Capacitors and calculus
Capacitors do not have a stable "resistance" as conductors do. However,
there is a definite mathematical relationship between voltage and
current for a capacitor, as follows:
The lowercase letter "i" symbolizes
instantaneous
current, which means
the amount of current at a specific point in time. This stands in contrast
to constant current or average current (capital letter "I") over an
unspecified period of time. The expression "dv/dt" is one borrowed
from calculus, meaning the instantaneous rate of voltage change over
time, or the rate of change of voltage (volts per second increase or
decrease) at a specific point in time, the same specific point in time that
the instantaneous current is referenced at. For whatever reason, the
letter
v
is usually used to represent instantaneous voltage rather than
the letter
e
. However, it would not be incorrect to express the
instantaneous voltage rateofchange as "de/dt" instead.
In this equation we see something novel to our experience thusfar with
electric circuits: the variable of
time
. When relating the quantities of
voltage, current, and resistance to a resistor, it doesn't matter if we're
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dealing with measurements taken over an unspecified period of time
(E=IR; V=IR), or at a specific moment in time (e=ir; v=ir). The same
basic formula holds true, because time is irrelevant to voltage, current,
and resistance in a component like a resistor.
In a capacitor, however, time is an essential variable, because current is
related to how
rapidly
voltage changes over time. To fully understand
this, a few illustrations may be necessary. Suppose we were to connect a
capacitor to a variablevoltage source, constructed with a potentiometer
and a battery:
If the potentiometer mechanism remains in a single position (wiper is
stationary), the voltmeter connected across the capacitor will register a
constant (unchanging) voltage, and the ammeter will register 0 amps. In
this scenario, the instantaneous rate of voltage change (dv/dt) is equal
to zero, because the voltage is unchanging. The equation tells us that
with 0 volts per second change for a dv/dt, there must be zero
instantaneous current (i). From a physical perspective, with no change
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 Spring '08
 pottsantone
 Electric charge, Farads, 2 volts, 0 Volts, 0 amps

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