A toroid is wound tightly and uniformly with turns on a non-magnetic former of…

PHYSICS · Multi-part · Fermi University Physics · Electricity, Magnetism & Electromagnetism · Magnetostatics · Electromagnetic induction and alternating-current circuits

Electricity, Magnetism and Electromagnetism, Ch Self and mutual inductance and magnetic field energy

Problem

A toroid is wound tightly and uniformly with

N = 500

turns on a non-magnetic former of rectangular cross-section: inner radius

a = 4.0\ \mathrm{cm}

, outer radius

b = 6.0\ \mathrm{cm}

, height

h = 2.0\ \mathrm{cm}

. The winding carries a current

I = 3.0\ \mathrm{A}

. Take the field to be confined to the interior of the toroid, circling the axis, with magnitude depending only on the distance

s

from the axis. ((a)) Using Ampère's law and the flux through the rectangular cross-section, derive an expression for the self-inductance

L

in terms of

\mu_0

N

h

a

and

b

, and evaluate it. ((b)) Evaluate the stored energy

\tfrac12 LI^2

. ((c)) Integrate the magnetic energy density

u = B^2/2\mu_0

over the volume of the toroid and show that the result equals your answer to (b).

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