**RC
Circuits**

An RC circuit is a circuit with both a resistor (R) and a capacitor (C). RC circuits are freqent element in electronic devices. They also play an important role in the transmission of electrical signals in nerve cells.

A capacitor can store energy and a resistor placed in series with it will
control the rate at which it charges or discharges. This produces a
characteristic time dependence that turns out to be exponential.
The crucial parameter that describes the time dependence is the
**"time constant" ** * R C *. The far-sighted
student might guess this just by observing that *R C * has
the dimensions of time: *(1 Ohm) x (1 Farad) = (1 second)*.

We will confine our studies to the following circuit, in which the
switch can be moved between positions *a* and *b*.

Let us begin by reviewing some facts about capacitors:

**The charge on a capacitor cannot change instantaneously**. The current is given by*I =**D**Q /**D***t**. Hence the change in charge*D**Q = I***D**goes to zero as the time interval*t**D*goes to zero.*t***The current flowing into a capacitor in the steady state that is reached after a long time interval is zero.**Since charge builds up on a capacitor rather than flowing through it, charge can build up until the point that the voltagebalances out the external voltage pushing charge onto the capacitor.**V=Q/C**

When a capacitor of capacitance * C*
is in series with a battery of voltage

,

which is a statement that the
voltage gained going across the battery must equal the voltage
drop across the capacitor plus the voltage drop across the
resistor. An equation where the rate of change of a quantitity **(D****Q/****D*** t*)
is proportional to the quantity

**Discharging the capacitor**: The capacitor initially is connected (switch in position) for a long time, and is then disconnected by moving the switch to**a**at time**b**. The capacitor then discharges, leaving the capacitor without charge or voltage after a long time.**t = 0****Charging the capacitor**: The switch is in positionfor a long time, allowing the capacitor to have no charge. At time**b**, the switch is changed to**t = 0**and the capacitor charges.**a**

Discharging | Charging | |

Charge | ||

Current | ||

Voltage |

Here, **Q**_{0},
**V**_{0}
and **I**_{0} refer to the charge, voltage and current of the
capacitor in the instant after the switch is thrown. The time t is the characteristic
time of the decay,** ****t***
= RC*. When confronted with an

Decide what the charge across the capacitor was just before the switch was thrown. Since the charge can not change instantly, this is the charge just after the switch is thrown.

Decide what the charge is long after the switch is thrown.

Pick the exponential form for the charge

**Q(t)**The voltage across the capacitor can be found through,

. The voltages across the other elements can be found with the help of Kirchhoff's first law.**V = Q/C**The current through a capacitor must always decay and end up at zero. The initial current can usually be ascertained with the help of Ohm's law,

**V=R**.**I**

The characteristic time **t** * =
RC* tells you that the charging/discharging is
slower with a larger resistor or capacitor. This makes sense,
because a larger resistor impedes the flow of current; thus
slowing the charging/discharging, and a larger capacitor holds
more charge; thus requiring more time to charge.

Every educated person should have a good feeling
for exponential functions. (Sketches of the charge
* Q(t)*
for charging and discharging capacitors could be shown here.)