Chapter 16CapacitorsBatteries

Parallel CircuitsSeries Circuits

Hint: Be able to do the homework (both theproblems to turn in AND the recommended ones)you’ll do fine on the exam!

Friday, February 26, 1999 in class

Ch. 15 - 16

You may bring one 3”X5” index card (hand-writtenon both sides), a pencil or pen, and a scientificcalculator with you.

U = INTERNAL ENERGY of the capacitor.

This is where the energy comes from to power many of our cordless, rechargeable devices…When it’s gone, we have to plug the devices into the wall socket to recharge the capacitors!

W QQ

U VC

CV1

2 2

1

2

22

Just so you know...

Insulating materials between capacitor plates are known as dielectrics. In the circuits we have dealt with, that material is air. It could be other insulators (glass, rubber, etc.).

Dielectric materials are characterized by a dielectric constant such that when placed between the plates of a capacitor, the capacitance becomes

C = Co Co is the vacuum (air) capacitance.

Dielectrics, therefore, increase the charge a capacitor can hold at a given voltage, since...

Q = C V = Co V

Co

It’s time to develop an understanding ofelectrical systems that are NOT in electrostaticequilibrium. In these systems, charges move,under the influence of an externally imposedelectric fields. Such systems provide us withthe useful electricity we get out of flashlightbatteries and rechargeable devices.

Current and Resistance

We’ve already used this concept, even though we haven’t formally introduced it.

What do you think of when you hear the word “current?”

Electrical Current is simply the flow of electrical charges.

A

The current is the number of charges flowing through a surface A per unit time.

charge carriersmoving charges.can be + or -

I Q

t

I Q

t

By convention, we say that the direction of the current is the direction in which the positive charge carriers move.

Note: for most materials we examine, it’s really the negative charge carriers that move. Nevertheless, we say that electrons move in a direction opposite to the electrical current.

Leftover from Ben Franklin!

Why do charges move?

Well, what happens when you put an electric field across a conductor?

E

electrical force on the charges in the conductor.

charges can move in conductors.

a current flows!

E

A

Vd t

The charge carriers, each with charge q, move with

an average speed vd in response to the electric field. If there are n charge carriers per volume in the conductor, then the number of charge carriers passing a surface A in a time interval t is given by

Q n Av tdq

I Q

tn Avdq

Electric fields exert an electrical force on charges given by

And I remember from last semester that Newton’s 2nd Law says

F ma

So shouldn’t the charges be accelerating

instead of moving with an average velocity vd?

F qE

E

A

Vd t

Let’s follow the path of one of the charge carriers to see what’s really going on...

The thermal motion of the charges in the conductor keep thecharges bouncing around all over the places, hitting thefixed atoms in the conductor.

The electric field exerts a force which “gently” guides thepositive charges toward the right so that over time, theyappear to drift along the electric field.

You might think of the collisions as a frictional forceopposing the flow generated by the electric field.

You are probably asking yourself, “So, just how long does it take the average electron to traverse a 1m length of 14 gauge copper wire if the current in the wire is 1 amp?

How long does it take the averageelectron to traverse a 1m length of14 gauge copper wire if the currentin the wire is 1 amp?

Let’s try to guess first:

a) yearsb) weeksc) daysd) hourse) minutesf) secondsg) microsecondsh) nanoseconds

14 gauge wire is a common size of wire having a radius of 0.0814 cm

Let’s assume that atom of copper is able tosupply one free charge to the current.

The mass density of copper is 8.92 g/cm3

1 mole of copper weighs 63.5 g.

So, the number density of charge carriers in the copper is...

n ( . )(8. )

( . )

6 02 10 92

635

23 atoms

mol

g

cm3

g

mol

n = 8.46 X 1022 atoms/cm3

q = 1.6 X 10-19 CA = r2 = 2.1 X 10-2 cm2

vn Ad

I

q

I Q

tn Avdq

t = 7.8 hours!

vd

1

46 10 16 10 21 1022 19 2

A

cmchgs

cm3C

chg2(8. )( . )( . )

vd 355 10 3. cm

s

td

vd

100

355 102 8 10

34 cm scm

s.

.

Describes the degree to which a current through a conductor is impeded.

In particular, if a voltage V is applied across a conductor, a current I will flow. The resistance R is defined to be:

R = V / I

R = V / I

[R] = [V] / [I]

[R] = Volt / Amp

= Ohm ()

Georg Ohm (early 19th century) systematically examined the electrical properties of a large number of materials. He found that the resistance of a large number of objects is NOT dependent upon the applied voltage. That is...

V = I R

V

I

Slope = RV

I

The objects for which Ohm’s Law holds are known as OHMIC.

Objects for which resistanceIS a function of the appliedvoltage (i.e., Ohm’s Law isinvalid) are known as NON-OHMIC.

Every material has its own characteristic resistivity to the conduction of electric charge.

On what does the Resistance (R) of an object depend? (OHMIC CONDUCTORS)

length cross-sectional area

Certainly, if two wires have the same cross-sectional area, the longer of the two will have the greater resistance.R ~ L

For two wires of the same length, the one with the larger cross-sectional area will have the smaller resistance. Think of water flowing in a pipe.

R ~ 1 / A

So if we plotted the resistance (R) versus theratio of the length (L) to the cross-sectionalarea (A)

R

L/A

Slope =

We define the slope to be theresistivity () of the material.

R = L / A

The proportionality constant, , is the resistivity.

= R A / L

[] = [R] [A] / [L]

= m2 / m

= m

Resisitivity and Resistance are also a function of Temperature.

= o [ 1 + ( T - To) ]

R = Ro [ 1 + ( T - To) ]

To is usually taken to be 20o C. is the temperature coefficient of resistivity

and is a characteristic of the particular material.

The temperature dependence of resistance holds for everyday to warm temperatures.

At very low temperatures, for some materials, the resistance can fall to zero. These materials are known as superconductors.

The temperature at which their resistance falls off rapidly is known as the critical temperature. T

R

Tc

ChargeInsulators/Conductors

Coulomb’s LawElectric Fields

Potentials & Potential EnergyCapacitors

Series & Parallel Circuits

Two Kinds of Charge+ -

Conservation of Charge.

Charge is Quantized

An electron carries a charge of -1e.A proton carries a charge of +1e.

1e = 1.6 X 10-19 C

GROUND

RUBBER

The charges remain near the end of the rubber rod--right where we rubbed them on!

GROUND

COPPER

Rub charges on here

They move down theconductor toward our hand

Eventually ending up in the ground.

GROUND

COPPER Bring negatively charged rubber ball close to the a copper rod. The copper rod is initially neutral.

Negative charges on the copper runaway from the rubber ball and into theground.

Rubber

GROUND

COPPER The copper rod is now positively charged. The electrons originally on it were forced away into the ground by the negative charges on the rubber ball.

Rubber

+++

++

++

GROUND

COPPER

Put a rubber glove on your hand to insolate the copper rod from ground.

Rubber

+++

++

++

Finally, remove the rubber ball...

GROUND

COPPER+

++

++

++

The excess positive charge is trapped on the copper rod with no path to ground. It redistributes itself uniformly over the

copper rod. We have taken an initially neutral copper rod and induced a positive charge on it!

F

kq q

r1 22

r

k = 9 X 109 N m2/C2

Superposition Principle

Electrostatic Forces

E =

k

r 2

q r

superposition principle applies

So….

E E E Etotal 1 2 3 ...

F = qE

+ -

Field linesFar apart.

Field linesclose together.

NOTE: d is the distance alongthe Electric field only!!!

Electrical work is “Quite Easily Done!”

Scalar quantity!No directions!W = q E d

Work in a UNIFORM electric field

Ed

r

q

As long as theelectric fieldis uniform, thisis the answer!

PE = -W = -q E d

V = Vb-Va = PE/q

The Electrical Potential:

V = - E d,Wilbur!It’s EASY!

In a uniform field

Ed+q

Va Vb

Vb - Va = - E d

It decreases in the direction of the electric field,REGARDLESS OF THE SIGN OF THE CHARGE!

The electrical potential ALWAYSdecreases in the direction of theelectric field! It does not dependupon the sign of the charge.

The electrical potential energydepends upon the sign of the charge.It decreases in the direction of theelectrical force.

What happens to the potential energy of a negative charge (-q) as it moves in the direction of the electric field?

Ed

-qA B

PE = - q E d = - (-q) E d = + q E d

Fe

What happens to the potential of a negative charge (-q) as it moves in the direction of the electric field?

Ed

-qA B

Fe

V = - E d

The superposition principle applies to potentials!

Vtot = V1 + V2 + V3 + ...

For pointcharges

V k

r

q

PEk

r1 2qq

Electrostatic Potential Energy

Equipotential surfaces are perpendicular to the electric field lines everywhere!

When in electrostatic equilibrium (i.e., no charges are moving around), all points on and inside of a conductor are at the same electrical potential!

Work is only done when a charge movesparallel to the electric field lines.

So no work is done by the electric field as a charge moves along an equipotential surface.

1) no electric field exists inside conductor.

2) Excess charges on an isolated conductor are found entirely on its surface.

3) The electric field just outside of aconductor must be perpendicularto the surface of the conductor.

Insulators and Conductors

C = oA/do = permittivity of free space = 8.85 X 10-12 C2/Nm2

C = Q / V W QQ

U VC

CV1

2 2

1

2

22

Capacitors

V C2

+

_C1

Ceq = C1 + C2 Capacitors in parallel ADD.

V C2, V2

+

_ C1, V1

+Q -Q +Q

-Q

1 1 1 = + Ceq C1 C2

Capacitors in seriesADD INVERSELY.