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2/9/17
Electric Potential (3)
A. B. Kaye, Ph.D.
Associate Professor of Physics
14 February 2017
Update
• Last class
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•
Continued Electric Potential
Homework #2 was due
• Today
•
•
Complete Electric Potential
Homework #3 Due
• Tomorrow
•
Exam I
ELECTRIC POTENTIAL
Potential Difference due to a
Charged Conductor
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V Due to a Charged Conductor
• Consider two points on
the surface of the
charged conductor in the
sketch
• E is always
perpendicular to the
displacement ds
• Therefore,
,
and the potential
difference between A
and B is also zero
V Due to a Charged Conductor, cont.
• V is constant everywhere on the surface of a
charged conductor in equilibrium
•
DV = 0 between any two points on the surface
• The surface of any charged conductor in
electrostatic equilibrium is an equipotential surface
•
I.e., Every point on the surface of a charged conductor in
equilibrium is at the same electric potential
• Because the electric field is zero inside the
conductor, we conclude that:
•
•
The electric potential is constant everywhere inside the conductor
The electric potential is equal to the value at the surface
Irregularly Shaped Objects
• For irregularly-shaped objects, the
charge density is:
•
•
high where the radius of curvature is small
low where the radius of curvature is large
• The electric field is:
•
•
large near the convex points having small radii of
curvature
reaches very high values at sharp points
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Cavity in a Conductor
• Create an irregularlyshaped cavity inside a
conductor
•
Assume no charges are
inside the cavity
• The electric field
inside the conductor
must be zero
Cavity in a Conductor, cont
• The electric field inside does not depend on the
charge distribution on the outside surface of the
conductor
• For all paths between A and B,
• A cavity surrounded by conducting walls is a
field-free region as long as no charges are inside
the cavity
Corona Discharge
• If the electric field near a conductor is sufficiently strong,
electrons resulting from random ionizations of air
molecules near the conductor accelerate away from their
parent molecules
• These electrons can ionize additional molecules near the
conductor
• This creates more free electrons
• The corona discharge is the glow that results from the
recombination of these free electrons with the ionized air
molecules
• The ionization and corona discharge are most likely to
occur near very sharp points
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ELECTRIC POTENTIAL
The Millikan Oil Drop
Experiment
Millikan Oil Drop Experiment
• In 1909, Robert Millikan and Harvey Fletcher
measured e, the magnitude of the elementary charge
on the electron
•
They also demonstrated the quantized nature of this
charge
• Their experiment entailed observing tiny charged
droplets of oil between two horizontal metal
electrodes and the droplets are illuminated by a
light to be observed
•
Let’s look at the experiment in more detail
Millikan Oil Drop Experiment – Experimental Set-Up
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Oil Drop Experiment
•
First, without any applied
electric field, the terminal
velocity of a droplet was
measured
•
•
•
At the terminal velocity, the drag force
is equal to the gravitational force
These depend on the radius in
different ways, so that the radius of the
droplet, and therefore the mass and
gravitational force, could be
determined
From Stokes’ Law and
Newton’s 2nd Law, the drop
reaches a terminal velocity and
we can compute its size
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Oil Drop Experiment – Drop Radius
• From Stokes’ Law, we know:
• For a spherical droplet, we know that the
gravitational force is:
• When the drop is not accelerating, the forces balance
(FD = FG), implying
Oil Drop Experiment
• In the second part of the
experiment, an adjustable
voltage was applied
between the plates to
induce an electric field
• An X-ray source was
turned on to induce a
charge on each oil drop
•
The voltage was adjusted
until the drops were
suspended in mechanical
equilibrium
New Force
• With the electric field, the force on the drop is now:
• For parallel plates with a potential difference V and a
separation d, the electric field is:
• To solve for q, simply adjust V until the drop is stationary,
at which point, FE and w are equal:
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Final Solution(s)…
• This would mean that
• If, instead, you adjusted V so that the drop has a new
terminal velocity, then
Oil Drop Experiment
•
The drop can be raised and allowed to fall numerous
times by turning the electric field on and off
•
After many experiments, Millikan and Fletcher
determined that
•
•
•
This experiment yielded conclusive evidence that
charge is quantized
ELECTRIC POTENTIAL
Examples of Applications of
Electrostatics
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Van de Graaff Generator
• Charge is delivered
continuously to a high-potential
electrode by means of a moving
belt of insulating material
• The high-voltage electrode is a
hollow metal dome mounted on
an insulated column
• Large potentials can be
developed by repeated trips of
the belt
• Protons accelerated through
such large potentials receive
enough energy to initiate
nuclear reactions
Electrostatic Precipitator
• An application of electrical discharge in
gases is the electrostatic precipitator
• It removes particulate matter from
combustible gases
• The air to be cleaned enters the duct
and moves near the wire
• As the electrons and negative ions
created by the discharge are accelerated
toward the outer wall by the electric
field, the dirt particles become charged
• Most of the dirt particles are negatively
charged and are drawn to the walls by
the electric field
CAPACITANCE AND DIELECTRICS
“Interesting Problems to Solve”
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Interesting Problem #11
• Suppose that near the ground directly
below a thundercloud, the electric field is
of a constant magnitude 2.0 x 104 V/m
and points upward. What is the potential
difference between the ground and a point
in the air 50 m above the ground?
Interesting Problem #12
Assume that the electron in a hydrogen
atom is 5.3 x 10–11 m from the proton, and
assume that the proton is a small ball of
charge with q′ = 1.60 x 10–19 C. Find the
electrostatic potential generated by the
proton at this distance and then determine
the potential energy of the electron.
Interesting Problem #13
A sphere of radius R
carries a total positive
charge Q distributed
uniformly throughout its
volume. Find the
electrostatic potential both
inside and outside the
sphere.
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Interesting Problem #14
A coaxial cable consists of a long,
cylindrical conductor of radius a
concentric with a thin cylindrical
shell of larger radius b. If the
central conductor has a charge per
unit length l = Q/L uniformly
distributed on its surface, what is
the potential difference between
the inner and outer conductors?
(Assume the space between them
is empty.)
Interesting Problem #15
Four charges, q1 = q,
q2 = 2q, q3 = –q, and
q4 = q, are at the corners
of a square with side
length a. If q = 2.0 µC
and a = 7.5 cm, what is
the total energy required
to assemble this system of
charges?
Interesting Problem #16
The metallic sphere on the top of a large
Van de Graaff generator has a radius of
3.0 m. Suppose that the sphere carries a
charge of 5.0 x 10–5 C uniformly distributed
over its surface. How much electric energy
is stored in this charge distribution?
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CAPACITANCE AND DIELECTRICS
Key Review Points
Key Review Points
•
The potential difference between points A and B in an electric field E
is defined as
•
The electric potential V = U/q is a scalar quantity with units of volts
where
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An equipotential surface is a surface on which all points have the
same electric potential
•
•
Equipotential surfaces are perpendicular to electric field lines
The potential difference between two points separated by a distance d
in a uniform electric field E is
Key Review Points
•
When a positive charge q is moved between points A and B in an
electric field E, the change in the potential energy of the charge-field
system is
•
If we define V = 0 at r = ∞, the electric potential due to a point charge
at any distance r from the charge is
•
The electric potential energy associated with a pair of point charges
separated by a distance r12 is
•
For a system of point charges, the total potential energy is the sum of the individual charges
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Key Review Points
•
The electric potential due to a continuous charge distribution is
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The individual components of the electric field can be found by:
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Potential energy of a conductor:
•
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For a system of conductors, the total energy is the sum of the components
Energy density in an electric field:
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