A coaxial cable consists of an inner solid cylindrical conductor of radius and an…

PHYSICS · Multi-part · A Level · Physics · Electricity · Electricity

Problem

A coaxial cable consists of an inner solid cylindrical conductor of radius

a

and an outer hollow cylindrical conductor of inner radius

b

and outer radius

c

. The space between

a

and

b

is filled with an insulator. At an instant in time, equal numbers of conduction electrons move to the right in the inner conductor and to the left in the outer conductor. The electron drift speed in the inner conductor is

v_{\text{in}}

and in the outer conductor is

v_{\text{out}}

. The number density of free electrons is the same in both conductors and equal to

n

. (a) Derive an expression for the magnitude of the instantaneous conventional current in the inner conductor in terms of

n

e

v_{\text{in}}

, and

a

. (b) Using your result from part (a), determine the value of

v_{\text{out}}

required for the net current along the cable axis to be zero. (c) Obtain an expression for the total kinetic energy per unit length associated with the moving electrons in both conductors when the condition found in part (b) is satisfied. (d) For

a=1.0\,\text{mm}

b=1.5\,\text{mm}

c=2.0\,\text{mm}

n=8.5\times10^{28}\,\text{m}^{-3}

, and

v_{\text{in}}=2.0\times10^{-3}\,\text{m}\,\text{s}^{-1}

, calculate the numerical value of the kinetic energy per unit length found in part (c). (e) Explain why the presence of this non-zero kinetic energy does not contradict the measurement of zero net current along the cable.

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