An a.c. source generating voltage e = e 0 sinωt is connected to a capacitor of capacitance C . Find the expression for the current i flowing through it. Plot a graph of e and i versus ω t .

Given the a.c source generating a voltage e = e 0 sin ωt . The capacitance of the capacitor = C . The current through the circuit i = ? The potential difference across the capacitor due to accumulated charge is, e = q C or q = eC = Ce 0 sinωt . The value of electric current at any time is, I = dq dt = d dt Ce 0 sin ωt = Ce 0 ω cosωt The impedance is given by X C = 1 ωC , hence, we get, = e 0 1 ωc cos ωt or I = I o cos ωt ................................(A This is the required expression., This can be written as I = I o sin ωt + π 2 This shows that the current is ahead of the voltage by π 2 . Therefore the graph between current I and voltage ς versus ω t is shown as:

Derive an expression for the impedance of a series LCR circuit, when an AC voltage is applied to it.

Reactance: The effective opposition offered by the component of the A.C. circuit to the flow of current in the circuit is referred to be Reactance. Impedance: The effective opposition offered by combination of resistors, inductors and capacitors to the flow of current is referred to be Impedance. Instantaneous current in L - C - R circuit: Suppose, I be the root-mean-square value of current flowing through all the circuit elements. Potential difference across Inductor L : V L = I X L ............................ 1 Potential difference across Capacitor C : V C = I X C ............................ 2 Potential difference across Resistor R : V R = I R ............................ 3 Since, V R and I are in phase. So, V R is represented by O A in the direction of I . The current lags behind the V L , by angles of π 2 . So, V L is represented by O B perpendicular to the direction of I . The current leads V C by angle of π 2 . So, V C is represented by O F perpendicular to the direction of I . Suppose, V L > V C . Then, resultant of V R and V L - V C is given by O H O H = O A 2 + O D 2 ⇒ V = V R 2 + V L - V C 2 ............................. 4 Substituting equation 1 , 2 and 3 in equation 4 we get, V = I 2 R 2 + I X L - I X C 2 ⇒ V I = R 2 + X L - X C 2 I = V R 2 + X L - X C 2 Here, Z = R 2 + X L - X C 2 is the impedance. Now, tan ϕ = V L - V C V R = I X L - X C I R = X L - X C R ⇒ ϕ = tan - 1 X L - X C R Hence the expression for impedance in series L - C - R has been established.