Week 9

PHYSICS 12 - General Physics 3

Capacitors

Two conductors, isolated electrically from

each other and from their surroundings,

form a capacitor.

◦ When the capacitor is charged, the charges

on the conductors, or plates as they are called,

have the same magnitude q but opposite signs.

Capacitance

When a capacitor is charged, its plates have charges of

equal magnitudes but opposite signs

The charge q and the potential difference V for a

capacitor are proportional to each other; that is,

𝑞 = 𝐶𝑉

C is the proportionality constant called the

capacitance.

Capacitance

The capacitance is a measure of how much

charge must be put on the plates to produce a

certain potential difference between them: The

greater the capacitance, the more charge is

required.

𝐶 =𝑞

𝑉◦ Unit: 1 farad = 1F = 1coulomb per volt = 1C/V

◦ Usually reported as 𝜇𝐹 (1 × 10−6) and 𝑝𝐹 (1 ×10−12)

Calculating the Capacitance

Recall:

◦ Gauss Law: 𝜀0 𝐸 ∙ 𝑑 𝐴 = 𝑞

◦ Potential Difference: 𝑉 = −+𝐸 𝑑𝑠

1. Parallel plate capacitor

𝐶 =𝜀0𝐴

𝑑2. Cylindrical capacitor

𝐶 = 2𝜋𝜀0𝐿/ ln 𝑟𝑎 𝑟𝑏3. Spherical capacitor

𝐶 = 4𝜋𝜀0𝑟𝑎𝑟𝑏𝑟𝑏 − 𝑟𝑎

Capacitors in Parallel

Capacitors connected in parallel can be

replaced with an equivalent capacitor that

has the same total charge q and the same

potential difference V as the actual

capacitors.

Capacitors in Series

When a potential difference V is applied across several

capacitors connected in series, the capacitors have

identical charge q. The sum of the potential differences

across all the capacitors is equal to the applied potential

difference V.

Capacitors that are connected in series can

be replaced with an equivalent capacitor

that has the same charge q and the same

total potential difference Vas the actual

series capacitors.

Energy Stored in an Electric Field

Voltage is related to capacitance

𝑉 =𝑄

𝐶 Potential energy stored in a capacitor

𝑈 =𝑄2

2𝐶

𝑈 =1

2𝐶𝑉2

𝑈 =1

2𝑄𝑉

Capacitor with a Dielectric

a non-conducting material between their

conducting plates.

Dielectric constant

𝐾 =𝐶

𝐶0

𝑉 =𝑉0

𝐾(when Q is constant)

Sample Problem

1. A parallel-plate air capacitor of capacitance 245 pF has a charge

of magnitude 0.148 µC on each plate. The plates are 0.328 mm

apart. (a) What is the potential difference between the plates? (b)

What is the area of each plate? (c) What is the electric field

magnitude between the plates? (d) What is the surface charge

density on each plate?

2. In the figure, each capacitor has 𝐶 = 4.00 𝜇𝐹 and 𝑉𝑎𝑏 =+ 28.0 𝑉. Calculate (a) the effective capacitance; and (b) the

charge on each capacitor.

Sample Problem

3. A parallel-plate air capacitor has a capacitance of The

charge on each plate is 2.55µC. (a) What is the potential

difference between the plates? (b) If the charge is kept

constant, what will be the potential difference between the

plates if the separation is doubled? (c) How much work is

required to double the separation?

4. A 12.5 µF capacitor is connected to a power supply that

keeps a constant potential difference of 24.0 V across the

plates. A piece of material having a dielectric constant of

3.75 is placed between the plates, completely filling the

space between them. (a) How much energy is stored in the

capacitor before and after the dielectric is inserted? (b) By

how much did the energy change during the insertion? Did

it increase or decrease?

Practice Problem

In the figure below, C1 = C5 = 8.4µF and C2 =

C3 = C4 = 8.4µF . The applied potential is Vab =

220 V. (a) What is the equivalent capacitance of

the network between points a and b? (b)

Calculate the charge on each capacitor and the

potential difference across each capacitor.

Current the net charge flowing through the area per unit

time

𝐼 =𝑑𝑄

𝑑𝑡Unit: 1𝐴 (𝑎𝑚𝑝𝑒𝑟𝑒) = 1𝐶/𝑠

Current density, 𝐽 =𝐼

𝐴= 𝑛 𝑞 𝑣𝑑

Resistivity

the ratio of the magnitudes of electric

field and current density:

𝜌 =𝐸

𝐽

(the reciprocal of 𝜌 is called conductivity)

Resistivity and temperature

𝜌 𝑇 = 𝜌0 1 + 𝛼 𝑇 − 𝑇0where 𝛼 is the temp coefficient of

resistivity

Resistance

Relationship between resistance and resistivity

𝑅 =𝜌𝐿

𝐴Unit: Ω (ohm)

Ohm’s Law

𝑉 = 𝐼𝑅

Color codes for resistor

Electromotive Force

The influence that makes current flow from

lower to higher potential, “driving force”

Abbreviated as “emf”

Examples are Batteries, electric generators,

solar cells, thermocouples, and fuel cells

For an ideal source of emf

𝑉𝑎𝑏 = 𝜀

In the presence of internal resistance, 𝑟𝑉𝑎𝑏 = 𝜀 − 𝐼𝑟