The L - R - C Parallel Circuit. A...

The L - R - C Parallel Circuit. A resistor, an inductor, and a capacitor are connected in parallel to an ac source with voltage amplitude V and angular frequency ω. Let the source voltage be given by v = V cos ωt. (a) Show that each of the instantaneous voltages v_R, v_L, and v_C at any instant is equal to v and that i = i_R + i_L + i_C, where i is the current through the source and i_R, i_L, and i_C are the currents through the resistor, inductor, and capacitor, respectively. (b) What are the phases of i_R, i_L, and i_C with respect to v? Use current phasors to represent i, i_R, i_L, and i_C. In a phasor diagram, show the phases of these four currents with respect to v. (c) Use the phasor diagram of part (b) to show that the current amplitude I for the current i through the source is (d) Show that the result of part (c) can be written as I = V/Z, with