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General Physics 2
Handout 1: Electric charge, force and field
Electric charge
The fundamental unit of electric charge is 𝑒
which is the magnitude of charge of single electron:
𝑒 = 1.6 × 10−19 C.
The unit of electric charge is coulomb (C). The electric
charges of objects in nature are found to be integral
multiples of 𝑒 such as 2𝑒, 3𝑒 and 150𝑒.
Figure 1: Very few or no free electrons
in insulator (left) and many free
electrons in conductor (right)
Conductor and insulator
Figure 1 shows the microscopic pictures of
insulators and conductors (metals). In insulators, the
electrons are tightly bound to the positive nuclei. There
is very little or no free electrons to move around. There
are many free (unbound) electrons in metals; these
electrons are weakly bound to positive nuclei and hence
are allowed to move. Therefore, metals are conductive;
they can conduct electrons.
Figure 2: Charge distribution on
insulator (left) and conductor (right)
Electric charge is distributed differently on
insulator and conductor as shown in Figure 2. In
insulator, the charge is located to points where the
charge is given. In conductors, the charge is spread
around the surface.
Electric charge transfer
Charging by friction By rubbing two substances
together, electrons from one substance may be
transferred to another. The one that loses electrons
becomes positively charged and the one that gains
electrons becomes negatively charged. The tendency of
a substance to acquire electrons is called “electron
affinity” as illustrated in Figure 3. For example, wool
gives up electron to silk shirt.
Charging by contact Electric charge can be
transferred by direct contact. In Figure 4, a negatively
charged rod touches an initially neutral metal sphere.
The negative charge is transferred to the sphere. After
the rod is removed, the sphere becomes negatively
charged.
Figure 3: Tendency to lose or gain
electrons of substances
Figure 4: Charging by contact
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Charging by induction In Figure 5, a negatively
charged rod is brought near a neutral metal sphere. The
electrons on the sphere are repelled to the far side from
the rod, inducing the positive charge on the near side.
The sphere is then grounded so that the electrons move
from the sphere to the ground. After removing the
grounding wire and the rod, the sphere are left with
positivity charge.
Figure 5: Charging by induction
Electroscope
The electroscope is a device used to detect excess
free charge (Figure 6). The metal ball on the top is
attached to gold leaves in a sealed chamber. When a
positively charged rod is brought near the metal ball of
the electroscope, the electrons are accumulated on the
ball, leaving the gold leaves positively charged. As the
result, the gold leaves repel each other. Therefore, the
deflection of the leaves indicates that the rod is
electrically charged.
Figure 6: Electroscope
Coulomb’s law
In Figure 7, two positive charge 𝑞1 and 𝑞2
separated by a distance 𝑟 repel. The force 𝐅21 acting on
𝑞1 and 𝐅12 acting on 𝑞2 are equal in magnitude and
opposite in directions according to Newton’s third law.
Coulomb’s law states that the magnitude of the force
between two electric charges is directly proportional to
the product of the two charges and inversely
proportional to the square of the separation:
𝐅21 = 𝐅12 = 𝐹 =
1 𝑞1 𝑞2
,
4𝜋𝜀0 𝑟 2
where 𝜀0 = 8.85 × 10−12 C2N-2m-2 is a constant known
as the permittivity of free space. The constant 1 4𝜋𝜀0
in the Coulomb’s law is called Coulomb’s constant 𝑘:
𝑘 =
1
≈ 9.0 × 109 Nm2 C−2 .
4𝜋𝜀0
Example 1 Calculate the electric force between nucleus
and electron of a hydrogen atom when the electron is at
distance 5.29 × 10−11 m from the nucleus.
Figure 7: Force between two
electric charge
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Example 2 Three electric charges are fixed at the
corners of a right triangle. Determine the size of electric
force on the negative charge.
Example 3 Two small metallic spheres, each of mass
0.28 g are suspended by light strings as shown. The
spheres are given the same electric charge. At
equilibrium, each string is at angle 5.1° with the vertical.
Each string is 22.5 cm long. Find the magnitude of the
charge on each sphere in unit of nC.
Electric field
Electric field (vector) is produced by electric
charge. The direction of electric field, indicated by
electric field lines, is away from positive charge and
toward negative charge as shown in Figure 8. Electric
field is defined as a region in space where an electric
charge experiences an electric force. When a charge 𝑞 is
placed in electric field 𝐸, the charge experiences the
electric force
Figure 8: Electric field lines of
positive and negative charges
𝐅 = 𝑞𝐄.
Note that if 𝑞 is positive, 𝐅 and 𝐄 are in the same
direction. If 𝑞 is negative, 𝐅 and 𝐄 are in the opposite
directions (Figure 9).
Figure 9: The directions of forces
on electric charges in electric field
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Consider Figure 10. A charge 𝑞 at point 𝑆
produces electric field 𝐸 at point 𝑃 given by
𝐸𝑃 =
1 𝑞
.
4𝜋𝜀0 𝑟 2
Another charge 𝑞0 located at point 𝑃 feels electric force
𝐹 = 𝑞0 𝐸𝑃 =
1 𝑞𝑞0
.
4𝜋𝜀0 𝑟 2
The above equation is the Coulomb’s law.
Example 4 Charges 𝑄1 = 5 nC and 𝑄2 = 2 nC are
separated by distance 15 cm. Determine the point along
the line joining the two charges where electric field
vanishes. This point is called the neutral point.
Example 5 An electron is accelerated from rest in a
uniform electric field 𝐸 = 1000 NC-1. How far does the
electron travel in 1 𝜇s?
Example 6 A charged particle mass 𝑚 = 1.2 × 10−6 kg is
at rest in a uniform electric field produced by two
charged plates as shown. The size of the electric field is
𝐸 = 2000 NC-1. Determine the nature of the charge on
the particle and its magnitude.
Figure 10: Charge 𝑞0 is in the presence of
electric field from charge 𝑞
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*Example 7 A point charge 𝑄 is located at the origin. A
straight rod of length 𝐿 has charge 𝑄 uniformly
distributed. The rod is aligned with 𝑥-axis with one end
at distance 𝑑 from the point charge.
𝑄
𝑄
𝑑
𝐿
Show that the force on the point charge is given by
1
𝑄2
𝐹 =
.
4𝜋𝜀0 𝑑(𝑑 + 𝐿)
*Example 8 In Thomson’s measurement of 𝑞/𝑚 of an
electron, the electron is projected horizontally with
speed 𝑣𝑥 in the vertical electric field 𝐸 between two
charged plates. Each plate has length 𝐿. The electron is
found to just clear plate and hits the screen at distance 𝐷
after the plates.
Show that the deflection
𝑦 =
𝑞𝐸𝐿 𝐿
+𝐷 .
𝑚𝑣𝑥2 2