4. You have seen how Kirchhoff's laws were used in your lectures to obtain a 2nd order differential equation where we solved for the current. This time we will use an even simpler concept: principle of conservation of energy to derive the 2nd order differential equation where we will solve for the charge. Take a look at the circuit below. 5H 2F In the circuit above, we have a capacitor with capacitance 2 F, an inductor of inductance 5 H and a resistor of 32 (a) The total energy that is supplied to the resistor is LI? E = Q? where L is the inductance, I is the current, C is the capacitance and Q is the charge. Write down the total energy supplied E in terms of Q and t only. dQ Remember that I = (b) Now you know that the power dissipation through a resistor is -1R. Use the conservation of energy (energy gain rate = energy loss rate) to derive the differential equation in terms Q and t only. (c) Solve the differential equation for initial charge to be 0C with a initial current of 2q/s, where q is some constant amount of charge measured in Coulombs. (d) Given that the coefficient of your sine function is the time-dependent amplitude (for example A(t) is the amplitude of the function A(t) sin t). At what time T will the amplitude of the charge oscillations in the circuit be 75% of its initial value? ப99-

4. You have seen how Kirchhoff's laws were used in your lectures to obtain a 2nd order differential equation where we solved for the current. This time we will use an even simpler concept: principle of conservation of energy to derive the 2nd order differential equation where we will solve for the charge. Take a look at the circuit below. 5H 2F In the circuit above, we have a capacitor with capacitance 2 F, an inductor of inductance 5 H and a resistor of 32 (a) The total energy that is supplied to the resistor is LI? E = Q? where L is the inductance, I is the current, C is the capacitance and Q is the charge. Write down the total energy supplied E in terms of Q and t only. dQ Remember that I = (b) Now you know that the power dissipation through a resistor is -1R. Use the conservation of energy (energy gain rate = energy loss rate) to derive the differential equation in terms Q and t only. (c) Solve the differential equation for initial charge to be 0C with a initial current of 2q/s, where q is some constant amount of charge measured in Coulombs. (d) Given that the coefficient of your sine function is the time-dependent amplitude (for example A(t) is the amplitude of the function A(t) sin t). At what time T will the amplitude of the charge oscillations in the circuit be 75% of its initial value? ப99-