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408 ⏐⏐⏐ CAPACITORS
10.12 THE CURRENT i C
The current i C associated with a capacitance C is related to the voltage
across the capacitor by
iC � C � dvC
�
dt
(10.27)
where dvC /dt is a measure of the change in vC in a vanishingly small
period of time. The function dvC /dt is called the derivative of the voltage
vC with respect to time t.
If the voltage fails to change at a particular instant, then
dvC � 0
and iC � C� dvC
� � 0
dt
In other words, if the voltage across a capacitor fails to change with
time, the current iC associated with the capacitor is zero. To take this a
step further, the equation also states that the more rapid the change in
voltage across the capacitor, the greater the resulting current.
In an effort to develop a clearer understanding of Eq. (10.27), let us
calculate the average current associated with a capacitor for various
voltages impressed across the capacitor. The average current is defined
by the equation
iCav � C � �vC
�
�t
(10.28)
where � indicates a finite (measurable) change in charge, voltage, or
time. The instantaneous current can be derived from Eq. (10.28) by letting
�t become vanishingly small; that is,
iCinst � lim C�
∆t→0 �vC
� � C�
�t
dvC
�
dt
In the following example, the change in voltage �vC will be considered
for each slope of the voltage waveform. If the voltage increases
with time, the average current is the change in voltage divided by the
change in time, with a positive sign. If the voltage decreases with time,
the average current is again the change in voltage divided by the change
in time, but with a negative sign.
EXAMPLE 10.13 Find the waveform for the average current if the
voltage across a 2-mF capacitor is as shown in Fig. 10.59.
Solutions:
a. From 0 ms to 2 ms, the voltage increases linearly from 0 V to 4 V,
the change in voltage �v � 4 V � 0 � 4 V (with a positive sign
since the voltage increases with time). The change in time �t �
2ms�0 � 2 ms, and
iCav � C� �vC
� � (2 � 10
�t
�6 F) � �
� 4 � 10 �3 4 V
���3
2 � 10 s
A � 4 mA