A piece of air-spaced coaxial cable is long

PHYSICS · Multi-part · Fermi University Physics · Electricity, Magnetism & Electromagnetism · Conductors, capacitance and boundary-value electrostatics · Electric current and circuits

Electricity, Magnetism and Electromagnetism, Ch Transients: RC, RL and LC circuits

Problem

A piece of air-spaced coaxial cable is

l = 25\,\mathrm{m}

long. The inner conductor has radius

a = 0.50\,\mathrm{mm}

and the inner surface of the outer conductor has radius

b = 3.5\,\mathrm{mm}

. Treat the space between the conductors as vacuum, treat the cable as a long cylindrical capacitor (end effects ignored), and ignore the resistance and inductance of the conductors and leads. The cable, initially uncharged and open at its far end, is charged from an ideal

60\,\mathrm{V}

supply through a

2.2\,\mathrm{M\Omega}

resistor; an ideal switch closes at

t = 0

. A separate air-cored solenoid is wound with

N = 500

turns evenly over a length

l_s = 0.20\,\mathrm{m}

on a cylinder of radius

r = 1.5\,\mathrm{cm}

. Treat it as a long solenoid (uniform field inside, negligible field outside) whose only imperfection is its winding resistance

2.4\,\Omega

. At

t = 0

an ideal switch connects it to an ideal

12\,\mathrm{V}

battery, the current then being zero. ((a)) Find the capacitance of the cable, in pF. ((b)) How long after the switch closes does the cable hold 90 per cent of its final charge? ((c)) How long after its switch closes does the current in the solenoid reach 95 per cent of its final value?

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