THE COPPERBELT UNIVERSITY
SCHOOL OF MATHEMATICS AND NATURAL SCIENCES
DEPARTMENT OF PHYSICS
2020/2021 ACADEMIC YEAR LECTURE NOTES
PHYSICS II
PH 210
CHAPTER 1: ELECTRIC CHARGE; COULOMB’S
LAW; ELECTRIC FIELD; ELECTRIC POTENTIAL
In this chapter, we study one of the fundamental forces of nature, the electric force.
We will discuss the existence of electric of electric charges and electric forces, the
basic properties of electrostatic forces, Coulomb’s law and its application to simple
charge distributions. In addition, we introduce the concept of an electric field
associated with a variety of charge distributions, and the electric potential.
1.1
1.1.1
•
•
•
Electric Charge
Properties of charges
like charges repel
unlike charges attract
charges can move but charge is conserved
1.1.2 Law of conservation of charge:
The net amount of electric charge produced in any process is zero.
Although there are two kinds of charged particles in an atom, electrons are the charges that
usually move around. A proton is roughly 2000 times more massive than an electron. Charges
are quantized (come in units of 𝑒 = 1.6 × 10−19 C). The charge of an electron is – 𝑒 =
– 1.6 × 10−19 coulombs. The charge of a proton is +𝑒 = +1.6 × 10−19 coulombs.
1.2 Coulomb’s Law
Coulomb’s law quantifies the magnitude of the electrostatic force. Coulomb’s law gives the
force (in newtons) between charges 𝑞1 and 𝑞2 , where 𝑟12 is the distance in meters between the
charges, and 𝑘 = 9 × 109 N·m2/C2.
𝐹12 = 𝑘
|𝑞1 𝑞2 |
2
𝑟12
(1.1)
Any correct variant of Equation (1.1) is “legal” and may be used.
Force is a vector quantity. Equation (1.1) gives the magnitude of the force. Use your diagram
for the problem to figure out the direction. If the charges are opposite in sign, the force is
attractive; if the charges are the same in sign, the force is repulsive.
Coulomb’s Law is valid for point charges. If the charged objects are spherical and the charge
is uniformly distributed, r12 is the distance between the centers of the spheres.
If more than one charge is involved, the net force is the vector sum of all forces (superposition).
For objects with complex shapes, you must add up all the forces acting on each separate charge
(calculus!!).
1.3
•
•
The Electric Field
A charged particle propagates (sends out) a "field" into all space.
Other charged particles sense the field, and “know” that the first one is there.
Figure 1.1
We define the electric field by the force it exerts on a small test charge 𝑞0 :
𝐸⃗ =
⃗⃗⃗
𝐹0
𝑞0
(1.2)
The subscript “0” reminds you the force is on the “test charge.”
If the test charge is "too big" it perturbs the electric field, so the “correct” definition is
⃗⃗⃗
𝐹0
𝑞0 →0 𝑞0
𝐸⃗ = lim
Any time you know the electric field, you can use Equation 1.2 to calculate the force on a
charged particle in that electric field. 𝐹 = 𝑞𝐸⃗.
Equation 1.2 tells you the direction of the electric field is the direction of the force exerted on
a POSITIVE test charge. The absence of absolute value signs around 𝑞 means you MUST
include the sign of 𝑞 in your work. The units of electric field are newtons/coulomb and can
also be expressed as volts/meter.
𝐸=
N V
=
C m
The electric field exists independent of whether there is a charged particle around to “feel” it.
The direction of the electric field is away from positive (and towards negative).
1.3.1 The Electric Field Due to a Point Charge
Coulomb's law says
𝐹12 = 𝑘
|𝑞1 𝑞2 |
2
𝑟12
This law tells us the electric field due to a point charge 𝑞 is
𝐸𝑞 = 𝑘
|𝑞|
𝑟2
If we define as a unit vector from the source point to the field point…
(1.3)
1.3.2 A Dipole
A combination of two electric charges with equal magnitude and opposite sign, separated by a
fixed distance, is called a dipole.
The charge on this dipole is 𝑞 (not zero, not +𝑞, not – 𝑞, not 2𝑞). The distance between the
charges is 𝑑. Dipoles are “everywhere” in nature. This is an electric dipole. Later in the course
we’ll study magnetic dipoles. Figure 1.2 shows the electric field lines of an electric dipole.
Figure 1.2 Electric field lines of an electric dipole.
1.3.3 Motion of a Charged Particle in a Uniform Electric Field
A charged particle in an electric field experiences a force, and if it is free to move, acceleration
is produced.
.
If the only force is due to the electric field, then
⃗ = 𝑚𝒂
⃗
⃗ = 𝑞𝑬
∑𝑭
If 𝐸 is constant, then 𝑎 is constant, and you can use the equations of kinematics.
There are two kinds of electric field problems in this unit:
1. Given an electric field, calculate the force on a charged particle.
𝑭 = 𝑞𝑬
2. Given one or more charged particles, calculate the electric field they produce.
𝐸=𝑘
|𝑞|
𝑟2
Make sure you understand which kind of problem you are working on!
1.4
Potential difference, electrical potential energy and electric potential
When a test charge is moved from A to B in an electric field, the work done on the charge is:
𝐹⃑ ∙ 𝑠⃑ = 𝑞 , 𝐸⃗⃑ ∙ 𝑠⃑
A charged particle that is free to move in an electric field is analogous to a massive object that
is free to move in a gravitational field. The exception is that massive objects move in only one
direction (left to their own devices) in a gravitational field
Just as an external force performs work in displacing an object uphill and the gravitational force
does work in displacing an object downhill in a gravitational field, an external force does work
in displacing a charge from a region of lesser electric potential to a region of greater electric
potential and vice versa in an electrical field.
We previously defined the work done by a conservative force in displacing an object as the
negative of the change in potential energy, ∆U. In the case of the electric force:
∆𝑈 = −𝑞 , 𝐸⃗⃑ ∙ 𝑠⃑
For the displacement A to B the change in potential energy is:
∆𝑈 = 𝑈𝐵 − 𝑈𝐴 = −𝑞 , 𝐸⃗⃑ ∙ 𝑠⃑
Since the force involved (the Coulomb force) is conservative, the work does not depend upon
the path taken between 𝐴 and 𝐵.
1.4.1 Potential difference
The potential difference between two points in an electric field, A and B, denoted 𝑉𝐵 − 𝑉𝐴 is
defined as the change in potential energy divided by a charge defined as a test charge, 𝑞 , :
𝑉𝐵 − 𝑉𝐴 =
𝑈𝐵 − 𝑈𝐴
= −𝐸𝑠
𝑞,
Potential difference is not the same as potential energy!
PD (ΔV) is proportional to PE (ΔU). The two are related by a factor of q’.
∆𝑉 =
∆𝑈
= −𝐸𝑠
𝑞′
• Because PE is scalar, PD is also a scalar.
• PD may be thought of as the potential energy per unit charge.
• The change in potential energy of a charge is the negative of the work done by the electric
force.
• The potential difference 𝑉𝐵 − 𝑉𝐴 is equal to the work per unit charge that an external agent
must perform to move a test charge from 𝐴 to 𝐵 without a change in kinetic energy.
• Notice that we have defined only changes in potential. That is because only differences in 𝑉
are meaningful.
• The electric potential function is often taken to be zero at some convenient reference point.
Usually (though not always) this reference point is chosen to be a point remote enough from
the charge or charge distribution producing the electric field that it may be taken as infinity. If
this is the case, the electric potential at any point P is:
𝑉𝑝 = − 𝐸𝑠
where 𝑉𝑝 is the work per unit charge required to bring a charge from infinity to point 𝑃.
• The unit of electrical potential is the Volt. 1𝑉 ≡ 1𝐽/𝐶
• The electric field, normally expressed in units of Newtons/Coulomb may be also expressed
conveniently in terms of volts/meter:
A unit of energy commonly used in physics is the electron volt (𝑒𝑉). An electron volt is the
change in energy that an electron (or proton) acquires when moving through a potential
difference of 1Volt.
1𝑒𝑉 = 1.6 × 10−19 𝐶 × 1𝑉 = 1.6 × 10−19 𝐽
1.4.2 The Potential of a Point Charge
Consider an isolated positive charge, 𝑞, as shown at left. Recall that such a charge produces an
electric field oriented radially away from the charge and that the equipotential surfaces are a
series of concentric shells enclosing 𝑞. The path between 𝐴 and 𝐵 is 𝑠. We wish to compute
the electrical potential between points 𝐴 and 𝐵.
Figure 1.3
Potential of a point charge
Recall 𝐸⃗⃑ = 𝑘
𝑞
𝑟̂ , this means that 𝐸𝑠 may be expressed as 𝑘
𝑞
𝑟̂ ∙ 𝑠 . Notice that any
displacement, 𝑑𝑠, produces a change 𝛥𝑟 in the magnitude of 𝑟 and the result is that 𝐸⃗⃑ ∙ 𝑠⃑ =
𝑞
𝑘 𝑟 2 ∆𝑟. It may be shown that
1 1
𝑉𝐵 − 𝑉𝐴 = 𝑘𝑞 [ − ]
𝑟𝐵 𝑟𝐴
𝑟2
𝑟2
Notice that the result depends only upon the radial coordinates 𝑟𝐴 and 𝑟𝐵 . If 𝐴 is taken to be
infinity, then the potential at point 𝐵 is:
𝑞
𝑉𝐵 = 𝑘
𝑟
𝑞
In general, the potential at some point 𝑃 due to a point charge is 𝑉𝑝 = 𝑘 𝑟 .
1.4.3 The Potential of an array of point charges
The electric potential of two or more point charges is merely the superposition of the individual
potentials at a given point in space
𝑞𝑖
𝑉𝑃 = 𝑘𝛴
𝑟𝑖
The potential energy of a system of two charged particles is:
𝑈=𝑘
𝑞1 𝑞2
𝑟12
• Notice that if the polarity of both charges is the same, 𝑈 is positive.
• This is consistent with the fact that like charges repel and so work must be done on the system
in order to bring the charges together.
The potential energy of a system of three charged particles is:
𝑈 = 𝑘[
𝑞1 𝑞2 𝑞1 𝑞3 𝑞2 𝑞3
+
+
]
𝑟12
𝑟13
𝑟23
• Imagine that 𝑞1 is fixed at some position and that 𝑞2 and 𝑞3 are at infinity.
• The work required to bring 𝑞2 from infinity to a position near 𝑞1 is represented by the first
term.
• The work required to bring 𝑞3 from infinity to its position near 𝑞1 and 𝑞2 is represented by
the last two terms.
• The result is independent of the order in which the charges are moved.
1.4.4 Potential Gradient (Determining Electric Field from Potential)
The electric field vector points from higher to lower potentials. More specifically, 𝐸⃗⃑ points
along shortest distance from a higher equipotential surface to a lower equipotential surface.
E
Figure 1.4
You can use 𝐸⃗⃑ to calculate 𝑉:
𝑏
𝑉𝑏 − 𝑉𝑎 = − ∫ 𝐸⃗⃑ ∙ 𝑑𝑙⃑
𝑎
You can use the differential version of this equation to calculate 𝐸⃗⃑ from a known 𝑉:
𝑑𝑉 = −𝐸⃗⃑ ∙ 𝑑𝑙⃑
⟹ 𝐸𝑙 = −
𝑑𝑉
𝑑𝑙
For spherically symmetric charge distribution:
𝐸𝑙 = −
𝑑𝑉
𝑑𝑙
𝐸𝑥 = −
𝑑𝑉
𝑑𝑥
In one dimension:
In three dimensions:
𝐸𝑥 = −
𝜕𝑉
𝜕𝑉
𝜕𝑉
, 𝐸𝑦 = − , 𝐸𝑧 = −
𝜕𝑥
𝜕𝑦
𝜕𝑧
Or
𝜕𝑉
𝜕𝑉
𝜕𝑉
⃗⃑𝑽
𝒊−
𝒋−
𝒌 = −∇
𝜕𝑥
𝜕𝑦
𝜕𝑧
Calculate −𝜕𝑉/𝜕(whatever) including all signs. If the result is +, 𝐸⃗⃑ points along the
+(whatever) direction. If the result is −, 𝐸⃗⃑ points along the –(whatever) direction.
𝐸⃗⃑ == −