N2018 P1 Q10

An electrical circuit comprises a power source of \(V\) volts in series with a resistance of \(R\) ohms, a capacitance of \(C\) farads and an inductance of \(L\) henries. The current in the circuit, \(t\) seconds after turning on the power, is \(I\) amps and the charge on the capacitor is \(q\) coulombs. The circuit can be used by scientists to investigate resonance, to model heavily damped motion and to tune into radio stations on a stereo tuner. It is given that \(R,C\) and \(L\) are constants, and that \(I=0\) when \(t=0\).

A differential equation for the circuit is \(L\frac{\mathrm{d}I}{\mathrm{d}t}+RI+\frac{q}{C}=V\), where \(I=\frac{\mathrm{d}q}{\mathrm{d}t}\).

1. Show that, under certain conditions on \(V\) which should be stated, \(L\frac{{{\mathrm{d}}^{2}}I}{\mathrm{d}{{t}^{2}}}+R\frac{\mathrm{d}I}{\mathrm{d}t}+\frac{I}{C}=0\)

   [2]

It is now given that the differential equation in part (i) holds for the rest of the question.

2. Given that \(I=\mathrm{A}t{{e}^{-\frac{Rt}{2L}}}\) is a solution of the differential equation, where \(\mathrm{A}\) is a positive constant, show that \(C=\frac{4L}{{{R}^{2}}}\).[5]
3. In a particular circuit, \(R=4,\,L=3\) and \(C=0.75\). Find the maximum value of \(I\) in terms of \(A\), showing that this value is a maximum.[4]
4. Sketch the graph of \(I\) against \(t\).[2]