NYJC EJC 2026 RLC Circuits (Oscillatory) Tutorial

Page 1 of 6 9814 (202 6) H3 Physics H305 Capacitors and Inductors (Oscillatory) – Tutorial H3 Physics Topic 5 - Capacitors and Inductors (Oscillatory) 11 A series circuit consists of a capacitor of capacitance C, an inductor of inductance L and a switch, as shown in Fig. 11.1. Fig. 11.1 Initially, the switch is open and the capacitor holds a charge Qo. (a) Compare how the energy is stored in a capacitor and in an inductor. C L

Page 2 of 6 9814 (202 6) H3 Physics H305 Capacitors and Inductors (Oscillatory) – Tutorial (b) The switch is closed at time t = 0 and the capacitor begins to discharge through the inductor. Show that the charge q on the capacitor obeys the equation 2 2 1 0dq q dt LC += .

Page 3 of 6 9814 (202 6) H3 Physics H305 Capacitors and Inductors (Oscillatory) – Tutorial (c) (i) Deduce an expression for the natural frequency fo of the circuit. (ii) Determine an expression for the charge q as a function of time, q(t). (iii) On Fig. 11.2, draw two separate graphs to show the variation with time of the charge q on the capacitor and the current I in the circuit. Fig. 11.2 0 time

Page 4 of 6 9814 (202 6) H3 Physics H305 Capacitors and Inductors (Oscillatory) – Tutorial (iv) Show that the total energy ET of the circuit for t > 0 is given by the expression 2 2 o T QE C = (d) A resistor of resistance R is now added in series with the capacitor and inductor as shown in Fig. 9.3. Fig. 11.1 The capacitor is again fully charged and the switch is closed at time t = 0. The charge q on the capacitor obeys the equation 2 2 1 0dq q dt LC R dq L dt += + Assuming that 2 LR C << , an expression of the form ( )cos t oq Qe t γ ωϕ − = + is a general solution of this equation. C L R

Page 5 of 6 9814 (202 6) H3 Physics H305 Capacitors and Inductors (Oscillatory) – Tutorial (i) Determine the expressions for the constants γ in terms of R and L, and ω in terms of γ and ωo where 2oo fωπ= and fo is the natural frequency of the circuit in (c)(i).

Page 6 of 6 9814 (202 6) H3 Physics H305 Capacitors and Inductors (Oscillatory) – Tutorial (ii) Derive an expression for the half -life time τ of the circuit, when the energy of the system drops to half of its initial value.

Page 1 of 7 9814 (202 6) H3 Physics H305 Capacitors and Inductors (Oscillatory) – Tutorial Solutions H3 Physics Topic 5 - Capacitors and Inductors (Oscillatory) Name: ____________________________________________ Class: ______________ 11 A series circuit consists of a capacitor of capacitance C, an inductor of inductance L and a switch, as shown in Fig. 11.1. Fig. 11.1 Initially, the switch is open and the capacitor holds a charge Qo. (a) Compare how the energy is stored in a capacitor and in an inductor. C L In a capacitor, the energy is stored in the electric field produced by the oppositely charged plates. In an inductor, the energy is stored in the magnetic field produced by the current flowing through it.

Page 2 of 7 9814 (202 6) H3 Physics H305 Capacitors and Inductors (Oscillatory) – Tutorial Solutions (b) The switch is closed at time t = 0 and the capacitor begins to discharge through the inductor. Show that the charge q on the capacitor obeys the equation 2 2 1 0dq q dt LC += . The total energy stored by the capacitor and inductor at any time 2 211 22 TU LI C q= + Since there are no energy losses: 2 2 22 2 2 2 2 0 11 0 22 11 0 22 0 1 0 TdU dt dq LI dt C d q dq L dt C dt q dq dq d qL C dt dt dt dq q dt LC = += += += =        +

Page 3 of 7 9814 (202 6) H3 Physics H305 Capacitors and Inductors (Oscillatory) – Tutorial Solutions (c) (i) Deduce an expression for the natural frequency fo of the circuit. By comparing to the defining equation for a simple harmonic motion: 2 2 2 1 oo dx x dt LC ωω= − ⇒= 1 2 2 o of LC ω π π ∴== (ii) Determine an expression for the charge q as a function of time, q(t). Since at t = 0, q = Q o, ( )cos 1cos o o oqQ t Qt LC ω∴= =    (iii) On Fig. 9.2, draw two separate graphs to show the variation with time of the charge q on the capacitor and the current I in the circuit. Fig. 9.2 0 time q(t) I(t) Q0

Page 4 of 7 9814 (202 6) H3 Physics H305 Capacitors and Inductors (Oscillatory) – Tutorial Solutions (iv) Show that the total energy ET of the circuit for t > 0 is given by the expression 2 2 o T QE C = ( )( ) ( ) ( )( ) ( ) 22 2 2 22 2 2 1The energy stored in the capacitor cos cos 22 2 The energy stored in the inductor sin sin 22 2 1, o Co o Lo QqU Qt t CC C QLLU Qt t C LC ωω ωω ω ω= = = = = = = I ( ) ( ) ( ) ( )( ) 22 22 2 22 2 cos sin 22 cos sin 2 (shown) 2 T CL oo o o UUU QQ tt CC Q tt C Q C ωω ωω = + = + = + = (d) A resistor of resistance R is now added in series with the capacitor and inductor as shown in Fig. 9.3. Fig. 9.3 The capacitor is again fully charged and the switch is closed at time t = 0. The charge q on the capacitor obeys the equation 2 2 1 0dq q dt LC R dq L dt += + Assuming that 2 LR C << , an expression of the form ( )cos t oq Qe t γ ωϕ − = + is a general solution of this equation. C L R