A long air-cored solenoid is wound in many layers

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 long air-cored solenoid is wound in many layers, so that the winding fills the space between the radii

a = 2.0\,\mathrm{cm}

and

b = 2.5\,\mathrm{cm}

. Counting all the layers together there are

n = 8000

turns per metre of length, spread uniformly across the thickness of the winding, and each turn carries a steady current

I = 1.2\,\mathrm{A}

in the same sense. Treat the solenoid as infinitely long, so the field inside is parallel to the axis and independent of position along it, and take as known the chapter's result that the field outside a long solenoid is zero. Let

s

be the distance from the axis. ((a)) Derive the field

B

in the hollow region

s < a

, naming the Ampèrian loop and stating what is enclosed, and evaluate it. ((b)) Derive an expression for

B(s)

at a radius inside the winding,

a \le s \le b

, in terms of

\mu_0

n

I

a

b

and

s

. ((c)) Evaluate

B

s = 2.25\,\mathrm{cm}

and at

s = 3.0\,\mathrm{cm}

, and check that your expression in (b) joins correctly onto the answers at

s = a

and

s = b

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