Chapter 18: Electric Forces and Fields Charges The electric force The electric field Electric flux and Gauss’s Law.

Chapter 18: Electric Forces and Fields

Charges

The electric force

The electric field

Electric flux and Gauss’s Law

Charges

Thales of Miletus, ~ 600 B.C.: a piece of amber, rubbed against fur, attracted bits of straw

“elektron” – Greek for “amber”

Charges

electric charge: an intrinsic property of matter

two kinds: positive and negative

net charge: more of one kind than the other

neutral: equal amounts of both kinds

Charges

charge is quantized: comes in integer multiples of a fundamental (“elementary”) charge

SI unit of charge: the coulombsymbol: C

Size of elementary charge: 1.60×10-19 C

Elementary charge: often written as “e”

Charges

Charge is a conserved quantity.

If a system is isolated, its net charge is constant.

Charges exert forces on each other, without touching.Attraction if charges are unlike (opposite sign)Repulsion if charges are like (same sign)

Charges

Motion of charges

Conductors: Charges can move freely on the surface or through

the material – loosely bound valence electrons Typically: metals

Insulators: Little movement of charge on or through the material Electrons are tightly bound Typically: rubber, plastic, glass, etc.

Charges

Separation of charges Sometimes possible by mechanical work (friction) Example: friction between hard rubber and fur or

hair electrons leave the fur and go to the rubber rubber acquires a net negative charge fur acquires a net positive charge net charge of total system remains zero

Charges

Transfer of charge By contact

Objects touch – net charge moves from one to the other

By induction Charged object brought near to another object Like charges driven from second object through path to

earth Path to earth taken away Original charged object withdrawn: opposite net charge

remains on second object

The Electric Force

Studied systematically by Charles-Augustin CoulombFrench natural philosopher, 1736-1806

The Electric Force: Coulomb’s Law

Attractive or repulsive – like or unlike charges

Magnitude:

Constant of proportionality:

221

r

qqkF

distance between charges

magnitudes of charges

constant of proportionality

space" free ofty permittivi" m /NC 1085.8

/Cm N 108.99 4

1

22120

229

0

k

The Electric Force: Coulomb’s Law

Coulomb’s Law (electric force)

Newton’s Law of Universal Gravitation (gravitational force)

221

r

qqkF

221

r

mmGF

The Electric Field

Field: the mapping of a physical quantity onto points in space

Example: the earth’s gravitational field maps a force per unit mass (acceleration) onto every point

Electric field: maps a force per unit charge onto points in the vicinity of a charge or charge distribution

The Electric Field

Place a test charge q0 at a point a distance r from a charge q

charge + qtest charge + q0

r

The Electric Field

Use Coulomb’s Law to calculate the force exerted on the test charge:


r

F

20

r

qqkF

The Electric Field

Divide the electric force by the magnitude of the test charge:


r

F

20 r

qk

q

F

The Electric Field

Take away the test charge and define the quantity E as the ratio F/q0:

20 r

qkE

q

F

charge + q

r

FE

The Electric Field

We calculated the magnitude of E, in terms of the magnitude of F :

Both E and F are vectors. For a positive test charge, E points in the same direction as F.

E always has the same direction as the electric force on a positive charge (opposite direction from the force on a negative charge).

20 r

qkE

q

F

The Electric Field

The electric field is “set up” in space by a charge or distribution of charges

The electric field produces an electric force on a net charge q1 :

If more than one charge is present, each charge produces an electric field vector at a given point in space. These vectors add according to the usual vector rules.

1EqF

The Electric Field

Parallel-Plate Capacitor two conducting plates each has area A each has net charge q (one +, one -) electric field magnitude between plates:

(where 0 is the permittivity of free space) field points from + plate to - plate

+

+

+

+

+

+

+

+

+

+

+

+

+

+

+

+

+

+

-

-

-

-

-

-

-

-

-

-

-

-

-

-

-

-

-

-

E

A

qE

0

The Electric Field: Field Lines

Electric Field Lines

Directed lines (curves, in general) that start at a positively-charged object and end at a negatively-charged one

Field lines are drawn so that the electric field vector is locally tangent to the field line

The Electric Field in Conductors

A net charge in a conducting object will move to the surface and spread out uniformly

mutual repulsive forces make the charges “want” to get as far from each other as possible

In the steady state, the electric field inside a conducting object is zero

because the charges in a conductor are free to move, if there is an electric field, the charges will move to a distribution in which the electric field is reduced to zero


Example: a conducting sphere is placed in a region where there is an electric field

Initially, the field is present inside the sphere

E

+ -

--

- -

--

-

--

+ +

+

+

+

+

+

++


The field causes the charges to separate, and

the separated charges produce their own field.

E

+

-

-

-

--

--

--

-

+

+

+

+

++

+

++

-

-

-

-

-+

+

+

+

+


The motion continues until the “internal” field

is equal and opposite to the “external” one …

E

+

-

-

-

--

--

--

-

+

+

+

+

++

+

++

-

-

-

-

-+

+

+

+

+


… and their sum is zero.

E

+

-

-

-

--

--

--

-

+

+

+

+

++

+

++

-

-

-

-

-+

+

+

+

+

Electric Flux

We define a quantity associated with the electric field:

SI unit of electric flux: Nm2/C

E

DA

cos AEE D

electric flux

area

angle between electric field vector and surface normal

Electric Flux

Consider a positive charge q … what is the electric field at a spherical surface centered on the charge and a distance r from it?

02

02 4

1

4 k

r

q

r

qkE

Electric Flux

Rearrange and substitute for the area of a sphere:

Note that the left side is the electric flux through the spherical surface. Since the field vectors are radial, = 0° everywhere.

0

0

22

0

4 4

q

EA

qrE

r

qE

Electric Flux: Gauss’ Law

Johann Carl Friedrich Gauss

German mathematician 1777 – 1855Mathematics, astronomy, electricity and

magnetism


Our result for the sphere enclosing the charge q :

is a statement of Gauss’ Law for a spherical surface, where is everywhere zero (the electric field vector is everywhere perpendicular to the surface).

The sphere is an example of a Gaussian (closed) surface.

0q

EA


In general, a Gaussian surface is any surface that continuously encloses a volume of space. Such a closed surface wraps continuously around the volume.

Think of a water balloon, hanging over your palm, assuming some strange, arbitrary shape.


Here is an arbitrary Gaussian surface, containing an arbitrarily-distributed net charge Q :

This is the general form of Gauss’ Law.

D0

cos

QAE

Gauss’ Law: Application

Calculating the electric field inside a parallel-plate capacitor

charge q, spread uniformly overplate area A

Gaussian cylinder radius = r

Flux through surfaces 1 and 2 zero



Flux through surface 3:

Net charge enclosed in cylinder:

Flux according to Gauss’ Law:

A

qrQ 2

ErareaEE2

0

2

0

A

qrQ

E



Equate the two expressions for and solve for E :

Sometimes is defined as a“charge density”:

Then:

E

A

qEA

qr

ErE00

2

2

Aq

A

q

0E

Chapter 18: Electric Forces and Fields Charges The electric force The electric field Electric flux and Gauss’s Law.

Documents

charges charge

charges electric charge

charges transfer of

charge q slide

charge distribution

unit charge

electric field field

charges separation of