Differential electricity
As a capacitor discharges, its charge changes continuously. Find
the differential equation governing this variation.
Age
16 to 18
Challenge level
Problem
V represents the voltage drop across the parallel combination of the capacitor (C) and resistor (R). Find the differential equation that V obeys in terms of $V_0$ and its derivatives.
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Make up some realistic constants for a capacitor. Can you solve the resulting differential equation?
Getting Started
V = L$\frac{dI}{dt}$
I = C$\frac{dV}{dt}$
I = C$\frac{dV}{dt}$
Student Solutions
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Kirchoff's Voltage Law:
$V_0 = V_L + V_R + V = L\frac{dI_1}{dt} + IR + V$
Current conservation at node X:
$I_1 = I_2 + I_3 $
$I_2 = I_C = C\frac{dV}{dt}$
$I_3 = \frac{V}{R}$
$I_1 = C\frac{dV}{dt} + \frac{V}{R} $
$\frac{dI_1}{dt} = C\frac{d^2V}{dt^2} +\frac{1}{R} \frac{dV}{dt}$
$V_0 = L(C\frac{d^2V}{dt^2} +\frac{1}{R} \frac{dV}{dt}) + (C\frac{dV}{dt} + \frac{V}{R})R + V = LC \frac{d^2V}{dt^2} + (\frac{L}{R} + CR)\frac{dV}{dt} + 2V$
The governing differential equation is:
$LC \frac{d^2V}{dt^2} + (\frac{L}{R} + CR)\frac{dV}{dt} + 2V = V_0 $