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 coulomb symbol: 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Original charged object withdrawn: opposite net charge remains on second object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 Coulomb
French natural philosopher, 1736-1806

The Electric Force: Coulomb’s Law
Attractive or repulsive – like or unlike charges magnitudes of charges
Magnitude: F
 k q
1
 r
2 q
2 constant of proportionality distance between charges
Constant of proportionality: k

1
4

0

8.99

10
9
N m
2
/C
2

0

8 .
85

10

12
C
2
/N m
2
" permittivi ty of free space"

The Electric Force: Coulomb’s Law
Coulomb’s Law (electric force)
F
 k q
1 q
2 r
2
Newton’s Law of Universal Gravitation (gravitational force)
F

G m
1 m
2 r
2

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
q q
0
r charge + q test charge + q
0 r

The Electric Field
charge + q test charge + q
0
F r
F
 k qq
0 r
2

The Electric Field
charge + q test charge + q
0
F r
F q
0
 k q r
2

The Electric Field
E
F/q
0
charge + q r
F q
0

E
 k q r
2

The Electric Field
We calculated the magnitude of magnitude of F :
F q
0

E

E , in terms of the k r q
2
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).

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 q
1
:
F

Eq
1

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.

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:
E


0 q
A
(where 
0
 is the permittivity of free space) field points from + plate to - plate
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
E
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-

The Electric Field: Field Lines
Electric Field Lines


Directed lines (curves, in general) that start at a positively-charged object and end at a negativelycharged 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

The Electric Field in Conductors
Example: a conducting sphere is placed in a region where there is an electric field
E
+
+
-
+
-
-
+
+
+
-
+
+
-
-
-
+
+
-
Initially, the field is present inside the sphere

The Electric Field in Conductors
The field causes the charges to separate, and
E
-
-
-
-
-
-
-
-
-
-
+
-
-
+
+
+
-
+
-
-
+
+
+
+
+
+
+
+
+
+ the separated charges produce their own field.

The Electric Field in Conductors
The motion continues until the “internal” field
E
-
-
-
-
-
-
-
-
-
-
+
-
-
+
+
+
-
+
-
-
+
+
+
+
+
+
+
+
+
+ is equal and opposite to the “external” one …

The Electric Field in Conductors
… and their sum is zero.
E
-
-
-
-
-
-
+
-
-
-
-
-
-
+
+
+
-
+
-
-
+
+
+
+
+
+
+
+
+
+

Electric Flux
We define a quantity associated with the electric field: electric flux E

E

E
D
A cos
 area angle between electric field vector and surface normal D
A
SI unit of electric flux: Nm 2 /C

Electric Flux
Consider a positive charge spherical surface centered on the charge and a distance from it?
q … what is the electric field at a r
E
 k q r
2
 q
4

0 r
2

 k

1
4

0



Electric Flux
E
Rearrange and substitute for the area of a sphere:
 q
4

0 r
2

E

4
 r
2  q

0
EA
 q

0
Note that the left side is the electric flux through the spherical surface. Since the field vectors are radial, everywhere.
f = 0°

Electric Flux: Gauss’ Law
Johann Carl Friedrich Gauss
German mathematician 1777 – 1855
Mathematics, astronomy, electricity and magnetism

Electric Flux: Gauss’ Law
Our result for the sphere enclosing the charge q :
EA
 q

0 is a statement of Gauss’ Law for a spherical surface, where f is everywhere zero (the electric field vector is everywhere perpendicular to the surface).
The sphere is an example of a Gaussian (closed) surface.

Electric Flux: Gauss’ Law
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.

Electric Flux: Gauss’ Law
Here is an arbitrary Gaussian surface, containing an arbitrarily-distributed net charge Q :
 
E cos f 
D
A

Q

0
This is the general form of Gauss’
Law.

Gauss’ Law: Application
Calculating the electric field inside a parallel-plate capacitor charge q , spread uniformly over plate area A
Gaussian cylinder radius = r
Flux through surfaces 1 and 2 zero

Gauss’ Law: Application
Calculating the electric field inside a parallel-plate capacitor
Flux through surface 3:

E

E
 area
  r
2
E
Net charge enclosed in cylinder:
Q
  r
2 q
A
Flux according to Gauss’ Law:

E

Q

0

 r
2 q

0
A

Gauss’ Law: Application
Calculating the electric field inside a parallel-plate capacitor
Equate the two expressions for

E and solve for E :

E
  r
2
E q

 r
2 q

0
A 
E


0 q
A
“charge density”:
Then: E



0
  q
A