Chapter 3.2, Problem 29P

Applications.

Electric Circuits. The theory of electric circuits, such as that shown in Figure 3.2.6, consisting of inductors, resistors, and capacitors, is based on Kirchhoff’s laws: (1) At any node (or junction), the sum of currents flowing into that node is equal to the sum of currents flowing into that node is equal to the sum of currents flowing out of that node, and (2) the net voltage drop around each closed loop is zero. In addition to Kirchhoff’s laws, we also have the relation between the current $i\left(t\right)$ in amperes through each circuit element and the voltage drop $v\left(t\right)$ in volts across the element:

$v-Ri$, $R=\text{resistance}\text{in}\text{ohms}$;

$C\frac{dv}{dt}=i$, $C=\text{capacitance}\text{in}{\text{farads}}^{\text{4}}$;

$L\frac{di}{dt}=v$, $L=\text{inductance}\text{in}\text{henries}$.

Kirchhoff’s laws and the current-voltage relation for each circuit element provide a system of algebraic and differential equations from which the voltage and current throughout the circuit can be determined. Problems 27 through 29 illustrate the procedure just described.

Consider the circuit shown in the Figure 3.2.8. Use the method outlined in problem 27 to show that the current $i$ through the inductor and the voltage $v$ across the capacitor satisfy the system of differential equations

$L\frac{di}{dt}=-R_{1}i-v,C\frac{dv}{dt}=i-\frac{v}{R_2}$.