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Physics for Engineers II
Lecture 3
Electric potential energy
Point charge potential
capacity for doing work
(arises from position or configuration)
any collection of charges
The change in potential is the work done per
unit charge against the Coulomb force, so
For a positive charge q the Coulomb force does positive work in moving it
from a to b, so that represents a drop in potential energy.
more convenient electric
potential or voltage
Potential reference at infinity
Taking the limit as rb → ∞ gives
for any arbitrary value of r. The choice of
potential equal to zero at infinity is logical
in this case because the electric field and
force approach zero there.
the work necessary to bring a test charge q from an
infinite distance to some distance r
M.Mulak / WUT
Zero potential
physical significance change in potential
zero point is arbitrary  like the origin of a
coordinate system
set for convenience
some physical or geometric logic to the choice of the zero
point
A single point charge or localized collection of charges
the zero point at infinity
Practical electrical circuits
the earth or ground potential is taken to be zero
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Physics for Engineers II
Lecture 3
Equipotential lines
Multiple point charges
any number of point charges
contour
V calculated from the point charge expression by simple
addition (a scalar quantity!)
in this case the "altitude" is electric potential or voltage
lines on a map which trace lines of equal altitude
- always perpendicular to the electric field
- 3D equipotential surfaces
- movement along an equipotential surface requires no work
simpler than the vector sum required to calculate the
electric field!
Electric dipole potential
Dipole moment
superposing the point charge potentials of the two charges
a pair of opposite charges of magnitude q
Useful concept:
- in atoms and molecules the effects of charge separation are
measurable, but the distances between the charges are too small to be
easily measurable
- in dielectrics and other applications in solid and liquid materials
M.Mulak / WUT
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Physics for Engineers II
Lecture 3
Potential of line charge
a continuous charge distribution
can be found by superposing the point charge potentials of
infinitesimal charge elements
Potential for ring of charge
a continuous charge distribution
each element of the ring has the same distance r from the point P
The ring potential can then be used as a
charge element to calculate the potential of
a charged disc.
If the charge is characterized by an area
density and the ring by an incremental
width dR', then:
Potential for disc of charge
by summing the potentials of charged rings
(the integral over the charged disc)
M.Mulak / WUT
Electric field from voltage
Great value of the scalar electric potential V
E
The component of electric field in any direction is the
negative of rate of change of the potential in that
direction.
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Physics for Engineers II
Lecture 3
Electric field as gradient
Voltage difference and electric field
the change in voltage = the work done per unit charge against the
electric field.
a constant electric field
more general case
M.Mulak / WUT
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Physics for Engineers II
Lecture 3
Potential of a charged conducting sphere
Gauss' law the electric field
outside the sphere is identical to
that of a point charge, so
Applications of electrostatics
SCIENTIFIC
Electrostatic generators (Van de Graaff generator):
used for acceleration of elementary particles
(e.g. in nuclear physics)
PRACTICAL
E=0 inside, so V=const at the
value it reaches at the surface
Lightning arrester
Electrostatic precipitators:
used to reduce atmospheric pollution
Xerography
The idea of a lightning arrester
Model of a sharp point conductor
at the same V
r1
Near the spike r 0
a strong E ionises the air and
make electrical breakdown
more likely to occur
q1
E=
M.Mulak / WUT
σ
ε0
q2
conducting wire
V = ke
E1 = k e
From Gauss’s Law
r2
q1
q
= ke 2
r1
r2
q1
2
r1
r2 → 0 ⇒
⇒
E2 = k e
E1
→0
E2
q1 r1
=
q2 r2
q2
E
r
⇒ 1 = 2
2
E2 r1
r2
when E1 is finite ⇒ E 2 → ∞
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Physics for Engineers II
Lecture 3
Van de Graaff Generator
Voltages of hundreds of
thousands of volts
Discharges do not represent a
serious shock hazard since the
currents attainable are small
Voltages enough to make our hair stand on end.
Like charges on individual hairs makes them repel
each other and stand away from Jennifer's head.
She is standing on a wooden chair to isolate her from
ground so that the charge will build up on her.
Electrostatic precipitator: an application of electrical discharge in
The xerographic process
gases
Idea: a photoconductive material
Used to remove dust from combustion gases (efficiency up to 99%))
THE BASIC IDEA:
High voltage (40 kV ÷100 kV) between the wire
running down the center of the duct and the outer
wall (grounded).
The wire is at negative potential with respect to
the walls ⇒ E toward the wire
Corona discharge around the wire (the highest E)
⇒ ionization of the air ⇒ negative ions
accelerated toward the outer wall
Due to collisions the dirt particles in the
streaming gas become charged (mostly
negatively) ⇒ they are attracted to the outer wall
Periodically shaking the duct ⇒ the ash and dust
particles fall down and are collected.
M.Mulak / WUT
(a) The photoconductive surface
is positively charged.
(b) Using light an image is
formed on the surface in the
form of hidden positive
charges.
(c) The surface containing the
image is covered with a
negatively charged powder
(toner), which adheres only to
the image area.
(d) A piece of paper is placed
over the surface and given a
charge. This transfers the
(e) A laser printer idea
visible image to the paper,
which is then heat-treated to
„fix” the powder to the paper.
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