Download Electric Fields

1/27/17
Electric Fields (3)
A. B. Kaye, Ph.D.
Associate Professor of Physics
31January 2017
Tentative Schedule
• Yesterday
•
•
Coulomb’s Law
First homework assigned
• Today
•
Conclude Electric Fields
• Tomorrow
•
•
Gauss’ Law
First homework is due
ELECTRIC FIELDS
Electric Field of a Continuous
Charge Distribution
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Electric Field – Continuous Charge Distribution
• The distances between charges in a group of
charges may be much smaller than the
distance between the group and a point of
interest.
• In this situation, the system of charges can be
modeled as continuous.
• The system of closely spaced charges is
equivalent to a total charge that is
continuously distributed along some line, over
some surface, or throughout some volume.
Electric Field – Continuous Charge Distribution, cont
How do we do this?
Electric Field – Continuous Charge Distribution, equations
• For the individual charge elements, we see that
• … but because the charge distribution is continuous:
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Charge Densities – Some Definitions
• Linear charge density: when a charge is
distributed along a line
•
l ≡ Q / ℓ with units of C/m
• Surface charge density: when a charge is
distributed evenly over a surface area
•
s ≡ Q / A with units of C/m2
• Volume charge density: when a charge is
distributed evenly throughout a volume
•
r ≡ Q / V with units of C/m3
Amount of Charge in a Small Volume
• If the charge is non-uniformly
distributed over a line, surface, or
volume, the amount of charge, dq, is
given by:
•
•
•
For a length element: dq = l dℓ
For a surface: dq = s dA
For a volume: dq = r dV
ELECTRIC FIELDS
Electric Field Lines
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Electric Field Lines
• Field lines give us a means of representing the
electric field pictorially
• The electric field vector is tangent to the
electric field line at each point
•
The line has a direction that is the same as that of the
electric field vector
• The number of lines per unit area through a
surface perpendicular to the lines is
proportional to the magnitude of the electric
field in that region
Electric Field Lines, General
• The density of lines
through surface A is
greater than through
surface B
• The magnitude of the
electric field is greater
on surface A than B
• The lines at different
locations point in
different directions
•
This indicates the field is
non-uniform
Electric Field Lines, Positive Point Charge
• The field lines radiate
outward in all
directions
•
In three dimensions, the
distribution is spherical
• The lines are directed
away from the source
charge
•
A positive test charge
would be repelled away
from the positive source
charge
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Electric Field Lines, Negative Point Charge
• The field lines are identical
to that of a positive point
charge, except for their
direction
•
The field lines radiate
inward (toward the point
charge) in all directions
• A positive test charge
would be attracted
toward the negative
source charge
Electric Field Lines – Rules for Drawing
• The lines must begin on a positive charge (source) and
terminate on a negative charge (drain)
•
In the case of an excess of one type of charge, some lines will
begin or end infinitely far away
• The number of lines drawn leaving a positive charge or
approaching a negative charge is proportional to the
magnitude of the charge
• No two field lines can cross
• Remember field lines are not material objects, they are a
pictorial representation used to qualitatively describe the
electric field
Examples
• Rules are great. How do we do draw these
field lines in practice?
• Let’s look at three examples:
1. A dipole arrangement of two charges with
opposite signs
2. A pair of equal charges with the same sign
3. An unequal pair of charges with opposite signs
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Electric Field Lines – Dipole
• The charges are equal
and opposite.
• The number of field
lines leaving the
positive charge equals
the number of lines
terminating on the
negative charge.
Electric Field Lines – Like Charges
•
The charges are equal and
positive
•
The same number of lines
leave each charge
•
They are equal in magnitude
•
At a great distance, the field is
approximately equal to that of
a single charge of 2q
•
Since there are no negative
charges available, the field
lines end infinitely far away
Electric Field Lines, Unequal Charges
• The positive charge is
twice the magnitude of
the negative charge
• Two lines leave the
positive charge for each
line that terminates on
the negative charge
• At a great distance, the
field would be
approximately the same
as that due to a single
charge of +q
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ELECTRIC FIELDS
Interesting Problems to Solve
Interesting Problem #4
• A total amount of
charge Q is distributed
uniformly along the
circumference of a thin
glass ring of radius R.
What is the electric
field on the axis of the
ring?
Interesting Problem #5
• Positive charge is
uniformly distributed over
an infinite flat horizontal
sheet, such as a very large
sheet of paper. Suppose
that the amount of charge
per unit area on this sheet is
σ.
•
Find the electric field in the
space above and below the
sheet.
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ELECTRIC FIELDS
Motion of a Charged Particle in a
Uniform Electric Field
Motion of Charged Particles
• When a charged particle is placed in an
electric field, it experiences an electrical force
• If this is the only force on the particle, it must
be the net force
• The net force will cause the particle to
accelerate according to Newton’s second law:
•
The acceleration is therefore:
Motion of Particles, cont
If the field is uniform, then the acceleration is constant
•
• A particle under constant acceleration model (from
Physics I) can be applied to the motion of the particle
•
The electric force causes a particle to move according to the
models of forces and motion
• If the particle has a positive charge, its acceleration is in
the direction of the field
• If the particle has a negative charge, its acceleration is in
the direction opposite the electric field
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Electron in a Uniform Field, Example
• An electron enters the region of a
uniform electric field of strength
E = 200 N/C with a velocity
vi = 3.00 x 106 m/s. The plates (blue
and red) have identical lengths of
0.100 m.
1. Find the acceleration of the electron
while it is in the electric field.
2. Assuming the electron enters the field
at a time t = 0, find the time at which it
leaves the field.
3. Assuming the vertical position of the
electron when it enters is yi = 0, what is
the vertical position when it leaves the
field?
ELECTRIC FIELDS
Interesting Problems to Solve
Interesting Problem #6
• An ion milling machine uses a beam of gallium ions (m = 70 u) to carve
microstructures from a target. A region of uniform electric field between parallel
sheets of charge is used for precise control of the beam direction. Singly ionized
gallium atoms with an initially horizontal velocity of 1.8 × 104 m/s enter a 2.0-cmlong region of uniform electric field which points vertically upward. The ions are
redirected by the field and exit the field region at the angle θ.
• If the field is set to a value of E = 90 N/C, what is the exit angle θ?
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ELECTRIC FIELDS
Key Review Points
Key Review Points (1)
•
The electric field at some point in space is defined as the electric force that acts
on a small, positive test charge at that point divided by the magnitude q0 of the test
charge:
•
Coulomb’s Law describes the force interacting between static (i.e., non-moving)
electrically charged particles:
•
The electric field due to a single point charge is
the direction of the field is outward (inward) for positive (negative) charges
•
The potential energy of a dipole can be written as
Key Review Points (2)
•
The electric field of an infinite, uniformly charged thin rod is
the direction is perpendicular to the rod and outward (inward) for positive
(negative) l
•
The electric field of an infinite, uniformly charged flat sheet is
the direction is perpendicular to the sheet and outward (inward) for positive
(negative) s
•
The electric field from a group of point charges can be computed by using
superposition; i.e., the total net field at some point is the vector sum of the electric
fields from all of the charges:
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Key Review Points (3)
•
The electric field of a continuous charge distribution has a magnitude
and the total field
contributions to each component (e.g.,
can be computed by summing the
) with
and where q is the angle between the electric field contribution and the x axis.
•
A linear charge distribution is given by dq = l dL, where l is in C/m
•
A surface charge distribution is given by dq = s dA, where s is in C/m2
•
The electric dipole moment is
from negative to positive
•
Torque on a dipole can be written as
; the direction of the dipole moment vector is
Key Review Points (4)
•
The electric field associated with a pair of oppositely charged, parallel flat sheets
is:
•
Motion in a uniform electric field is governed by
•
The Coulomb constant ke has a value of 8.9876 x 109 N・m2/C2 and can be written
where e0 is the permittivity of free space with a value 8.8542 x 10–12 C2/N・m2
•
The value of the fundamental free charge is e = 1.60218 x 10–19 C
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