Question #330210

A 100-volt electromotive force is applied to an RC series circuit in which the resistance is 200 ohms and the capacitance is 10−4 farad. Find the charge q(t) on the capacitor if q(0) = 0. Find the current i(t).



1
Expert's answer
2022-04-19T03:15:32-0400

Due to Kirchhoff's Law:

UR+UC=EU_R+U_C=E (1)

where UR=iR=dqdtRU_R=iR=\frac{dq}{dt}R , UC=qCU_C=\frac qC , E=100VE=100V.

Substitution URU_R and UCU_C into (1) gives:

dqdtR+qC=E\frac {dq}{dt}R+\frac qC=E

dqdt+qRC=ER\frac {dq}{dt}+\frac q{RC}=\frac ER

1RC=1200104=50Ω1F1\frac{1}{RC}=\frac{1}{200\cdot 10^{-4}}=50\Omega^{-1}F^{-1}

ER=100200=0.5VΩ\frac ER=\frac{100}{200}=0.5\frac V\Omega

dqdt+50q=0.5\frac {dq}{dt}+50q=0.5 (2)

The solutions to a nonhomogeneous equation are of the form

q(t)=qh(t)+qp(t)q(t) = q_h(t) + q_p(t) ,

where qhq_h is the general solution to the associated homogeneous equation and qpq_p is a particular solution.

The associated homogeneous equation:

dqdt+50q=0\frac {dq}{dt}+50q=0

The general solution of this equation is determined by the roots of the characteristic equation:

λ+50=0\lambda+50=0

λ=50\lambda=-50

qh(t)=Ae50tq_h(t)=Ae^{-50t} .

The particular solution of the differential equation:

qp(t)=Bq_p(t)=B

qp(t)=0q_p'(t)=0

Substituting these into (2) we get:

0+50B=0.50+50B=0.5

B=0.01B=0.01.

q(t)=Ae50t+0.01q(t)=Ae^{-50t}+0.01

Now we can apply initial condition to find A:

q(0)=0q(0)=0

q(0)=A+0.01=0q(0)=A+0.01=0

A=0.01A=-0.01

q(t)=0.01(1e50t)q(t)=0.01(1-e^{-50t}) ;

i(t)=q(t)=0.5e50ti(t)=q'(t)=0.5e^{-50t} .


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