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Chapter 21 Electric Charge and Electric Fields
• What is a field?
• Why have them?
• What causes fields?
Field Type Caused By
gravity mass
electric charge
magnetic moving charge
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Electric Charge• Types:
– Positive• Glass rubbed with silk • Missing electrons
– Negative• Rubber/Plastic rubbed with fur• Extra electrons
• Arbitrary choice – convention attributed to ?
• Units: amount of charge is measured in [Coulombs]
• Empirical Observations:– Like charges repel– Unlike charges attract
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Charge in the Atom
• Protons (+)
• Electrons (-)
• Ions
• Polar Molecules
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Charge Properties• Conservation
– Charge is not created or destroyed, only transferred.
– The net amount of electric charge produced in any process is zero.
• Quantization– The smallest unit of charge is that on an electron or
proton. (e = 1.6 x 10-19 C)
• It is impossible to have less charge than this
• It is possible to have integer multiples of this charge
Q Ne
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Conductors and Insulators
• Conductor transfers charge on contact• Insulator does not transfer charge on contact• Semiconductor might transfer charge on contact
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Charge Transfer Processes
• Conduction
• Polarization
• Induction
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The Electroscope
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Coulomb’s Law• Empirical Observations
• Formal Statement
1 2F q q
2
1F
r
Direction of the force is along the line joining the two charges
1 212 122
12
kq qˆF r
r
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Active Figure 23.7
(SLIDESHOW MODE ONLY)
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Coulomb’s Law Example
• What is the magnitude of the electric force of attraction between an iron nucleus (q=+26e) and its innermost electron if the distance between them is 1.5 x 10-12 m
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Hydrogen Atom Example
• The electrical force between the electron and proton is found from Coulomb’s law– Fe = keq1q2 / r2 = 8.2 x 108 N
• This can be compared to the gravitational force between the electron and the proton– Fg = Gmemp / r2 = 3.6 x 10-47 N
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Subscript Convention
1 212 122
12
kq qˆF r
r
12 2 1F force on charge q due to charge q
12 1 1r distance from charge q to charge q
12 1 2r̂ unit vector oriented from charge q to charge q
+q1 +q2
12r̂ 12F
12r
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More Coulomb’s Law
Coulomb’s constant:2 2
9 92 2
o
N m N m 1k 8.988x10 9.0x10
C C 4
permittivity of free space:2
12o 2
1 C8.85x10
4 k N m
1 212 122
12
kq qˆF r
r
1212
12
rr̂
r
Charge polarity: Same sign Force is rightOpposite sign Force is Left
Electrostatics --- Charges must be at rest!
+q1 +q2
12r̂ 12F
12r
12r
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Superposition of Forces
0 10 20 30F F F F ....
0 1 0 2 0 30 10 20 302 2 2
10 20 30
kq q kq q kq qˆ ˆ ˆF r r r ....
r r r
N31 2 i
0 0 10 20 30 0 i02 2 2 2i 110 20 30 i0
qq q qˆ ˆ ˆ ˆF kq r r r .... kq r
r r r r
+Q3
+Q2
+Q1
+Q0
10F
30F
20F
10r
20r
30r
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Coulomb’s Law Example
• Q = 6.0 mC• L = 0.10 m• What is the magnitude
and direction of the net force on one of the charges?
F1
+
F2
F3
Q
x
y
L
++
+
L
Q
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Zero Resultant Force, Example
• Where is the resultant force equal to zero?– The magnitudes of the
individual forces will be equal
– Directions will be opposite
• Will result in a quadratic
• Choose the root that gives the forces in opposite directions
q1 = 15.0 C
q2 = 6.0 C
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Electrical Force with Other Forces, Example
• The spheres are in equilibrium
• Since they are separated, they exert a repulsive force on each other– Charges are like charges
• Proceed as usual with equilibrium problems, noting one force is an electrical force
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Electrical Force with Other Forces, Example cont.
• The free body diagram includes the components of the tension, the electrical force, and the weight
• Solve for |q| • You cannot
determine the sign of q, only that they both have same sign
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The Electric Field• Charge particles create forces on each other without ever
coming into contact. » “action at a distance”
• A charge creates in space the ability to exert a force on a second very small charge. This ability exists even if the second charge is not present.
• We call this ability to exert a force at a distance a “field”• In general, a field is defined:
• The Electric Field is then:
test quantity 0
ForceField lim
test quantity
q 0
FE lim
q
N
C
Why in the limit?
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Electric Field near a Point Charge
oq 0o
FE lim
q
o2
kq qˆF r
r
o
o2
2q 0o
kqqr̂ kqr ˆE lim r
q r
+Q -Q
Electric Field Vectors
Electric Field Lines
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Active Figure 23.13
(SLIDESHOW MODE ONLY)
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Rules for Drawing Field Lines• The electric field, , is tangent to the field lines.• The number of lines leaving/entering a charge is
proportional to the charge.• The number of lines passing through a unit area normal
to the lines is proportional to the strength of the field in that region.
• Field lines must begin on positive charges (or from infinity) and end on negative charges (or at infinity). The test charge is positive by convention.
• No two field lines can cross.
E
# of electric field linesE
Area
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Electric Field Lines, General
• The density of lines through surface A is greater than through surface B
• The magnitude of the electric field is greater on surface A than B
• The lines at different locations point in different directions– This indicates the field is
non-uniform
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Example Field Lines
+ + + + +
Dipole Line Charge
Q dq
dx
Linear Charge Density:
For a continuous linear charge distribution,
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Active Figure 23.24
(SLIDESHOW MODE ONLY)
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More Field Lines
Q dq
A dA Surface Charge Density:
Q dq
V dV Volume Charge Density:
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Superposition of Fields
P 1P 2P 3PE E E E ....
31 2P 10 20 302 2 2
10 20 30
kqkq kqˆ ˆ ˆE r r r ....
r r r
N31 2 i
P 10 20 30 i02 2 2 2i 110 20 30 i0
qq q qˆ ˆ ˆ ˆE k r r r .... k r
r r r r
+q3
+q2
+q1
10E
30E
20E
10r
20r
30rP
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Superposition Example
• Find the electric field due to q1, E1
• Find the electric field due to q2, E2
• E = E1 + E2
– Remember, the fields add as vectors
– The direction of the individual fields is the direction of the force on a positive test charge
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Electric Field of a Dipole
p
2a-q +q
E
y
E
E E E
22
kqE E
y a
x xE E E 2E cos
32 2 2 22 2 2
kq a kpE 2
y a y a y a
y 2a3
kpE
yp 2aq
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Example
Three point charges are arranged as shown in Figure P23.19.
(a) Find the vector electric field that the 6.00-nC and –3.00-nC charges together create at the origin.
(b) (b) Find the vector force on the 5.00-nC charge.
Figure P23.19
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Example
Three point charges are aligned along the x axis as shown in Figure P23.52. Find the electric field at
(a) the position (2.00, 0) and
(b) the position (0, 2.00).
Figure P23.52
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9 2 2 9
1 2 2
8.99 10 N m C 4.00 10 Cˆˆ
2.50 m
ˆ5.75 N C
ek q
r
E r i
i
9 2 2 9
2 2 2
8.99 10 N m C 5.00 10 Cˆ ˆˆ 11.2 N C
2.00 mek q
r
E r i i
9 2 2 9
3 2
8.99 10 N m C 3.00 10 Cˆ ˆ18.7 N C
1.20 m
E i i
1 2 3 24.2 N CR E E E E
FIG. P23.52(a)
in +x direction.
1 2ˆ ˆˆ 8.46 N C 0.243 0.970ek q
r E r i j
2 2
3 2
1 3 1 2 3
ˆˆ 11.2 N C
ˆ ˆˆ 5.81 N C 0.371 +0.928
ˆ ˆ4.21 N C 8.43 N C
e
e
x x x y y y y
k q
rk q
r
E E E E E E E
E r j
E r i j
i j
9.42 N CRE 63.4 above axisx
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P23.19
9 91 3
1 2 21
8.99 10 3.00 10ˆ ˆ ˆ2.70 10 N C
0.100
ek q
r
E j j j
9 92 2
2 2 22
2 32 1
8.99 10 6.00 10ˆ ˆ ˆ5.99 10 N C
0.300
ˆ ˆ5.99 10 N C 2.70 10 N C
ek q
r
E i i i
E E E i j
9 ˆ ˆ5.00 10 C 599 2700 N Cq F E i j
6 6ˆ ˆ ˆ ˆ3.00 10 13.5 10 N 3.00 13.5 N F i j i j
(a)
(b)
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Motion of Charged Particles in a Uniform Electric Field
-e
-Q +Q
q 0
FE lim
q
F qE ma
qEa
m
2 20 0v v 2a x x
x x
e Ev 2a x 2 x
m
x
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Example
• A proton accelerates from rest in a uniform electric field of 500 N/C. At some time later, its speed is 2.50 x 106 m/s.– Find the acceleration of the
proton.– How long does it take for the
proton to reach this speed?– How far has it moved in this time?– What is the kinetic energy?
e
-Q +Q
x
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Motion of Charged Particles in a Uniform Electric Field
-e
+Q
-Q
q 0
FE lim
q
F qE ma
y
e Ea
m
x x0 x x0v v a t v
y y0 y
e Ev v a t t
m
x0v
2
2 2 2x y x0
e E tv v v v
m
v
1
x0
e E ttan
mv
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Active Figure 23.26
(SLIDESHOW MODE ONLY)
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Motion of Charged Particles in a Uniform Electric Field
-e-e
-Q +Q
+Q
-Qx
Phosphor Screen
This device is known as a cathode ray tube (CRT)
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DipolesThe combination of two equal charges of opposite sign, +q and –q, separated by a distance l
p q2a qL p
2a L
-q +q
1p
2p
1 2p p p
p qL
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Dipoles in a Uniform Electric Field
p
2a L
-q
+q F qE
F qE
E
F a sin F a sin qEa sin qEa sin q2aEsin pEsin
x y z
ˆ ˆ ˆi j k
r F x y z
F F F
r F sin
p E
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Work Rotating a Dipole in an Uniform Electric Field
p
2a L
-q
+q F qE
F qE
E
fielddW d pEsin d dU
oLet: U( =90 ) 0 U pE cos
U p E
0U pE sin d pE cos U
0U 0