Show that the instantaneous voltages vr vl and vc at any

Question

A resistor, inductor, and 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 = Vcos ωt.

(a) Show that the instantaneous voltages v_R, v_L, and v_C at any instant arc each 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, the inductor, and the 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 given by I = √(I_R² + (I_C - I_L))².

(d) Show that the result of part (c) can be written as I = V/Z with 1/Z = √(1/R² + (ωC -1/ωL))².