An ideal, resistance-free circuit with and carries charge on the capacitor obeying ,…

PHYSICS · Multi-part · Fermi University Physics · Mathematical Methods for Physics · Electricity, Magnetism & Electromagnetism · Ordinary differential equations

Conditions of Validity, Ch Linear ordinary differential equations: existence, the Wronskian, and a kernel in disguise

Problem

An ideal, resistance-free

LC

circuit with

L=5.0\,\mathrm{mH}

and

C=2.0\,\mu\mathrm{F}

carries charge

q(t)

on the capacitor obeying

L\ddot{q}+q/C=V(t)

, with

q(0)=0

\dot{q}(0)=0

. A voltage

V(t)=V_0=5.0\,\mathrm{V}

is applied at

t=0

and held on for

\tau=0.20\,\mathrm{ms}

, then switched off. ((a)) Write down the initial-value Green function

G(t,\xi)

of the operator

L\,\mathrm{d}^2/\mathrm{d}t^2+1/C

in terms of

\omega_0=1/\sqrt{LC}

, and verify that its derivative jump at

t=\xi

equals

1/L

. ((b)) Using

q(t)=\int_0^tG(t,\xi)V(\xi)\,\mathrm{d}\xi

, derive a closed form for

q(t)

valid while the voltage is on,

0\le t\le\tau

, and show it can be written as

q(t)=CV_0(1-\cos\omega_0t)

. ((c)) Evaluate

q(\tau)

and compare it with the static charge

CV_0

about which the charge oscillates while the voltage is held.

Related theory

Ordinary differential equations

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