagnetism, Ch Ampère's law, magnetic flux and the shape of the magnetic field

Problem

A toroid of rectangular cross-section is wound with

N = 600

turns on an air former whose inner radius is

4.0\,\mathrm{cm}

, outer radius

6.0\,\mathrm{cm}

and height

2.0\,\mathrm{cm}

. The winding carries a steady current

I = 0.80\,\mathrm{A}

. Along the symmetry axis of the toroid (the vertical axis through the hole) runs a long straight solid conductor of radius

R = 1.0\,\mathrm{cm}

, carrying a steady current

I_0 = 40\,\mathrm{A}

vertically upwards, distributed uniformly over its cross-section. The conductor is insulated from the winding, and both the winding's leads and the conductor's return path are so far away that their fields may be ignored here. On the inner side of each turn of the winding (at radius

4.0\,\mathrm{cm}

) the current also flows upwards. Let

s

be the distance from the axis, and work in the horizontal mid-plane of the core. ((a)) Explain why one family of Ampèrian loops serves for both currents, and derive expressions for

B(s)

in the four regions

s < R

R < s < 4.0\,\mathrm{cm}

4.0\,\mathrm{cm} < s < 6.0\,\mathrm{cm}

and

s > 6.0\,\mathrm{cm}

. ((b)) Evaluate

B

s = 2.5\,\mathrm{mm}

s = 2.0\,\mathrm{cm}

s = 5.0\,\mathrm{cm}

and

s = 8.0\,\mathrm{cm}

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