A long non-magnetic former of length and cross-sectional area carries two windings

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

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

Problem

A long non-magnetic former of length

l = 0.25\ \mathrm{m}

and cross-sectional area

A = 5.0\ \mathrm{cm^2}

carries two windings, each spread uniformly over the whole length: winding 1 with

N_1 = 400

turns and winding 2 with

N_2 = 600

turns, wound in the same sense. The two windings are joined in series, so the same current flows in both. End effects are ignored: inside the former the field is uniform and along the axis, outside it the field is zero. ((a)) Derive an expression for the self-inductance

L

of the series combination in terms of

\mu_0

N_1

N_2

A

and

l

, and evaluate it. ((b)) Evaluate the separate self-inductances

L_1

and

L_2

and the mutual inductance

L_{12}

, and verify

L = L_1 + L_2 + 2L_{12}

. ((c)) The current is

2.0\ \mathrm{A}

. Evaluate the stored energy from

\tfrac12 L I^2

and check it against

B^2/2\mu_0

times the interior volume.

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