Resource Lesson
Kirchhoff's Laws: Analyzing DC Circuits with Capacitors
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Capacitors are used in DC circuits to provide "bursts of energy." Typical examples would be a capacitor to jump start a motor or a capacitor used to operate a camera's flash.
When the switch is closed, charges immediately start flowing onto the plates of the capacitor. As the charge on the capacitor's plates increases, this
transient current
decreases; until finally, the current ceases to flow and the capacitor is fully charged. In the diagram shown above, the right plate of the capacitor would be positively charged and its left plate negatively charged since the plates are arbitrarily assigned as + and - according to their proximity to the nearest battery terminal.
Graphs of
current vs time
and
charge vs time
are shown below. Mathematically, both of these graphs are exponential functions - current is an example of exponential decay, while charge is an example of exponential growth.
Charging Capacitor Graphs
current vs time
charge vs time
In the circuit shown below, charges immediately start flowing off of the plates of the capacitor as soon as the switch is closed. As the charge on the capacitor's plates decreases, the current decreases; until finally, the current ceases to flow and the capacitor is fully discharged.
In this situation, graphs of
current vs time
and
charge vs time
will both be decay functions since the current flowing through the resistor will fall off according to the flow of charge off of the capacitor's plates.
Discharging Capacitor Graphs
current vs time
charge vs time
Steady-State Conditions
In a network containing one or more capacitors,
steady-state conditions
means that there are NO CURRENTS flowing through any branches in which a charged capacitor is located. Charged capacitors have voltage but not resistance: V = IR is not applicable since no currents flow THROUGH a capacitor. When a "loop" contains a capacitor, the capacitor is treated like a "battery." That is, if the loop approaches the capacitor from "positive to negative" or "high to low" then the potential difference across the capacitor is written as -V
_{C}
. Similarly, if the loop approaches the capacitor from "negative to positive" or "low to high" then the potential difference across the capacitor is written as +V
_{C}
. Any resistors on the same branch of a circuit as a capacitor receive no current, and therefore do NOT lose any voltage.
The rules for assigning SIGNS to the voltages changes across capacitors in a closed loop for
Kirchoff’s loop rule
are:
V
_{C}
= - Q/C if the direction of the loop crosses the capacitor from its positive to its negative plate (high to low)
V
_{C}
= + Q/C if the direction of the loop crosses the capacitor from its negative to its positive plate (low to high)
Refer to the following information for the next two questions.
After the switch is closed and steady state conditions have been reached, no current will be flowing through the 3-ohm resistor since the capacitor will be fully charged.
(a) What would be the voltage across the 2 µF capacitor?
(b) How much charge does it hold?
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