Question

Differential Equations

d In a series circuit we have an inductor of 100 mH, a resistor of 1012 and a capacitor of 1000uF. At time t=0 the capacitor carries no charge and 9(0) = 0. An external voltage of E(t) = cos(100t) V is applied to the circuit. Compute the steady state charge of the capacitor. dt R E(t) = HE L

Answer 1

1 Answer: Given that, Inductance (L) - 100mH, Resistance (P) = 104, capacitance (C) = 1000MF and em.f.(E )- Colloo t) V By Kirchhott +12=E voltage law, di de where i is the current and is Here 'q is the change day dt dt we have di + ĭ R riezt di 10] i= cos(1004) 100x103 -) + dt IDDH -3 * Since inH = 10 h

2. di + 10 GO -) dt Cosloot Goh 10 di dt + (100) i = locosloot 0 The equation 0 ms a lineas difterential equation ob first older interms of 'P. + Pi=B. Pand if are functions of t cols constants. Now, Integrating bactos (IF) = 2 Skeidt ie di dt - esloode loot I.F. = e loot i e こ S Edt 10 Cosloot e Now, the solution for it is given by 8(t) (I-F) de loot, 10 selon le look cosscot dit .: Seat Cosht dit arthr[a cosht tb simbt tc (10o Cosloot + 100 Sin loot 1+c loot ie = 10 pak 1000 =) lieloot 10. 10084 loon loot = ict) = [100 co 100 Cofloot + looSmloot + cot]+ et 100E 20000 =) (t) = -1100 Cosloot +100. Smioot]+ celot 2000

3 According to given data in the problem, At time t=0 the capacitos carries no charge and 960) = 0 ie current ict)=0 at t=0 de i (0)=0 from ①, 0 = 2000 (too Cos 1000) + leo Sin 100)+c.cº Ecoso=1 [100-1+0+e.! O = 2000 Sino=0 e-1 -) e- 100 2000 auf an obitrary =) c= // constant - loot - i(t) = 20 2000 100 Ces boot + 100 Smloot 7+ Now to find the charge (9) of the capacitos, integrate it) W. v. to t . q = lindt S (ou cestoot + 10o fin love) dt + to Selected Cosroot + 100 2000 100 Sin loot + 100 (Cosloot) - loot e -100 you 2000 100 100 - loot charged 7) = 2000 (Sin 100t - cosloot) - ē 2000