PHYSICS 3 (ELECTRICITY AND MAGNETISM) - CHAPTER 1 pot

Electricity and Magnetism
1

PHYSICS 3 (ELECTRICITY AND MAGNETISM)
By Assoc. Prof. Dr. Dương Hoài Nghĩa
Email : -
Web site : www4.hcmut.edu.vn/~dhnghia
phone: 0918 416 425
1. Code of module: PH015IU
2. Level:
3. Credits: 3 Credits – 2 hours per week of lectures
4. Prerequisites: Basic Mathematical Analytics, High School Physics
5. Evaluation & grading: 2-hour examination
6. Objectives of module:
• Know and understand basic physical processes and phenomena.
• Solve basic physics problem by applying both theoretical and experimental techniques.
• Understand and acquire skills needed to use physical laws governing real process and to solve
them in the engineering environment.

7. Synopsis of module:

Chapter 1: Electric Fields
• Properties of Electric Charges
• Insulators and Conductors
• Coulomb’s Law
• The Electric Field. Electric Field Lines
• Electric Field of a Continuous Charge Distribution
• Motion of Charged Particles in a Uniform Electric Field
• Electric Flux. Gauss’s Law
• Conductors in Electrostatic Equilibrium
• Insulator with Uniform Charge Density

Chapter 2: Electric Energy and Capacitance
• Potential Difference and Electric Potential
• Potential Differences in a Uniform Electric Field
• Electric Potential and Potential Energy Due to Point Charges
• Electric Potential Due to Continuous Charge Distributions
• Electric Potential Due to a Charged Conductor
• Capacitance.
• Combinations of Capacitors
• Energy Stored in a Charged Capacitor
• Capacitors with Dielectrics

Chapter 3: Current and Resistance, Direct Current Circuits
• Electric Current
• A Model for Electrical Conduction
• Resistance and Ohm’s Law
• Electrical Energy and Power
• Electromotive Force
• Kirchoff’s Rules
• Resistors in Series and in Parallel
• RC Circuits
Electricity and Magnetism
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Chapter 4: Magnetism
• The Magnetic Field
• Magnetic Force Acting on a Current-Carrying Conductor
• Torque on a Current Loop in a Uniform Magnetic Field
• Motion of a Charged Particle in a Uniform Magnetic Field

• The Hall Effect
• The Biot–Savart Law
• Ampere’s Law
• The Magnetic Field of a Solenoid
• Magnetic Flux. Gauss’s Law in Magnetism
• Displacement Current and the General Form of Ampère’s Law
• Magnetism in Matter
• The Magnetic Field of the Earth

Chapter 5: Electromagnetic Induction
• Faraday’s Law of Induction
• Motional emf
• Lenz’s Law
• Induced emf and Electric Fields
• Self-Inductance
• RL Circuits
• Energy in a Magnetic Field
• Mutual Inductance

Chapter 6: Alternating-Current Circuits
• AC Sources and Phasors
• Resistors in an ac Circuit
• Inductors in an ac Circuit
• Capacitors in an ac Circuit
• The RLC Series Circuit
• Power in an ac Circuit
• Resonance in a Series RLC Circuit
• The Transformer and Power Transmission

Chapter 7: Electromagnetic Waves

• Maxwell’s Equations and Hertz’s Discoveries
• Plane Electromagnetic Waves
• Energy Carried by Electromagnetic Waves
• Momentum and Radiation Pressure
• Production of Electromagnetic Waves by an Antenna
• The Spectrum of Electromagnetic Waves

8. References:
1. Halliday D., Resnick R. and Merrill, J. (1988). Fundamentals of Physics. Extended third edition.
John Willey and Sons, Inc.
2. Alonso M. and Finn E.J. (1992). Physics. Addison-Wesley Publishing Company
3. Hecht, E. (2000). Physics. Calculus. Second Edition. Brooks/Cole.
4. Faughn/Serway (2006). Serway’s College Physics. Thomson Brooks/Cole.
Electricity and Magnetism
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Chapter 1 ELECTRIC FIELDS

1.1 Properties of Electric Charges

Every object contains a vast amount of electric charge. Object which contains equal amounts of the two
kinds of charge is call electrically neutral. One with an imbalance is electrically charged. The net charge
of an object is the difference between the amount of positive charge and negative charge of the object.

Experiment (Fig. 1.1 and Fig. 1.2): Rub one end of a glass rod with silk → electrons are transfered to silk
→ the glass rod contains a positive net charge. Rub one end of a plastic rod with fur → electrons are
transfered to the plastic rod → the plastic rod contains a negative net charge.

Fig. 1.1 : Charges with the same electrical sign repel each other





Fig. 1.2 : Charges with opposite electrical signs attract each other


Electric charge is conserved. The net charge of any isolated system can not change.

Electric charge is quantized. Elementary charge is e = 1.602 x 10
-19
C. The charge of an electron is –e.
The charge of a proton is +e.

Electric current

dq
i =
dt
[A] (1.1)


Electricity and Magnetism
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1.2 Insulators and Conductors

Conductors
: materials through which charge can move rather freely (metal, tap water, human body, …).
Insulators (nonconductor)
: materials through which charge can not move freely (plastic, glass,
chemically pure water, … ).
Superconductors
: materials that are perfect conductors, allowing charge to move without any hindrance.

Experiment
(Fig. 1.3) : Put a plastic rod with negative net charge near a neutral copper rod. Conduction
electrons on the copper rod are repelled to the far end of the copper rod by the negative charge on the
plastic rod. Then the negative charge on the plastic rod attracts the remaining positive charge on the near
end of the copper rod.

Fig. 1.3
1.3 Coulomb’s Law
1) The electrostatic force of attraction or repulsion between two charged particles (point charges) which are
at rest and in vacuum


1 2
2
|q |.|q |
F = k
r
[N] (1.2)

where
o
1

k =
4
πε
= 8.99 x 10
9
Nm
2
/C
2

and
ε
o
= 8.85 x 10
-12
C
2
/Nm
2
: the permittivity constant



Fig. 1.4 Fig. 1.5




Fig. 1.6 Fig. 1.7

2) The force on any one charge due to a collection of other charges is the vector sum of the forces due to
each individual charge (Fig. 1.7).

i
i
F= F
∑
r r
(1.3)
Electricity and Magnetism
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Experiment
(Fig. 1.8)
a) An aluminum ball with zero net charge assumes a vertical position at the end of a thread or string.
b) A negatively charged ball is brought close to the neutral ball which becomes polarized.
c) The positive pole of the aluminum ball is attracted to the negatively charged ball up to contact.
d) After contact the aluminum ball becomes negatively charged by
charge transfer
through the point of
contact and is repelled by the negatively charged ball.
e) The aluminum ball will stop at equilibrium in a position deviated from vertical at an angle

determined by the charges of the balls.

Fig. 1.8

3)
Properties

A shell of uniform charge attracts or repels a charged particle that is outside the shell as if all the shell’s
charge were concentrated at its center.

If a charged particle is located inside a shell of uniform charge, there is no net electrostatic force on the
particle from the shell.


1.4 The Electric Field. Electric Field Lines
1) Electric field is defined as the electric force per unit charge. The direction of the field is taken to be the
direction of the force it would exert on a positive test charge. The electric field is radially outward from a
positive charge and radially in toward a negative point charge.

2) To find the electric field at point P near a charged object : Place a positive charge q
o
(called test charge) at
P. Measure the electrostatic force
F
r
that acts on the test charge. The electric field at point P due to the
charged object

o

F
E =
q
r
r
[V/m, N/C] (1.4)
Electricity and Magnetism
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3) Field line diagrams : A convenient way to visualize the electric field due to any charge distribution is to
draw a field line diagram. At any point the field line has the same direction as the electric field vector.
Electric field lines diverge from positive charges and converge into negative charges. Rules for
constructing filed lines:
a) Field lines begin at positive charge and end at negative charge
b) The number of field lines shown diverging from or converging into a point is proportional to the
magnitude of the charge.
c) Field lines are spherically symmetric near a point charge
d) If the system has a net charge, the field lines are spherically symmetric at great distances
e) Field lines never cross each other.

4) The electric field of a point charge

2
o
r4πε
=
|
q
|

E [N/C] (1.5)



Fig. 1.9 : Electric field lines of a point charge





Fig. 1.10 : Electric field of 2 charges

Electricity and Magnetism
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5) The electric field of an electric dipole : an electric dipole consists of two charges + q and –q, of equal
magnitude but opposite sign, that are separated by a distance d (Fig. 1.11).
E = E
+
- E
-
=
2
o
r4
+
πε
q
-
2

o
r4
−
πε
q
=
2
o
)2/dz(4 −πε
q
-
2
o
)2/dz(4 +πε
q

E
≈

3
o
z2πε
qd
=
3
o
z2πε
p
(1.6)
p = qd : dipole moment [Cm] (1.7)

The vector p points from the negative charge to the positive charge (Fig. 1.12).



Fig. 1.11 Fig. 1.12




1.5 Electric Field of a Continuous Charge Distribution
1) The electric field of a charged ring
(Fig. 1.13)
λ
: linear charge density [C/m]
⇒
dq =
λ
ds (1.8)

dE =
2
o
r4
ds
πε
λ
=
)Rz(4
ds
22

o
+πε
λ
(1.9)

E =
∫
θ)cos(dE
=
( )
∫
+πε
λ
2/3
22
o
Rz4
zds
=
( )
∫
π
+πε
λ
R2
0
2/3
22
o
ds

Rz4
z


E =
( )
2/3
22
o
Rz4
zR2
+πε
λ
π
=
( )
2/3
22
o
Rz4
qz
+πε
(1.10)
(q = 2
π
R
λ
: the total charge on the ring)

if z >> R then E =

2
o
z4
q
πε
: from a large distance, the ring looks like a point charge.

if z = 0 then E = 0.
Electricity and Magnetism
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Fig. 1.13 : A ring of uniform positive charge Fig. 1.14 : A disk of uniform positive charge


2) The electric field of a charged disk
(Fig. 1.14)
σ
: surface charge density [C/m
2
]
dq =
σ
dA =
σ
2
π
rdr : the charge on the ring with radius r (1.11)
dE =
( )

2/3
22
o
Rz4
rdr2z
+πε
π
σ
(1.12)
E =
∫
dE =
( )
∫
+ε
σ
2/3
22
o
Rz2
rdrz
=









+
−
ε
σ
22
o
Rz
z
1
2
(1.13)
As R
→

∞
: E
→

o
2ε
σ
: electric field produced by an infinite sheet of uniform charge


1.6 Motion of charged particles in a uniform electric field
1) Point charge in an electric field : The electrostatic force on a point charge q


EqF
r

r
=
[N] (1.14)


Fig. 1.15

2) Fig. 1.16 describes the essential features of an ink-jet printer. Drops are shot out from generator G and
receive a (negative) charge in a charging unit C. An input signal from a computer controls the charge
given to each drop and thus the effect of field E on the drop and the position on the paper at which the
drop lands. About 100 tiny drops are needed to form a single character.
Electricity and Magnetism
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Let m be the mass of the drop. The acceleration of the drop along the vertical axis is
a
y
= qE/m
→
y =
m
2
qEt
2

The speed of the drop along the horizontal axis
v
x
= constant
→

x = v
x
t
Let L be the length of the deflecting plate, the vertical deflection of the drop is y =
2
x
2
mv2
qEL



Fig. 1.16 Fig. 1.17

3) A dipole in an electric field : Fig. 1.17 shows an electric dipole in a uniform external electric field E. Two
centers of equal but opposite charge are separated by distance d. The line between them represents rigid
connection. The magnitude of the net torque

τ
= -Eqsin(
θ
)d = -pEsin(
θ
) [Nm] (1.15)
(by convention,
τ
< 0 because it tends to rotate the dipole in the clockwise direction). The torque acting on
a dipole tends to rotate it into the direction of the field E.

If we choose the potential energy to be zero when

θ
= 90
o
then the potential energy U at any angle
θ
is
U =
∫
θ
θτ−
o
90
d = -pEcos(
θ
) = Ep
r
r
− [J] (1.16)

4) In a water molecule, the two hydrogen atoms and the oxygen atom do not lie on a straight line but form an
angle of about 105
o
. Moreover the 10 electrons of the molecule tend to remain closer to the oxygen
nucleus than to the hydrogen nuclei. This makes the oxygen side of the molecule slightly more negative
than the hydrogen side and creates an electric dipole moment p that points along the symmetry axis of the
molecule. If the water molecule is placed in an external electric field, it is rotated into the direction of the
electric field (as shown in Fig. 1.17).


Fig. 1.18: a H

2
O molecule
Electricity and Magnetism
10

Example
: A neutral water molecule H
2
O in its vapor state has an electric dipole moment of magnitude
6.2x10
-30
Cm. How far apart are the molecule’s centers of positive and negative charge ? If the molecule is
placed in an electric field of 1.5x10
4
N/C, what maximum torque can the field exert on it ? How much
work much an external agent do to rotate this molecule by 180
o
in this field, starting from its fully aligned
position ?

Since there are 10 electrons and 10 protons in a neutral water molecule, the magnitude of its dipole
moment is
p = qd = 10ed
⇒
d = 3.9 pm
The torque on a dipole is maximum when the angle between E and p is 90
o
.

τ

max
= pEsin(90
o
) = 9.3x10
-26
Nm
The work done by an external agent
-pEcos(180
o
) - [ -pEcos(0
o
) ] = 1.9x10
-25
J



1.7 Electric flux. Gauss’ law
1) Flux of an Electric Field
(Fig. 1.19)

The electric flux through an area is defined as the electric field multiplied by the area of the surface
projected in a plane perpendicular to the field.


Φ
=
A.dE
r
r

= E.dA.cos(
θ
) [Nm
2
/C] (1.17)




Fig. 1.19 Fig. 1.20 Fig. 1.21


It is often simpler to find the flux through one surface of an object than through another. In the case of the
cone the flux through the base (Area =
R
2
) is the same as the flux through the lateral surface, but it is
much easier to calculate the flux through the base.

= E A
lateral
cos( ) = E ( R
2
)


2) Gauss’ Law
The total of the electric flux out of a closed surface is equal to the charge enclosed divided by the
permittivity

Φ
=
o
ε
Q
(1.18)
Electricity and Magnetism
11


Fig. 1.22

Example 1
: A cylindrical Gaussian surface, closed by end caps, is immersed in a uniform electric field.
The cylinder axis is parallel to the field direction (Fig. 1.23)

Φ
=
∫
A.dE
r
r
=
∫
a
A.dE
r
r
+

∫
b
A.dE
r
r
+
∫
c
A.dE
r
r
= 0
(a : left cap, b : right cap, c : cylinder surface)



Fig. 1.23 Fig. 1.24

Example 2
: A spherical Gaussian surface (of radius r) centered on a point charge q (Fig. 1.24)

2
o
r4πε
=
q
E
[N/C] (1.19)

Φ

=
∫
A.dE
r
r
=
∫
πε
dA
r4
2
o
q
=
2
o
2
r4
r4
πε
π
q
=
o
ε
q
[Nm
2
/C] (1.20)

1.8 Conductors in electrostatic equilibrium
1) The net electric charge of an isolated conductor is located entirely on the outer surface of the conductor.
Because the mutual repulsion of like charges from Coulomb's law demands that the charges are as far
apart as possible, hence on the outer surface of the conductor. Using Gauss’ law and this fact, we deduce
that the electric field inside the conductor is zero. Any net electric field inside the conductor would cause
charge to move since it is abundant and mobile.

2) The external electric field near the surface of a charged conductor is perpendicular to the surface. Because
if there were a field component parallel to the surface, it would cause mobile charge to move along the
surface. This violates the assumption of equilibrium. Using Gauss’ law and the fact that the electric field
inside the conductor is zero, we deduce the external electric field
E =
o
ε
σ
(1.21)
where
σ
: surface charge density.

Electricity and Magnetism
12

1.9 Insulator with uniform charge density
1) Infinite nonconducting line of charge : the electric field at any point due to an infinite line of charge with
uniform linear charge density
λ
is perpendicular to the line of charge and has magnitude
E =
r2

o
πε
λ
(1.22)
where r is the perpendicular distance from the line of charge to the point.

2) Infinite nonconducting sheet of charge : the electric field due to an infinite nonconducting sheet with
uniform surface charge density
σ
is perpendicular to the plane of the sheet and has magnitude
E =
o
2ε
σ
(1.23)

3) Nonconducting spherical shell : The electric field outside a spherical shell of charge with radius R,
uniform volume charge density and total charge q, is directed radially and has magnitude
E =
2
o
r4
q
πε
, for r
≥
R (1.24)
The charge behaves, for external points, as if it were all located at the center of the sphere. The field inside
the shell is
E =

3
o
R4
qr
πε
, for r < R (1.25)
r is the distance from the center of the shell to the point at which E is measured.

Problems
Electric charges
1.1) What is the magnitude and direction of the electrostatic force on each charge in Fig. P1.1 ? The charges
are q
1
= 10e, q
2
= -20e, where e = 1.602 x 10
-19
C is the elementary charge, and r = 1mm.



Fig. P1.1 Fig. P1.2

1.2) In Fig. P1.2, q
1
= 10e, q
2
= -20e, q
3
= -10e, where e = 1.602 x 10

-19
C, r = 0.1mm. What is the magnitude
and direction of the electrostatic force on each charge ?

1.3) A proton and two electrons form three corners of an equilateral triangle with sides of length 3x10
-6
m.
What is the magnitude of the net electrostatic force at each corner?

1.4) Two equally charged particles are held 3.2x10
-3
m apart and then released from rest. The initial
acceleration of the first particle is 7m/s
2
and that of the second is 9m/s
2
. The mass of the first particle is
6.3x10
-7
kg. Find the mass of the second particle and the magnitude of the charge of each particle.

1.5) The magnitude of the electrostatic force between the two identical ions that are separated by a distance of
5x10
-10
m is 3.7x10
-9
N. What is the charge of each ion ? How many electrons are missing from each ion.

1.6) In Fig. P1.3, q
1

= 10e, q
2
= -20e, q
3
= -10e, q
4
= 20e, where e = 1.602 x 10
-19
C, r = 0.1mm. What is the
magnitude and direction electrostatic force on each charge ?

Electricity and Magnetism
13

1.7) In Fig. P1.4, two tiny conducting balls of identical mass m and identical charge q hang from
nonconducting threads of length L. Assume that
θ
is so small that tan(
θ
)
≈
sin(
θ
).
a) Find the equilibrium separation x of the balls.
b) Explain what happens to the balls if one of them is discharged.


Fig. P1.3 Fig. P1.4 Fig. P1.5

1.8) In crystals of the salt cesium chloride, cesium ions Cs
+
form the eight corners of a cube and a chlorine ion
Cl
-
is at the cube’s center (Fig. P1.5). The edge length of the cube is r = 0.40 nm. The Cs
+
ions are each
deficient by one electron (and thus each has a charge of +e).
a) What is the magnitude of the net electrostatic force exerted on the Cl
-
ion by the eight Cs
+
ions at the
corners of the cube ?
b) If one of the Cs
+
ions is missing, the crystal is said to have a defect. What is the magnitude of the net
electrostatic force exerted on the Cl
-
ion by the seven remaining Cs
+
ions ?

Electric fields
1.9) In Fig. P1.6, q
1
= q, q
2
= -2q, q

3
= -q, q
4
= 2q, where q is the elementary charge, r = 0.1mm. What is the
electric field at the center of the square ?



Fig. P1.6 Fig. P1.7 : Plastic rod of charge –Q Fig. P1.8

1.10) What is the electric field due to the plastic rod with uniformly distributed charge -Q at point P (Fig. P1.7)?

1.11) A thin glass is bent into a semi-circle of radius a as shown below. A charge +q is uniformly distributed
along one half of the glass, and a charge -q is uniformly distributed along the other half of the ring. Use
Coulomb’s law to determine the magnitude and direction of the electric field strength at the point P (Fig.
P1.8).

1.12) Find the electric field a distance z above the midpoint of a straight line segment of length 2L which carries
a uniform linear charge density
λ
.

1.13) Find the electric field a distance
z
above one end of a straight line segment of length L, which carries a
uniform linear charge density
λ
.

Electricity and Magnetism

14

1.14) Find the electric field of a dipole at B, C, D in Fig. P1.9



Fig. P1.9 Fig. P1.10

1.15) An electric dipole consists of two charges q
1
= +2e and q
2
= -2e separated by a distance d = 10
-9
m. The
electric charges are placed along the y-axis as shown in Fig. P1.10. Suppose a constant external electric
field
j3i3E
ext
r
r
r
+=
N/C is applied.
(a) What is the magnitude and direction of the dipole moment?
(b) What is the magnitude and direction of the torque on the dipole?
(c) Do the electric fields of the charges q
1
and q
2

contribute to the torque on the dipole? Briefly explain
your answer.

1.16) An electric dipole with dipole moment
-30
10 1.24 )j4i3(p
xx
r
r
r
+= Cm is in an electric field
i4000E
r
r
=
N/C
(a) What is the potential energy of the electric dipole ?
(b) What is the torque acting on it ?
(c) If an external agent turns the dipole until its electric dipole moment is
-30
10 1.24 )j3i-4(p
xx
r
r
r
+= Cm. How much work is done by the agent.

1.17) The electric field at point P (x,y) of an electric dipole is E
x
= x and E

y
= -y where E
x
and E
y
are the
components of the electric field vector
E
r
in x and y axis respectively (Fig. P1.11). Find and draw the
electric field lines (the curves of electric force). Hint : dx / E
x
= dy / E
y



Fig. P1.11 Fig. P1.12

1.18) In fig. P1.12, a uniform, upward electric field
E
r
of magnitude E = 2,000 N/C has been set up between
two horizontal plates by charging the lower plate positively and the upper plate negatively. The plate has
length L = 10 cm and separation d = 2 cm. An electron is then shot between the plates from the left edge
of the lower plate. The initial velocity
o
v
r
of the electron makes an angle

α
= 45
°
with the lower plate and
has magnitude 6 x 10
6
m/s. Will the electron strike one of the plates ? If so which plate and how far
horizontally from the left edge will the electron strike ?


Electricity and Magnetism
15


Gauss’ law
1.19) Find the electric field at all points due to a long, solid cylinder of radius R and uniform linear charge
density
λ
.

1.20) A solid non-conducting sphere of radius R has a uniform charge distribution of volume density
ρ
s
Cm
-3
.
Determine an expression for the electric field inside and outside the sphere as a function of the distance
from the center of the sphere.

1.21) Consider an uncharged metal shell of inner radius a and outer radius b. If a charge +Q is placed within the

center of the shell, draw a diagram of the electric field around the charge +Q and within the shell. Using
Gauss' law, determine the strength of the electric field inside, within and outside the shell.

1.22) Consider a metal shell of inner radius a and outer radius b. What is the charge distribution on the inner
surface and outer surface of the shell
a) if negative charge is added to the outer surface of the shell from an external source.
b) if electrons are extracted from outer surface of the shell.
Determine the strength of the electric field inside, within and outside the shell.

1.23) Find the electric field of a long, nonconducting, solid cylinder of radius 4 cm which has a nonuniform
volume charge density
ρ
= Ar
2
where A = 2.5
µ
C/m
5
and r is the radial distance from the cylinder axis.

1.24) A charge distribution that is spherically symmetric but not uniform radially produces an electric field of
magnitude E = Kr
4
, directed radially outward from the center of the sphere. Here r is the radial distance
from that center and K is a constant. What is the volume density
ρ
of the charge distribution ?

1.25) Use Gauss's Law to find the electric field everywhere due to a uniformly charged insulator shell (Fig.
P1.13). The shell has a total charge Q, which is uniformly distributed throughout its volume.

Fig. P1.13 Fig. P1.14

1.26) In Fig. P1.14, a solid nonconducting sphere of radius a = 2 cm is concentric with a spherical conducting
shell of inner radius b = 1.5a and outer radius c = 1.7a. The sphere has a net uniform charge q
1
= +5 fC.
Determine an expression for the electric field as a function of the distance from the center of the sphere.

1.27) In Fig. P1.14, a conducting sphere of radius a = 2 mm is concentric with a spherical conducting shell of
inner radius b = 3 mm and outer radius c = 3.5 mm. The sphere has a net charge q
1
= +5 pC. Initially the
net charge of the conducting shell is zero.
a) Determine the charge distribution on the conducting shell.
b) Determine the electric field as function of the distance r from the center of the sphere.
c) Determine the electric potential as function of the distance r from the center of the sphere.

Electricity and Magnetism
16

Homeworks 1
H1.1 Four charges q
1
, q
2
, q
3

, q
4
form four corners of a square with side r [mm].
a) What is the magnitude of the net electrostatic force at each corner ?
b) What is the magnitude and direction of the electric field at the center of the square ?


Fig. H1.1 Fig. H1.3

n 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16
q
1
2e 2e 4e -2e 2e -4e 6e 6e 6e -6e 6e -8e e e 3e -e
q
2
-2e -4e -2e 2e 4e -2e -6e -8e -8e 6e 8e -6e -e -3e -e e
q
3
4e 4e -4e -4e -4e 4e 8e 8e -6e -8e -8e 8e 3e 3e -3e -3e
q
4
-4e -2e 2e 4e -2e 2e -8e -6e 8e 8e -6e 6e -3e -e e 3e
r 0.2 0.4 0.6 0.1 0.3 0.5 0.7 0.8 0.9 0.1 0.2 0.3 0.4 0.4 0.6 0.7

n 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32
q
1
3e 3e 5e -3e 3e -5e 7e 7e 7e -7e 7e -9e e e 3e -e
q
2

-3e -5e -3e 3e 5e -3e -7e -9e -9e 7e 9e -7e -e -3e -e e
q
3
5e 5e -5e -5e -5e 5e 9e 9e -7e -9e -9e 9e 3e 3e -3e -3e
q
4
-5e -3e 3e 5e -3e 3e -9e -7e 9e 9e -7e 7e -3e -e e 3e
r 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9

n 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48
q
1
2e 2e 8e -2e 2e -8e 4e 4e 4e -4e 4e -6e e e 3e -e
q
2
-2e -8e -2e 2e 8e -2e -4e -6e -6e 4e 6e -4e -e -3e -e e
q
3
8e 8e -8e -8e -8e 8e 6e 6e -4e -6e -6e 6e 3e 3e -3e -3e
q
4
-8e -2e 2e 8e -2e 2e -6e -4e 6e 6e -4e 4e -3e -e e 3e
r 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9

n 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64
q
1
4e 4e -4e -4e -4e 4e 8e 8e -6e -8e -8e 8e 3e 3e -3e -3e
q
2

-4e -2e 2e 4e -2e 2e -8e -6e 8e 8e -6e 6e -3e -e e 3e
q
3
3e 3e 5e -3e 3e -5e 7e 7e 7e -7e 7e -9e e e 3e -e
q
4
-3e -5e -3e 3e 5e -3e -7e -9e -9e 7e 9e -7e -e -3e -e e
r 0.2 0.4 0.6 0.1 0.3 0.5 0.7 0.8 0.9 0.1 0.2 0.3 0.4 0.4 0.6 0.7





Electricity and Magnetism
17

H1.2 Find the electric field a distance z [mm] above one end of a straight line segment of length L [mm], which
carries a uniform linear charge density
λ
[
µ
C/m].
n 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16
L 6 7 8 9 10 11 12 13 14 6 7 8 9 10 11 12
z 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9
λ

1 2 3 4 5 1 2 3 4 5 1 2 3 4 5 1

n 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32

L 13 14 6 7 8 9 10 11 12 13 14 6 7 8 9 10
z 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9
λ

2 3 4 5 1 2 3 4 5 1 2 3 4 5 1 2

n 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48
L 11 12 13 14 6 7 8 9 10 11 12 13 14 6 7 8
z 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9
λ

3 4 5 1 2 3 4 5 1 2 3 4 5 1 2 3

n 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64
L 9 10 11 12 13 14 6 7 8 9 10 9 10 11 12 13
z 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9
λ

4 5 1 2 3 4 5 1 2 3 4 5 1 2 3 4

H1.3 In Fig. H1.3, a solid nonconducting sphere of radius a [mm] is concentric with a spherical conducting
shell of inner radius b [mm] and outer radius c [mm]. The sphere has a net uniform charge q [fC].
Determine an expression for the electric field as a function of the distance from the center of the sphere.

n 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16
a 15 16 17 18 19 20 21 22 23 24 25 15 16 17 18 19
b 20 21 22 23 24 25 26 27 28 29 30 19 20 21 22 23
c 22 23 24 25 26 27 28 29 30 31 32 22 23 24 25 26
q 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23

n 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32
a 20 21 22 23 24 25 15 16 17 18 19 20 21 22 23 24
b 24 25 26 27 28 29 21 22 23 24 25 26 27 28 29 30
c 27 28 29 30 31 32 23 24 25 26 27 28 29 30 32 33
q 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39

n 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48
a 25 15 16 17 18 19 20 21 22 23 24 25 15 16 17 18
b 31 22 23 24 25 26 27 28 29 30 31 32 23 24 25 26
c 34 26 27 28 29 30 31 32 33 34 35 36 26 27 28 29
q 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55

n 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64
a 19 20 21 22 23 24 25 15 16 17 18 19 20 21 22 23
b 27 28 29 30 31 32 23 24 25 26 27 28 29 30 31 32
c 30 31 32 33 34 35 36 26 27 28 29 30 31 32 33 34
q 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39