Transcript Chapter 18: Electric Forces and Fields

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
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Sometimes possible by mechanical work (friction)
Example: friction between hard rubber and fur or hair
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


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



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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Original charged object withdrawn: opposite net charge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
q1  q2
F k
r2
Magnitude:
constant of proportionality
distance between charges
Constant of proportionality:
k
1
40
 8.99  109 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)
q1q2
F k 2
r
Newton’s Law of Universal Gravitation (gravitational force)
m1m2
F G 2
r
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 + q
test charge + q0
r
The Electric Field
Use Coulomb’s Law to calculate the force
exerted on the test charge:
charge + q
test charge + q0
F
r
qq0
F k 2
r
The Electric Field
Divide the electric force by the magnitude of
the test charge:
charge + q
test charge + q0
F
r
F
q
k 2
q0
r
The Electric Field
Take away the test charge and define the
quantity E as the ratio F/q0:
charge + q
E
F
r
F
q
Ek 2
q0
r
The Electric Field
We calculated the magnitude of E, in terms of the
magnitude of F :
F
q
q0
Ek
r
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
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
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 : 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
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two conducting plates
each has area A
each has net charge q (one +, one -)
electric field magnitude between plates:
q
E
0 A
(where 0 is the permittivity of free space)
 field points from + plate to - plate
+
-
+
-
+
-
+
-
+
-
+
-
+
-
+
-
+
E
-
+
-
+
-
+
-
+
-
+
-
+
-
+
-
+
-
+
-
The Electric Field: Field Lines
Electric Field Lines
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
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 DA cos
area
angle between
electric field vector
and surface normal
SI unit of electric flux: Nm2/C
DA
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?
q
q
Ek 2 
2
r
40r

1 
 k 

40 

Electric Flux
Rearrange and substitute for the area of a
sphere:
E
q
40 r
EA 
 E  4r 
2
2
q
0
q
0
Note that the left side is the electric flux
through the spherical surface. Since the
field vectors are radial, f = 0°
everywhere.
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 cosf DA  
Q
This is the general form of Gauss’
Law.
0
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
Q  r  
 A
2
Flux according to Gauss’ Law:
q
r
Q
A
E 

2
0
0
Gauss’ Law: Application
Calculating the electric field inside a
parallel-plate capacitor
Equate the two expressions for
and solve for E :
 E  r E 
r 2
2
q
0
q
A
q
 E
0 A
A
Sometimes
is defined as a
q
“charge density”:

Then:

E
0
E
A