Question #116887
simple series circuit inductor 1 henry capacitor of 10 power minus 6 Faradas and a resistor of 1000 ohms . the initial charge on a capacitor is zero if a 12 volt battery is connected to the circuit and the circuit is closed at t = 0 find the charge on the capacitor 1 second later and the steady state charge
1
Expert's answer
2020-05-19T08:59:40-0400

The voltage and the current relationships in circuit analysis

i=dqdti= \frac{dq}{dt} ​

voltage drop across the inductor

VL=Ldidt=Ld2qdt2V_L =L\frac{di}{dt}=L\frac{d^2q}{dt^2}​

voltage drop across the resistor

VR=Ri=RdqdtV_R=Ri=R\frac{dq}{dt}

voltage drop across the capacitor

VC=qCV _C=\frac{q}{C}

Kirchhoff's Second Law

Ld2qdt2+Rdqdt+qC=E(t)L\frac{d^2q}{dt^2}+R\frac{dq}{dt}+\frac{q}{C}=E(t)

Given that L=1L=1H, R=1000R=1000ohm, C=106C=10^{-6} F, E(t)=12E(t)=12 V

1d2qdt2+1000dqdt+1106q=12d2qdt2+1000dqdt+1000000q=121\frac{d^2q}{dt^2}+1000\frac{dq}{dt}+\frac{1}{10^{-6}}q=12\\ \frac{d^2q}{dt^2}+1000\frac{dq}{dt}+1000000q=12\\

The characteristic equation is

r2+1000r+1000000=0D=(1000)241000000=3000000r1=1000+i100032=500+5003ir2=1000i100032=5005003iqC=e500t(c1cos(5003t)+c2sin(5003t))r^2+1000r+1000000=0\\ D=(1000)^2-4\cdot1000000=-3000000\\ r_1=\frac{-1000+i1000\sqrt3}{2}=-500+500\sqrt3i\\ r_2=\frac{-1000-i1000\sqrt3}{2}=-500-500\sqrt3i\\ q_C=e^{-500t}(c_1\cos(500\sqrt3t)+c_2\sin(500\sqrt3t))


Find the particular solution of the non-homogeneous differential equation using the method of the undetermined coefficients.

Assume that Q(t)=AQ(t)=A is a solution of the non-homogeneous differential equation where E(t)=12E(t)=12

Differentiate the assumption with respect to tt

Q=0,Q=0Q'=0, Q''=0

Substitute in the original equation

0+0+1000000A=12A=1.2105Q(t)=1.2105q(t)=e500t(c1cos(5003t)+c2sin(5003t))++1.21050+0+1000000A=12\\ A=1.2\cdot10^{-5}\\ Q(t)=1.2\cdot10^{-5}\\ q(t)= e^{-500t}(c_1\cos(500\sqrt3t)+c_2\sin(500\sqrt3t))+\\ +1.2\cdot10^{-5}

The initial charge of the capacitor is zero

q(0)=e5000(c1cos(50030)++c2sin(50030))+1.2105=0c1=1.2105q(0)= e^{-500\cdot 0}(c_1\cos(500\sqrt3\cdot 0)+\\ +c_2\sin(500\sqrt3\cdot 0))+1.2\cdot10^{-5}=0\\ c_1=-1.2\cdot10^{-5}

The circuit is closed at t=0:i(0)=q(0)=0t=0: i(0)=q'(0)=0\\

q(t)=500e500t(1.2105)(cos(5003t)500e500t(c2sin(5003t))++1.2105(5003)e500t(sin(5003t)++c25003e500tcos(5003t))q(0)=500(1.2105)+c25003=0c2=0.43105q(t)=e500t(1.2105cos(5003t)0.43105sin(5003t))+1.2105q'(t)= -500e^{-500t}(-1.2\cdot10^{-5})(\cos(500\sqrt3t)-\\ -500e^{-500t}(c_2\sin(500\sqrt3t))+\\ + 1.2\cdot10^{-5}(-500\sqrt3)e^{-500t}(\sin(500\sqrt3t)+\\ +c_2\cdot500\sqrt3e^{-500t}\cos(500\sqrt3t))\\ q'(0)=-500(-1.2\cdot10^{-5})+c_2\cdot500\sqrt3=0\\ c_2=-0.4\sqrt3\cdot10^{-5}\\ q(t)=e^{-500t}(-1.2\cdot10^{-5}\cos(500\sqrt3t)-\\ -0.4\sqrt3\cdot10^{-5}\sin(500\sqrt3t))+1.2\cdot10^{-5}

Coulomb:

q(1)=e500(1)(1.2105cos(5003(1))0.43105sin(5003(1)))+1.21051.2105(C)q(1)=e^{-500\cdot(1)}(-1.2\cdot10^{-5}\cos(500\sqrt3\cdot(1))-\\ -0.4\sqrt3\cdot10^{-5}\sin(500\sqrt3\cdot(1)))+1.2\cdot10^{-5}\approx\\ \approx1.2\cdot10^{-5}(C)

Find the steady state charge.

t:qp(t)1.2105Ct\to\infty:q_{p}(t)\to1.2\cdot10^{-5}C


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