Question #105352
A simple series circuit has an inductor of 1henry , a capacitor of 10^-6 farads and a resistor of 1000ohms . The initial charge on the capacitor is zero. If a 12volt battery is connected to the circuit and the circuit is closed at t=0, find the charge on the capacitor 1sevond later and the steady state charge.
1
Expert's answer
2020-03-16T10:56:31-0400

The voltage and the current relationships in circuit analysis


i=dqdti={dq \over dt}

voltage drop across the inductor


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

voltage drop across the resistor


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

voltage drop across the capacitor


VC=qCV_C={q \over C}

Kirchhoff's Second Law


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

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


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

d2qdt2+1000dqdt+1000000q=12{d^2q\over dt^2}+1000{dq \over dt}+1000000q=12

The characteristic equation is


r2+1000r+1000000=0r^2+1000r+1000000=0

D=(1000)24(1000000)=3(1000000)D=(1000)^2-4(1000000)=-3(1000000)

r=1000±i100032r={-1000\pm i1000\sqrt{3} \over 2}

qc=e500t(c1cos(5003t)+c2sin(5003t))q_c=e^{-500t}(c_1\cos{(500 \sqrt{3}t)}+c_2\sin{(500 \sqrt{3}t)})

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)=12.E(t)=12.

Differentiate the assumption with respect to tt


Q=0, Q=0Q'=0,\ Q''=0

Substitute in the original equation


0+0+1000000A=120+0+1000000A=12

A=1.2×105A=1.2\times10^{-5}

Q(t)=1.2×105Q(t)=1.2\times10^{-5}

q(t)=e500t(c1cos(5003t)+c2sin(5003t))+1.2×105q(t)=e^{-500t}(c_1\cos{(500 \sqrt{3}t)}+c_2\sin{(500 \sqrt{3}t)})+1.2\times10^{-5}

The initial charge of the capacitor is zero


q(0)=e500(0)(c1cos(5003(0))+c2sin(5003(0)))+1.2×105=0q(0)=e^{-500(0)}(c_1\cos{(500 \sqrt{3}(0))}+c_2\sin{(500 \sqrt{3}(0))})+1.2\times10^{-5}=0

c1=1.2×105c_1=-1.2\times10^{-5}

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


q(t)=500e500t(1.2×105)cos(5003t)q'(t)=-500e^{-500t}(-1.2\times10^{-5})\cos{(500 \sqrt{3}t)}-500e500tc2sin(5003t)+-500e^{-500t}c_2\sin{(500 \sqrt{3}t)}++1.2×105(5003)e500tsin(5003t)++1.2\times10^{-5}(500 \sqrt{3})e^{-500t}\sin{(500 \sqrt{3}t)}++c25003e500tcos(5003t)+c_2500 \sqrt{3}e^{-500t}\cos{(500 \sqrt{3}t)}

q(0)=500(1.2×105)0+0+c25003=0q'(0)=-500(-1.2\times10^{-5})-0+0+c_2500 \sqrt{3}=0

c2=31.2×105c_2=-{\sqrt{3} \over 1.2}\times10^5

q(t)=e500t(1.2×105cos(5003t)31.2×105sin(5003t))+q(t)=e^{-500t}\big(-1.2\times10^{-5}\cos{(500 \sqrt{3}t)}-{\sqrt{3} \over 1.2}\times10^5 \sin{(500 \sqrt{3}t)}\big)++1.2×105+1.2\times10^{-5}

Coulomb:[C]Coulomb: [C]


q(1)=e500(1)(1.2×105cos(5003(1))31.2×105sin(5003(1)))+q(1)=e^{-500(1)}\big(-1.2\times10^{-5}\cos{(500 \sqrt{3}(1))}-{\sqrt{3} \over 1.2}\times10^5 \sin{(500 \sqrt{3}(1))}\big)++1.2×1051.2×105(C)+1.2\times10^{-5}\approx1.2\times10^{-5}(C)


Find the steady state charge.

t:qp(t)1.2×105 Ct\to\infin: q_p(t)\to1.2\times10^{-5} \ C



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