Capacitance A capacitor consists of two conductors that are close but not touching. A capacitor has the ability to store electric charge.

a) Parallel-plate capacitor connected to battery. (b) is a circuit diagram.

Some Uses of Capacitors

•  Energy storage •  Pulsed power and weapons •  High-pass and low-pass filters •  Sensing

When a capacitor is connected to a battery, the charge on its plates is proportional to the voltage:

The quantity C is called the capacitance.

Unit of capacitance: the farad (F)

1 F = 1 C/V

Q= Charge (C), and V= voltage (V)

The capacitance does not depend on the voltage; it is a function of the geometry and materials of the capacitor.

For a parallel-plate capacitor:

C= Capacitance, A= area, d= distance between the plates, and = permittivity of free space

= 8.85x10-12 C2/N·m2 ∈0

Example: a)   Calculate the capacitance of a capacitor whose plates are 20

cm x 3.0 cm and are separated by a 1.0 mm air gap. b)  What is the charge on each plate if the capacitor is

connected to a 12 V battery? c)   What is the electric field between the plates? (recall

electrostatics)

Solution: a) = 5.31x10-11 F = 53.1 pF

b) = 6.4x10-10 C

c) E = Vd

= 1.2x104 V/m

Dielectrics

A dielectric is an insulator, and is characterized by a dielectric constant K.

Capacitance of a parallel-plate capacitor filled with dielectric:

Dielectric strength is the maximum field a dielectric can experience without breaking down.

The molecules in a dielectric tend to become oriented in a way that reduces the external field.

This means that the electric field within the dielectric is less than it would be in air, allowing more charge to be stored for the same potential. The capacitance increases with the addition of the dielectric.

Storage of Electric Energy in a Capacitor

A charged capacitor stores electric energy; the energy stored is equal to the work done to charge the capacitor.

The energy density, defined as the energy per unit volume, is the same no matter the origin of the electric field:

The sudden discharge of electric energy can be harmful or fatal. Capacitors can retain their charge indefinitely even when disconnected from a voltage source – be careful!

Circuits Containing Capacitors in Series and in Parallel

Capacitors in parallel have the same voltage across each one:

In this case, the total (equivalent) capacitance is the sum:

Capacitors in series have the same charge:

In this case, the reciprocals of the capacitances add to give the reciprocal of the equivalent capacitance:

Example: Find the total (equivalent) capacitance, in terms of C, for the circuit below. Each capacitor has capacitance C.

Answer: 9/10 C

Activity: Construct the circuit below using the sim “Circuit Construction Kit (AC+DC)” on the website phet.colorado.edu. a) Predict what will happen to current and voltage measured by the meters once the switch is closed. Close the switch and see how your predictons faired. b) Examine how the resistance, capacitance, and voltage of the battery effect the time for charging the capacitor. State conclusions.

Battery

resistor

switch

capacitor

Activity: An uncharged capacitor is in a circuit as shown. What happens to the currents measured by ammeters A1 and A2 when the switch is closed? Predict what will happen.. Explain your observations.

A1

A2

uncharged!capacitor

swtich

R R

R

Solution: Kirchoff’s laws must apply at every stage even if there is a varying current or voltage.

Charging and Discharging of a Capacitor When the switch is closed, the capacitor will begin to charge.

How do you think the value of resistor and the capacitor might affect how rapidly a capacitor charges?

The voltage across the capacitor increases with time:

This is a type of exponential.

This curve has a characteristic time constant:

The charge follows a similar curve:

If an isolated charged capacitor is connected across a resistor, it discharges:

V =V0e−tRCAlso

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