Answer to Question #330210 in Differential Equations for jim

Question

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).

Answer 1

Due to Kirchhoff's Law:

$U_R+U_C=E$ (1)

where $U_R=iR=\frac{dq}{dt}R$ , $U_C=\frac qC$ , $E=100V$.

Substitution $U_R$ and $U_C$ into (1) gives:

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

$\frac {dq}{dt}+\frac q{RC}=\frac ER$

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

$\frac ER=\frac{100}{200}=0.5\frac V\Omega$

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

The solutions to a nonhomogeneous equation are of the form

$q(t) = q_h(t) + q_p(t)$ ,

where $q_h$ is the general solution to the associated homogeneous equation and $q_p$ is a particular solution.

The associated homogeneous equation:

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

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

$\lambda+50=0$

$\lambda=-50$

$q_h(t)=Ae^{-50t}$ .

The particular solution of the differential equation:

$q_p(t)=B$

$q_p'(t)=0$

Substituting these into (2) we get:

$0+50B=0.5$

$B=0.01$.

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

Now we can apply initial condition to find A:

$q(0)=0$

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

$A=-0.01$

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

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