Lecture Notes in Differential Equations - Bruce E. Shapiro

48 LESSON 6. EXPONENTIAL RELAXATION
Figure 6.3: Schematic of RC circuit used in example 6.2.
C
a
V
+
R
b
6. The total current through any node (a point where multiple wires
come together) must sum to zero (Kirchoff’s Law).
If we were to measure the voltage between points a and b in an RC circuit
as illustrated in fig 6.3, these rules tell us first that the capacitor should
cause the voltage along the left branch to fluctuate according to
i C = C dV ab
dt
(6.25)
where i C is the current through the capacitor. If i R is the current through
the resistor, then the voltage drop through the lower side of the circuit is
Solving for the current through the resistor,
V ab = V batt + i R R (6.26)
i R = V ab − V batt
R
Since the total current around the loop must be zero, i C + i R = 0,
(6.27)
Dropping the ab subscript,
0 = i R + i C = V ab − V batt
R
dV
dt = V batt − V
RC
+ C dV ab
dt
(6.28)
(6.29)