Download Electric Fields

1/27/17
Electric Fields (2)
A. B. Kaye, Ph.D.
Associate Professor of Physics
26 January 2017
Tentative Schedule
• Yesterday
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History, etc.
Definition and conservation of charge
• Today
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Coulomb’s Law
First homework becomes available (due 2 Feb)
• Tomorrow
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Electric Fields (continued)
ELECTRIC FIELDS
Coulomb’s Law
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Charles-Augustin de Coulomb
14 June 1736 – 23 August 1806
French physicist
Major contributions were in areas
of electrostatics and magnetism
Also investigated in areas of
§ Strengths of materials
§ Structural mechanics
§ Ergonomics
Coulomb’s Law
• Charles Coulomb measured the
magnitudes of electric forces
between two small charged spheres
• The force is inversely proportional
to the square of the separation r
between the charges and directed
along the line joining them
• The force is proportional to the
product of the charges, q1 and q2,
on the two particles
• The electrical force between two
stationary point charges is given by
Coulomb’s Law
Point Charge
• The term point charge refers to a particle of
zero size (or zero dimensions) that carries an
electric charge (positive or negative)
•
The electrical behavior of electrons and protons is
well-described by modeling them as point charges
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Coulomb’s Law, cont.
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The force is attractive if the charges are of opposite sign and
repulsive if the charges are of like sign
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The force is a conservative force
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Mathematically,
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The SI unit of charge is the coulomb, C
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ke is called the Coulomb constant
ke = 8.9875517873681764 x 109 N・m2/C2
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eo is the permittivity of free space
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eo = 8.85418782 x 10–12 C2 / N・m2 = 8.85418782 x 10–12 F・m–1
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Coulomb's Law, Notes
• Remember the charges (typically) need to be in
coulombs
e is the smallest unit of charge and is equal to 1.602 x 10–19 C
Therefore, 1 C is equivalent to 6.24 x 1018 electrons or
protons
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This can be a relatively large charge – when we charge things by induction,
we typically have µC-range charges
•
This can also be a relatively small charge – a 1 cm3 block of aluminum has
~1023 free electrons in it – roughly 0.1 MC
• Remember that force is a vector quantity (hence,
the boldface text for the vector quantities in the
equation)
Vector Nature of Electric Forces
• In vector form,
•
is a unit vector
directed from q1 to q2
• The like charges
produce a repulsive
force between them
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Vector Nature of Electrical Forces, cont.
• Electrical forces obey Newton’s Third Law
• That means that the force on q1 is equal in
magnitude and opposite in direction to the
force on q2
• With like signs for the charges, the product
q1・q2 is positive, and the force is repulsive
Vector Nature of Electrical Forces, 3
• Two point charges are
separated by a distance r.
• The unlike charges produce
an attractive force between
them.
• With unlike signs for the
charges, the product q1・q2
is negative and the force is
attractive.
Two Final Notes about Directions
1. The sign of the product of q1・q2 gives
the relative direction of the force
between q1 and q2
2. The absolute direction is determined by
the actual location of the charges
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Multiple Charges
• The resultant force on any one charge (e.g., q1)
equals the vector sum of the forces exerted by the
other individual charges that are present.
• For example, if four charges are present, the
resultant force on one of these equals the vector
sum of the forces exerted on it by each of the
other charges:
Zero Resultant Force, Example
• From the diagram, find the
point at which the resultant
force equal to zero
•
•
The magnitudes of the
individual forces will be
equal
Directions will be opposite
• The answer should result in
a quadratic equation;
choose the root that gives
the forces in opposite
directions
Electrical Force with Other Forces, Example
• Two identical small
charged spheres, each with
a mass of 30 g, hang in
equilibrium (see sketch).
The length L of each string
is 0.15 m and the angle q is
5º.
• Find the magnitude of the
charge on each sphere.
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ELECTRIC FIELDS
A Particle in an Electric Field
Electric Field – Introduction
• You should have studied the concept of a “force”
and “field” in Physics I (e.g., the gravitational
force and the field it produces).
• Here, will will examine the electric force as a
field force.
• Remember that field forces can act through space,
even if there is no physical contact between
objects.
Cf. gravity
•
Electric Field – Definition
• An electric field is a vector field that exists in
the region of space around a charged object
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This charged object is the source charge
• This field defines the relationship between
every point in space via the Coulomb
interaction
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When another charged object, the test charge, enters this
electric field, an electric force acts on it. This is how we
evaluate the effects of the field
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Electric Field – Definition, cont
• The electric field is defined as the electric force
on the test charge per unit charge.
• The electric field vector, E, at a point in space is
defined as the electric force F acting on a positive
test charge, qo, placed at that point divided by the
test charge:
Electric Field, Notes
• E is the field produced by some charge or charge
distribution, separate from the test charge.
• The existence of an electric field is a property of the source
charge
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The presence of the test charge is not necessary for the field to
exist
• We can also say that an electric field exists at a point if a
test charge at that point experiences an electric force
•
This means that our test charge serves as a kind of “detector” of
the field
Electric Field Notes, Final
• The direction of E is that of the force on a positive
test charge
• The SI units of E are N/C
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Relationship Between F and E
§ This is valid for a point charge only
§ One charge (test or source) must be of zero size
§ For larger objects, the field will likely vary over the size of the object
• If q is positive, the force and the field are in the same
direction.
• If q is negative, the force and the field are in opposite
directions.
• This is the electric field equivalent of the particle in a
gravitational field:
Electric Field, Vector Form
• Remember Coulomb’s law, between the
source and test charges, can be expressed as
• Therefore, the electric field will be
More About Electric Field Direction
• If q is positive, the force is directed away from q:
• The direction of the field is also away from the positive
source charge:
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More About Electric Field Direction
• If q is negative, the force is directed toward q:
• The field is also toward the negative source charge:
Electric Fields from Multiple Charges
• At any point P, the total electric field due to a
group of source charges equals the vector sum
of the electric fields of all the charges:
Example
Given: Four point charges labeled 1 through 4, all with the same
magnitude, q, are placed around the origin as shown. Charge 2 is
negative; all others are positive.
What is the direction of the E-field at the origin?
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Example 2
Two charges Q1 = +2e and Q2 = –3e are placed as
shown in the diagram. What is the x-component of the
electric field at the origin?
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