Download Dipole Electric Field

Key Ideas in Chapter 14: Electric Field
 A charged particle makes an electric field at every location in space
(except its own location).
 The electric field due to one particle affects other charged particles.
 The electric force on a charged particle is proportional to
the net electric field at the location of that particle.
 The Superposition Principle:
 The net electric field at any lecation is the vector sum
of the individual electric fields of all charge particles at other locations.
 The field due to one charged particle is not changed
by the presence of other charged particles.
 An electric dipole consists of two particles with charges equal in magnitude
and opposite in sign, separated by a short distance.
 Changes in electric fields travel at the speed of light ("retardation").
1
The Coulomb Force Law
Key Idea: Charges exert forces on each other
Q1
Q2
F
F
1 Q1Q2
F=
r̂
2
4pe 0 r
Direction of Force is:
ATTRACTIVE if charges have OPPOSITE sign
REPULSIVE if charges have SAME sign
Always acts along a line connecting the charges
2
Which way does the field point?
Key Idea: A charged particle makes an electric field at every location in space
q1
E1 =
r̂
2
4pe 0 r
1
Point Particle
Here is a useful mnemonic:
+
Positive people
care about others
--
Negative people
care only about
3
themselves
iClicker
A penny carrying a positive charge Qp exerts an electric
force of magnitude F on a nickel carrying positive charge
Qn that is a distance d away (Qn > Qp ). Which one of the
following is not true?
A. The electric force exerted on the penny by the nickel is
also equal to F.
B. The number of electrons in the penny is less than the
number of protons in the penny.
C. The number of electrons in the penny is larger than
the number of protons in the penny.
1 Qp Q n
D. F ≈
, if d is large compared to the size of the
2
4 pe0 d
coins.
4
How Strong is the Coulomb Force?
Example: Hydrogen Atom = proton and electron
Felec =
Fgrav =
Felec
Fgrav
Electricity is much stronger
than Gravity
5
Electric Field of Point Charges
Key Idea: A charged particle makes an electric field
at every location in space (except its own location).
1 Q1Q2
F=
r̂
2
4pe 0 r
F2 = q2 E1
What happens as
q1
E1 =
rˆ
2
4pe 0 r
1
?
Can the particle exert a force on itself?
No.
E is undefined at the origin, since
is self-contradictory.
6 would it point?
If there were a force at the origin, which way
Electric Field of a
Uniformly Charged Spherical Shell
q1
E1 =
rˆ
2
4pe 0 r
1
for a point particle
What should the electric field
look like from far away?
Esphere
Q
=
rˆ
2
4pe 0 r
1
for r >> R
Which direction does the field point close to the sphere?
Spherical symmetry
The field is in the
direction
even close up to the sphere
7
Electric Field of a
Uniformly Charged Spherical Shell
We'll calculate the whole thing in Ch. 16
using the principle of superposition:
Esphere
Q
=
rˆ
2
4pe 0 r
1
Esphere = 0
for r >R (outside)
for r <R (inside)
8
The Superposition Principle
Key Idea: The net electric field at a location in space is a
vector sum of the individual electric fields contributed by all
charged particles located elsewhere.
Key Idea: The electric field contributed by a charged particle
is unaffected by the presence of other charged particles.
9
The Superposition Principle
The electric field of a dipole:
Electric dipole:
Made of two equally but oppositely
charged point-like objects
s
-q
+q
Example of electric dipole: HCl molecule
What is the E field far from the dipole (r >> s)?
10
Calculating Electric Field
Choice of the origin
y
s
-q
+q
x
z
Choice of origin: use symmetry
11
1. E along the x-axis
q1
E1 =
rˆ
2
4pe 0 r
1
E1, x = E+, x + E-, x=
E1, x =
E1, x =
Point Particle
q
1
4pe 0 ( r - s 2 )
2
+
-q
1
4pe 0 ( r + s 2 )
2
1 qr + qrs + qs / 4 - qr + qrs - qs / 4
2
4pe 0
1
2
2
2
( r - s 2) ( r + s 2)
2
2qrs
4pe 0 ( r - s 2 ) ( r + s 2 )
2
2
2
Dipole Electric Field
(Along
12 X-Axis)
Far from the Dipole along the x-axis
E1, x =
2 srq
1
2
2
4pe 0 æ
sö æ
sö
çr - ÷ çr + ÷
2ø è
2ø
è
Dipole Electric Field
(Along X-Axis)
To find the Electric field far from the dipole (r >> s):
2
2
sö æ
sö
æ
2
çr - ÷ » çr + ÷ » r
2ø è
2ø
è
1 2 sq
E1 =
,0,0
3
4pe 0 r
Electric Field of Point Particle: ~ 1/r2
Electric Field of Dipole: ~ 1/r3
E1, x
1 2 sq
=
4pe 0 r 3
Dipole Electric Field
(Far Away Along X-Axis)
13
2. E along the y-axis
q1
E1 =
rˆ
2
4pe 0 r
1
for a point particle
ü
s
s
r+ = 0, y ,0 - ,0,0 = - , y ,0 ï
2
2
ï
ý Þ r+ = r- =
s
s
r- = 0, y ,0 - - ,0,0 = , y ,0 ï
ïþ
2
2
E+ =
æsö
y +ç ÷
è2ø
1
q
4pe 0
æsö
y2 + ç ÷
è2ø
s
- , y ,0
2
2
2
æsö
y2 + ç ÷
è2ø
2
2
E- =
1
-q
4pe 0
æsö
2
y +ç ÷
è2ø
s
, y,0
2
2
æsö
2
y + ç14÷
è2ø
2
æsö
y2 + ç ÷
è2ø
2
2. E along the y-axis
E+ =
1
4pe 0
s
- , y ,0
2
q
æsö
y2 + ç ÷
è2ø
2
æsö
y2 + ç ÷
è2ø
E- =
2
E2 = E + + E - =
æsö
y +ç ÷
è2ø
2
1
4pe 0
s
- ,- y,0
2
1
q
4pe 0
æsö
y2 + ç ÷
è2ø
qs
q
33
22 22
é 22 æ s ö
êy + ç ÷
è2ø
êë
2
æsö
y2 + ç ÷
è2ø
2
- 1s,0,0
ù
ú
úû
2
-1 qs
,0, 0
if r>>s, then E2 »
3
4pe 0 r
15
at <0,r,0>
3. E along the z-axis
- 1 qs
E2 =
,0,0
3
4pe 0 r
1 2 sq
E1 =
,0,0
3
4pe 0 r
at <r,0,0>
at <0,r,0>
or <0,0,r>
By symmetry,
E along the z-axis must be the same as
E along the y-axis!
16
Shape of Dipole Electric Field
Wikimedia Commons
17
Summary: Point Charge and Dipole
Point Charge:
q1
E1 =
rˆ
2
4pe 0 r
1
Dipole:
for r>>s :
y
s
-q
+q
z
x
+
-
1 2qs
E=
,0,0
3
4pe 0 r
at <r,0,0>
-1 qs
E=
,0,0
3
4pe 0 r
at <0,r,0>
-1 qs
E=
,0,0
3
4pe 0 r
at <0,0,r>
18
Definition of "Dipole Moment"
x:
y, z:
1 2qs
p
E1 =
,0,0,0,0
33
4pe 0 rr
r>>s
- 1 qs
p
E2 =
,0,0
3
4pe 0 r
The electric field of a dipole is proportional to the
Dipole moment: p = qs
p = qs, direction from –q to +q
Dipole moment is a vector pointing
from negative to positive
19 charge
Example Problem
y
E=?
A dipole is located at the origin, and is composed of particles with
charges e and –e, separated by a distance 210-10 m along the xaxis. Calculate the magnitude of the E field at <0,210-8,0> m.
sq
4 0 r 3

2 
10
19


Nm
2

10
m

1
.
6

10
C

E1, y   9 109 2  
3
8


C


 2 10 m 

N
E1, y  7.2 104
C
Since r>>s: E1, y 
200Å
x
2Å
1
Using exact solution:
4 N
E1, y  7.1999973 10
C
20
Interaction of a Point Charge and a Dipole
Edipole
F
F
F = QEdipole
- 1 2qs
=Q
,0,0
3
4pe 0 d
• Direction makes sense?
- negative end of dipole is closer, so its net contribution is larger
• What is the force exerted on the dipole by the point charge?
- Newton’s third law: equal but opposite sign
21
Dipole in a Uniform Field
F = qE
Forces on +q and –q have the same
magnitude but opposite direction
Fnet = + qE - qE = 0
What happens in this case?
22
Dipole in a Uniform Field
F = qE
Forces on +q and –q have the same
magnitude but opposite direction
Fnet = + qE - qE = 0
Dipole experiences a torque about its
center of mass.
What is the equilibrium position?
Electric dipole can be used to measure
the direction of electric field.
23
A Fundamental Rationale
• If know E at some location, then we know the electric force on
any charge:
F = qE
• Can describe the electric properties of matter in terms of
electric field – independent of how this field was produced.
Example: if E>3106 N/C air becomes conductor
• Retardation
Nothing can move faster than light c
c = 300,000 km/s = 30 cm/ns
Coulomb’s law is not completely correct – it does not
contain time t nor speed of light c.
1 q1q2
F=
rˆ
2
4pe 0 r
q
E=
rˆ
2
4pe 0 r
1
v <<c !!!
24
Key Ideas in Chapter 14: Electric Field
 A charged particle makes an electric field at every location in space
(except its own location).
 The electric field due to one particle affects other charged particles.
 The electric force on a charged particle is proportional to
the net electric field at the location of that particle.
 The Superposition Principle:
 The net electric field at any lecation is the vector sum
of the individual electric fields of all charge particles at other locations.
 The field due to one charged particle is not changed
by the presence of other charged particles.
 An electric dipole consists of two particles with charges equal in magnitude
and opposite in sign, separated by a short distance.
 Changes in electric fields travel at the speed of light ("retardation").
25
iClicker
Which one of the following is not true?
The electric force exerted by an electron on an electron:
A. decreases by a factor of 25 if the distance is
increased by a factor of 5.
B. has the same magnitude as the electric force exerted
by a proton on a proton at the same distance.
C. has the same direction as the electric force exerted by
a proton on a proton at the same distance.
D. is much weaker than the gravitational force between
them.
26