Question

Suppose a circuit contains an electromotive force (a battery) that produces a voltage of E(t) volts (V), a capacitor with a capacitance of C farads (F), and a resistor with a resistance of Rohms (N). The voltage drop across the capacitor is where Q is the charge (in coulombs), so in this case Kirchhoff's Law gives RI + 8 = E(t). Since I we have er et de 2 – EC). ae dt Suppose the resistance is 3082, the capacitance is 0.15F, a battery gives a constant voltage of 60V, and the initial charge is Q(0) = 0 coulombs. Preview Find the charge and the current at time t. Q(t) - 9e^(-2/9(t))-9 I(t) = Preview

Answer 1

sol - Girun :- In a circuit containing an electromotive force ca battery) produces a voltage ECt) volto (), having a capacitor with capacitance of c frado (f) and a resistor with a resistance of Rohms (-2). The voltage drop accrous the capacitor is g, where oto the charge (in coulombs). According to the Kirchoffe law :- RI + 9 = RdQ c dt Ect) I=d4 dt . In a circuit with R = 30.12, C=0.15 F, Elt) = 600, 9 COD=o coulombs, To find :- Plt) and Ilt). → According to the equation by Kirchoff law a and from the given information, we have 30 I + = 60 — 0 0.15 ive. (0.15830) I + 9 = (0.15X60) ie. 4.5 I +9 = 9 I Let ylt) denote the charge in iicuit at time . t i.e. yct) = g(t) yco) = 9(0)=o. eginen) : I = d 9 = I= y' (t). — * Equation ® becomes, 4.5 y'lt) + y(t) =9. -☺ ie 4.5dy + yet) = 9 dt

wege Taking Laplace on both the sides of eq? we set 4.5L{y'ltyy + l{y(t)} = 9L21). - © ..4.5 fil fyltyy ycosy+ 4141633 = .recess **) - FC (since, LYFct)y = fss) [{1} = 1/4 and J ...::yco=o. .: 4.5/4 L?yct)3-oj +{4663 - .4:52{l? 4.64)33+ l{4Lt)} = 1 . [4:58 +1].64468}} = . :: Lly Lt)y = _9 (4.5 4+1) on both the sides Taking Laplace Inverse transform of above equation, we get ... (1+P5.7)?. 4.5 - (4.5at1) | By partial s fraction усе - чL I (4 зачуу - ва барои -- Yles - 91'} £ en ny yets = 9204+3-91"{sta per diventang yet) = 9 - 9. list = q qeyat o * yuey = yeah) = (12-2*) - 4) 9,4 (246)

.: yllt) = 0 i at at ... idé = ae dt y'lt) = { 249€ © Plt) = 9-9e 9- 1439 ..... from (12), © ¢ © -49 E Ilt) = le Thank You