Download C_Fields Notes 2009

Physics C
Electric Fields
Name:____________________
When we say an object is “charged”, what
do we really mean?
How are the electrical and gravitational
forces similar?
How do we represent the charge on a
proton?
How are the electrical and gravitational
forces different?
How do we represent the charge on an
electron?
What is the fundamental unit of charge?
Sample problem: Using unit vector notation, write an expression for the
force exerted on the upper right charge by the other two charges.
-2.0 C
-
What do we mean when we say that a law is
a spherical distribution law?
5.0 C
0.10 m
+
0.10 m
+ 5.0 C
Write the formula for the surface area of a
sphere.
Sample problem: Two identical balls of
mass m and charge q are hanging from
strings of length L. Derive an expression
for q in terms of m, , L, and fundamental
constants.
Write the spherical distribution law for
gravitational force. What do we call this law?

L
L
Write the spherical distribution law for
electric force. What do we call this law?
m,q
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m,q
What is meant by a “force field”?
Draw the electric field around a positive
charge. Do this by giving the direction of the
force on a small test charge (which is also
positive) at various locations in the space
around the positive charge.
Draw the gravitational field around the earth.
Do this by giving the direction of the force on
a small test mass at various locations in the
space around the earth.
Draw the electric field around a negative
charge. Do this by giving the direction of the
force on a small test charge (which is
positive) at various locations in the space
around the negative charge.
How can you calculate the gravitational
force on a small mass in a gravitational field
from the magnitude of the gravitational field?
How can you calculate the electric force on
a small charge in a electric field from the
magnitude of the electric field?
What is an equation that can be used for
gravitational field magnitude calculations?
What is an equation that can be used for
electric field magnitude calculations?
What is the value of the gravitational field at
the surface of the earth?
What is the limitation of the equation shown
above for electric field magnitude
calculation?
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General Procedure for Calculating Electric Fields
around Non-Spherical Charge Distributions.
Sample problem: Determine where the electric field
is zero if a 2.0 mC charge is located at the origin and
a -3.0 mC charge is located at x = 1.0 meter.
1.
2.
3.
4.
Find a common (spatial) variable that r
and/or dq both depend on.
See if symmetry (and trig) can be used
to simplify the problem by elimination of
off-axis components of E.
Find the appropriate limits to the
integral.
Don’t skip set-up steps. The physics is
in the setup!
• Sample Problem: Determine the electric field magnitude and
direction a distance y away from an extremely long, straight
wire of charge density .
+++ + ++ + ++ + ++ + ++ + ++ + ++ + ++ + ++ + ++ + ++ + ++ + ++ + ++ + +
y
Linear Charge Distribution
++
+
+
• When charge resides on a long + + +
++
thin object such as a wire or a
++
++
ring, we call that a linear charge
++
++
distribution.
+++++
• It is sometimes convenient for us
to define a linear charge density,
, which is charge per unit
length.
Surface Charge Distribution
• When charge resides on larger
surface, we call it a surface
charge distribution.
• It is sometimes convenient for us
to define a surface charge
density, σ, which is charge per
unit area.
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
• Sample Problem: Determine the electric field magnitude and
direction a distance x away from a ring of radius R bearing
charge Q.
+++++
++ + + + ++
++ + + + ++
++ + + + ++
+ + + + ++
+
+ +++
+
+
+
x
R
Volume Charge Distribution
• When charge resides distributed
within a solid object, we have a
volume charge distribution.
• It is sometimes convenient for us
to define a surface charge
density, , which is charge per
unit area.
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• Sample Problem: Determine the electric field magnitude and
direction at point P in the figure shown. The semicircular ring
bears charge -2.0C and has a radius of 0.50 m.
What is the meaning of the word flux?
P
R
• Problem: Which of the wire frames has maximum flux (or
flow) of the field lines through it?
Consider
a Vector
Field, v
Define the “Area Vector”
Sample problem: What is the speed and position of an
electron released from rest in this electric field after 3.0
ns?
Draw Area Vectors on the figure above.
e-
E = 320 N/C
Write the Flux Equation below:
Sample Problem: What is the velocity and position of this electron
3.0 ns after it enters the field?
E = 320 N/C
y
Area Vectors For a Closed Shape
e-
This rectangular
prism has six
surfaces. Each
surface has an
area vector that
points outward
from center of the
prism, and is
normal to the
surface.
x
v = 20,000 m/s
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Mathematical Representation for Flux
Another Example
This cylinder is a bit
more complicated. The
top and bottom have
areas that can easily be
calculated, and the
corresponding vectors
point outward. On the
sides, we must define
and infinite number of
infinitesimally small
areas, each of which
defines a little vector
(dA) that points
outward.
•
For a general vector field, v:
•
For an electric field, E:
Sample Problem: Calculate the electric flux
through a spherical surface of radius 2.0 m
containing a point charge of 3mC at its
center.
The Calculation of Flux Over a Closed Surface in
a Vector Field
• At each point on the closed surface, we must
take the dot product with the vector field to get
the flux for that small area. Then we add all
these dot products up together to get the flux
for the entire surface. This leads to some
interesting observations.
• If there is a “source” of the vector field in the
closed shape, the flux over its surface is
positive.
• If there is a “sink” of the vector field in the
closed shape, the flux over its surface is
negative.
• If there is neither a source or sink of the
vector field in the closed shape, the flux over
its surface is zero.
Sample Problem: Draw an electric dipole,
and sketch three Gaussian surfaces for
which one has positive electric flux, one has
negative electric flux, and one has zero
electric flux.
What do we mean by “source” of an electric
field?
What do we mean by a “sink” of an electric
field?
– If a closed shape encloses a positive
charge, the flux is ________.
– If a closed shape encloses a negative
charge, the flux is ________.
– If the closed shape encloses no net
charge, the flux is ________.
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Gauss’ Law of Electricity
Sample Problem: Consider two Gaussian
surfaces, a sphere of radius R and a cube of
side 4R. In each is a positive point charge of
+q. How does the electric flux compare for
the two surfaces?
Other forms of Gauss’s Law
Sample Problem: Derive the electric field
outside a charged non-conducting cylinder
with an even volume charge distribution .
What is a “Gaussian Surface”?
What’s Gauss’s Law Good For?
Sample Problem: A point charge q is located
a distance d from a long infinite wire.
Determine the electric flux through the plane
due to the point charge.
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Sample Problem: Derive the electric field
INSIDE a charged non-conducting cylinder
with an even volume charge distribution .
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