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. dl A differential equation for the circuit is L- +RI+- + 2=V, where I dq dt C dt (i) Show that, under certain conditions on V which should be stated, d²I dt² I C L +R- +- = 0. dt It is now given that the differential equation in part (i) holds for the rest of the question. (ii) Rt [2] Given that I = Ate 2L is a solution of the differential equation, where A is a positive constant, show that C = 4L R² [5] (iii) In a particular circuit, R = 4, L = 3 and C = 0.75. Find the maximum value of I in terms of A, show that this value is a maximum. (iv) Sketch the graph of I against t. [4] [2]
