Download Chapter 27: 4-5

Chapter 27: 4-5
Electric Field formulas for several
continuous distribution of charge.
The Electric Field of a Dipole
We can represent an electric dipole by two opposite charges
±q separated by the small distance s.
The dipole moment is defined as the vector
The dipole-moment magnitude p = qs determines the
electric field strength. The SI units of the dipole moment
are C m.
The Electric Field of a Dipole
The electric field at a point on the axis of a dipole is
where r is the distance measured from the center of the
dipole.
The electric field in the plane that bisects and is
perpendicular to the dipole is
This field is opposite to the dipole direction, and it is only
half the strength of the on-axis field at the same distance.
Problem-Solving Strategy: The electric
field of a continuous distribution of
charge
Problem-Solving Strategy: The electric field of a continuous
distribution of charge
Problem-Solving Strategy: The electric field of a continuous
distribution of charge
Derivation
on pages
827-828
Example 26.3
Derivation
on pages
827-828
See appendix A-3
for integral
Electric Field of a line of charge
E rod
Q
1

4 0 r r 2  L 2 2
L   then radical reduces to L 2
Electric Field of a infinite line of charge
Electric Field on Ring
Eiz  Ei cos  i 
Q
cos  i
40 ri 2
1
E 
ri  z 2  R 2
cos  i 
ring x
z

ri
z
z 2  R2
1
Q
z
Eiz 
40 z 2  R 2 z 2  R 2
1
z
Eiz 
Q
40 z 2  R 2 3 / 2
N




E z   Eiz 
i 1
E 
ring z

1
z

40 z 2  R 2
1
zQ

40 z 2  R 2

3/ 2
N
Q


3/ 2
i 1

1
zQ
40 z 2  R 2 3 2
A Disk of Charge
Electric Field of a disk of charge
N
z
i 1
40
Edisk z   Ei z 

i 1
Q  Ai   2ri r
Edisk z
z

2 0
ri r
N

i 1
z
2
 ri

2 3/ 2
N  , r  dr
z
( Edisk ) z 
2 0
R
 z
0

1 
2 0 
Edisk z  z
rdr
2
 r2

3/ 2


z2  R2 
z
Q
N
z
2
 ri

2 3/ 2
A Disk of Charge
The on-axis electric field of a charged disk of radius R,
centered on the origin with axis parallel to z, and surface
charge density η = Q/πR2 is
NOTE: This expression is only valid for z > 0. The field for z <
0 has the same magnitude but points in the opposite
direction.
NOTE 2: The formula reduces to a plane of
charge relationship at R>>z
A Plane of Charge
The electric field of an infinite plane of charge with
surface charge density η is:
Formula above can be derived from charge disk formula, see page 830.
For a positively charged plane, with η > 0, the electric
field points away from the plane on both sides of the
plane.
For a negatively charged plane, with η < 0, the electric
field points towards the plane on both sides of the
plane.
The Parallel-Plate Capacitor
•
The figure shows two electrodes, one
with charge +Q and the other with –
Q placed face-to-face a distance d apart.
• This arrangement of two electrodes,
charged equally but oppositely, is called
a parallel-plate capacitor.
• Capacitors play important roles in many
electric circuits.
The Parallel-Plate Capacitor
The electric field inside a capacitor is
where A is the surface area of each electrode. Outside the
capacitor plates, where E+ and E– have equal magnitudes
but opposite directions, the electric field is zero.
Use two infinite size disks to create the
relationship for a capacitor. If z <<R,
second fraction reduces to 0. Since there
are 2 disks, then formula is doubled.
EXAMPLE 27.7 The electric field inside
a capacitor
QUESTIONS:
EXAMPLE 27.7 The electric field inside
a capacitor
EXAMPLE 27.7 The electric field inside
a capacitor
EXAMPLE 27.7 The electric field inside
a capacitor
A Sphere of Charge
A sphere of charge Q and radius R, be it a uniformly charged
sphere or just a spherical shell, has an electric field outside
the sphere that is exactly the same as that of a point charge
Q located at the center of the sphere: