Download Chapter 21 Electric Charge and the Electric Field 1 Electric Charge

Chapter 21
Electric Charge and the Electric Field
1
Electric Charge
Electrostatics is the study of charges when they are stationery.
Figure 1: This is Fig. 21.1 and it shows how negatively charged objects repel each other, and
positively charged objects repel each other. Likewise, oppositely charged objects attract each other.
1
1.1
Electric Charge and Structure of Matter
Figure 2: This is Fig. 21.3 showing the structure of the atom
Figure 3: This is Fig. 21.4 showing the structure of neutral and ionized Lithium atoms.
2
2
Conductors, Insulators, and Induced Charges
• Conductors permit electric charge to move easily from one region of the
material to another.
• Insulators inhibit the motion of electric charge from one region of the material
to another
2.1
Charging by Induction
Figure 4: This is Fig. 21.7 showing the charging of a metal ball by induction.
3
Coulomb’s Law
The magnitude of the electric force between two point charges is directly proportional to the product of the charges and inversely proportional to the square of the
distance between them. This can be written as:
|q1 q2 |
(magnitude only)
r2
where k = 1/4πo = 8.988 × 109 N · m2 /C2 .
F =k
The electric permittivity of a vacuum:
o = 8.85 × 10−12 C2 /(N · m2 )
3
(1)
3.1
The force between two point charges
The general rule for the magnitude of the Coulomb force (i.e., the electrostatic
force) is:
F =k
q1 q2
r2
(magnitude only)
(2)
where a positive result represents “repulsion” and a negative result represents “attraction.”
3.2
Charge of the electron and proton
qp = −qe = e = 1.602 176 565(35)×10−19 C the fundamental quantum of charge
3.3
Electrical Force versus Gravitational Force
Fe = k
q1 q2
r2
Fg = −G
m1 m2
r2
(3)
Figure 5: This is Fig. 21.11 showing the electrical and gravitational forces between to α particles
(i.e., helium nuclei).
4
Figure 6: This is Fig. 21.12 showing the forces between two point charges q1 and q2 .
3.4
Forces between two point charges
N.B. The electrical forces between two stationary point charges is along
a straight line joining their positions, either towards each other
(i.e., attractive), or away from each other (i.e., repulsive).
5
4
Electric Field and Electric Forces
A charged body creates an “electric field” in the space around it.
~ is determined by measuring the force
Figure 7: This is Fig. 21.15 showing how the electric field E
~
Fo on a test charge qo at various locations around a charged body.
~ is known, the force on any charge q “at that location” can
Once the electric field E
be calculated using the equation:
~
F~ = q E
(Force on q in an electric field)
(4)
This is similar to the equation we used for measuring the gravitational force on any
mass m near the surface of the earth:
F~g = m ~g
(Force on m in a gravitational field)
6
4.1
Electric field due to a Point Charge
~ produced at the field point P . The
Figure 8: This is Fig. 21.17 showing the electric field vector E
~ points away from the + charge while the electric field E
~ point toward the −charge.
electric field E
~ = k q r̂
E
r2
(5)
where r̂ points in the direction from the source S “to” the field point P . The
~ is determined solely by the sign of the charge q.
direction of the electric field E
4.2
Electric-Field Vector due to a Point Charge
~ produced by a point charge q at a field point P .
Figure 9: This is Fig. 21.19 showing electric field E
7
4.3
Electron in a Uniform Field
Figure 10: This is Fig. 21.20 showing the acceleration of an electron in a uniform electric field. The
electric field in this example is E = 1.00 × 104 N/C.
Some of the physical quantities that can be calculated from the above figure
include:
• acceleration
• velocity
• kinetic energy
K = 21 mv 2
• time
8
5
Electric Field Calculations
“Real life” is more than a single point-charge. Let’s take a look at the electric field
produced by multiple point charges, or by a continuous distribution of charges.
5.1
Electric field due to multiple point charges
~ produced by two charges, one positive, the
Figure 11: This is Fig. 21.21 showing the electric field E
other negative.
~ = E
~1 + E
~2
E
N.B. The picture above illustrates the principle of superposition for multiple
electric field vectors.
9
5.2
Field due to an Electric Dipole
~ produced by two charges, one positive,
Figure 12: This is Fig. 21.22 showing the electric field E
the other negative. For an electric dipole, the two charges must be equal and opposite, q2 = −q1 .
In this figure, point c is the vertex of an isosceles triangle, and therefore the same distance r from
both of the charges. q1 = 12 nC.
~a = 1
E
4πo
|q2 |
q1
î
+
î
(0.06m)2
(0.04m)2
~c = E
~ 1c + E
~ 2c =
E
1
(q1 r̂1 + q2 r̂2 )
4πo r2
10
(6)
5.3
Field due to a ring of charge
~ produced by a uniform ring of charge
Figure 13: This is Fig. 21.23 showing the electric field E
having a charge per unit length λ = Q/2πa..
~ =
E
Question:
Qx
1
î
4πo (x2 + a2 )3/2
(7)
What is the electric field when the field point P is at a distance
far away from the charged ring?
~ =
E
1 Q
î
4πo x2
11
(8)
5.4
Field due to a Charged Line Segment
~ produced by a finite line segment of charge
Figure 14: This is Fig. 21.24 showing the electric field E
Q uniformly distributed over a length 2a. The charge per unit length is λ = Q/2a.
~ =
E
Question:
1
Q
√
î
4πo x x2 + a2
(9)
What is the electric field if the line segment becomes infinitely
long, but λ remains the same?
~ =
E
λ
î
2πo x
12
(10)
5.5
Field due to a Uniformly Charge Disk
~ produced by a uniform disk of charge
Figure 15: This is Fig. 21.25 showing the electric field E
2
where the charge per unit area is σ = Q/πR .
1 2π σ rx dr
4πo (x2 + r2 )3/2
"
#
σ
1
Ex =
1− p
2o
(R2 /x2 ) + 1
dEx =
Question:
(11)
(12)
~ when the disk becomes very large?
What is the electric field E
Ex =
13
σ
2o
(13)
5.6
Field due to Two Oppositely Charged Infinite Plates
~ inside and outside of two charged plates
Figure 16: This is Fig. 21.26 showing the electric field E
having uniform but opposite charges. The charge densities of the two plates are +σ and −σ, where
σ = Q/area.
Inside the plates
~ = σ ĵ
E
o
Outside the plates
~ = ~0
E
~ is uniform everywhere inside the plates
N.B. The electric field vector E
as long as you are not close to the edges.
14
6
Electric Field Lines
Figure 17: This is Fig. 21.28 showing the field lines due to three different charge distributions. The
positive charges are the “source” of the electric field while the negative charges are the “sinks” for
the electric field.
7
Electric Dipoles
Electric dipoles are commonly found in atomic and molecular systems. A dipole
moment is described by two equal-but-opposite charges +q and −q, separated by
a distance d. The Dipole Moment is is defined as follows:
p~ = q d~
(14)
where d~ is the displacement vector pointing from the negative charge to the positive
charge.
15
7.1
Torque on an Electric Dipole
The torque on an electric dipole is defined as follows:
7.2
~
~τ = ~r × F~ = p~ × E
(15)
|~τ | = p E sin θ
(16)
Potential Energy of an Electric Dipole
The potential energy for an electric dipole in an electric field is:
~
U = −~p · E
(17)
U = −p E cos θ
(18)
Figure 18: This is Fig. 21.31 showing the torque on an electric dipole p~ in a uniform electric field.
16