An air-spaced coaxial cable has a solid inner conductor of radius and a thin…

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

An air-spaced coaxial cable has a solid inner conductor of radius

a = 0.80\ \mathrm{mm}

and a thin cylindrical outer sheath of inner radius

b = 4.0\ \mathrm{mm}

, coaxial with it. The space between them is vacuum. The cable is long enough for end effects to be ignored, and the flux and field energy inside the metal are neglected. A steady current

I = 6.0\ \mathrm{A}

flows along the inner conductor and returns along the sheath, and the conductors carry line charges

+\lambda

and

-\lambda

. Between the conductors (

a < s < b

, with

s

the distance from the axis) the fields are

B = \mu_0 I / 2\pi s

circling the axis and

E = \lambda/2\pi\varepsilon_0 s

pointing radially outwards; both fields are zero outside the sheath. ((a)) By finding the flux through a rectangle of length

1.0\ \mathrm{m}

lying between the conductors, with the inner and outer conductors as its long edges, obtain the inductance per unit length

L'

and evaluate it. ((b)) By integrating the magnetic energy density

u = B^2/2\mu_0

over the volume between the conductors, find the magnetic field energy per metre of cable, and check that it equals

\tfrac12 L' I^2

. ((c)) From the field

E

, obtain the capacitance per unit length

C'

and evaluate it; then find the electric field energy per metre when the potential difference between the conductors is

100\ \mathrm{V}

. ((d)) Evaluate the product

L'C'

, state its SI unit, and compute

1/\sqrt{L'C'}

with its unit.

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