A spherical capacitor is made of a metal ball of radius inside a concentric…

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 spherical capacitor is made of a metal ball of radius

a = 20.0\,\mathrm{mm}

inside a concentric conducting shell of inner radius

b = 25.0\,\mathrm{mm}

, with vacuum between them. A toroidal coil has

N = 800

turns wound on a ring of square cross-section: inner radius

4.0\,\mathrm{cm}

, outer radius

5.0\,\mathrm{cm}

, height

1.2\,\mathrm{cm}

, air-filled. Inside the winding the field is

B = \mu_0 N I/(2\pi s)

at distance

s

from the axis, and outside it is zero. ((a)) Find the capacitance of the spherical capacitor, in pF. ((b)) The capacitor is charged and then discharged through a

4.7\,\mathrm{M\Omega}

resistor with ideal leads of no resistance. How long does it take for its charge to fall to 10 per cent of its initial value? ((c)) The toroid carries a steady

1.5\,\mathrm{A}

. At

t = 0

a make-before-break switch (so the coil's current is never interrupted) connects the coil across a resistor, the total resistance of coil plus resistor being

6.8\,\Omega

, and the source is removed. How long does it take the current to fall to half of

1.5\,\mathrm{A}

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