Gauss`s Law - UCF Physics

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PHY2054 – Gauss’s Law
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Section 4.1 - GAUSS’ LAW
Objectives

to understand that the area of a flat surface can be represented by a vector

to understand the concept of electric flux qualitatively and quantitatively

to understand Gauss’ Law qualitatively
HOMEWORK ASSIGNMENT: COMPLETE READING CHAPTER
18.
1.1
a. Obtain a sheet of graph paper and an index card.
Determine the area of the graph paper and the index card. Is there a unique way to
represent the area of a flat surface by a vector? Which direction would it point? How
long would it be? Explain.
b. Think about a sealed box. Is it possible to represent each surface of the box
unambiguously by a vector? Explain your reasoning. Is it possible to represent a flat
surface unambiguously by a vector? Explain.
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PHY2054 – Gauss’s Law
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c. Write a rule for representing the area of a surface by a vector.
Discuss your rule with the class (instructor will guide this).
d. Roll the graph paper into a cylinder. Is it now possible to represent the area of the
surface by a vector? Explain your reasoning.
e. Could you represent the area of a small box on the graph paper by a vector when it is
rolled into a cylinder? Explain.
(Please do NOT proceed to or read the next pages until this activity is completed.)
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PHY2054 – Gauss’s Law
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1.2 The area of a flat surface can be represented by a single vector perpendicular to the
plane of the surface. The length of the vector is proportional to the area. It is useful to
define a quantity called electric flux.
Electric flux is defined as a measure of the number of electric field lines passing through
a surface. Consider a positive charge. The electric field lines all come out of the charge
and then spread out. But how many lines should we use? Whatever we choose, the
number should be proportional to the charge because we want all of this to make sense.
So how many? Consider the following drawing. Let’s assume that there are N lines
coming out from the charge q. We want N to be
proportional to q.
Think about this and provide a brief answer: must N be
an integer?
We know that if we consider a sphere of radius R, the
Electric Field on the surface is the same everywhere
and is given by Coulomb’s Law E  1 q2 . Consider a
4 0 R
sphere drawn around the charge with a radius R. The number of lines that we count
2
should increase with both the area and the Electric Field. For the sphere A  4R .
Consider a closed surface and define the flux as being positive if the lines are leaving the
closed surface. So N~AE which for our sphere is 4R 2 1 2 q  q . So the number of lines
4 0 R
o
leaving a closed surface is proportional to q as we wanted!
What can you say if q is negative? Can we have a negative number of lines? Are they
entering or leaving the surface (closed) if they exist at all? Prepare to discuss this.
Let’s get back to our diagram:
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PHY2054 – Gauss’s Law
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A
B
C
There are three closed surfaces shown surrounding the same charge. How many lines are
leaving each? (Count them.)
A______________________ B _______________________ C ___________________
Going back to our applet:
A
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PHY2054 – Gauss’s Law
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Red represents the positive charges and blue the negative. The diameter of the charge is
proportional to the charge and an advanced calculation is done to present these results.
Count the number of lines leaving (positive) or entering (negative) each charge and verify
that the number entering or leaving is proportional to the charge. Notice that one line may
start on one charge and end on the other so it is entering or leaving depending on the
surface that you construct.
Consider the “closed” surface labeled “A”. How many lines are leaving the surface?
Remember that a line entering is a “negative” line leaving. (Does this blow your mind??)
How many lines are leaving the volume?
How much charge is in the volume?
______
______
Here is a simple two charge example where the charges are known:
Draw a surface around the negative charge and around the positive charge and count the
number of lines leaving these surfaces. Draw a surface that encloses BOTH charges and
do the same thing. Fill in the following table:
Surface
Surrounds - chg
Surrounds + chg
Surrounds BOTH
Lines Leaving
Charge inside
What do you conclude:
5
Lines/Charge
PHY2054 – Gauss’s Law
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You have hopefully been able to state something close to Gauss’s Law: The number of
lines leaving a closed volume is proportional to the charge enclosed. Instead of dealing
with abstract “lines” it is more convenient to deal with the electric field itself and state
Gauss’s law in a cleaner way. But you should have the idea via the previous exercises.
The electric flux through a surface is defined as the magnitude of the electric field times
the area of the surface times the cosine of the angle between the direction of the electric
field and the area vector of surface
  EAcos
The units of flux are Nm2/C.
Qualitatively, flux is the number of field lines passing through a surface. When the angle
between the area vector of the surface and the direction of the electric field is greater than
90, the flux is negative.
2.1
a. Consider the situations shown below. Each picture contains some charge and an
imaginary box. In each case, determine if there is net flux through the box. The total
electric flux through the imaginary box is the sum of the electric flux through each
surface of the box. Remember that when the angle between the area vector of the surface
and the direction of the electric field is greater than 90, the flux is negative.
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b. Qualitatively, what does the flux through the box depend on? When is there more flux
and when is there less flux? When is there zero flux?
Can you discuss your reasoning with the class?
At this point your instructor will knit this together and show some applications.
Gauss’ Law states that the net flux through an enclosed surface is proportional to the
amount of charge enclosed by the surface. The constant of proportionality is 0. The value
of 0 is 8.85 10-12 C2/ N m2. This can be written as
 net 
qenclosed
0
SUMMARY
You should understand that the area of a flat surface can be represented by a vector. You
should understand the concept of electric flux qualitatively and quantitatively and
understand Gauss’ Law qualitatively.
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PHY2054 – Gauss’s Law
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At this point you should be able to do the following problems:
UNIT 4 EXERCISES
1) Consider the following boxes. The electric field vectors at the surface of the box are
shown.
Determine if there is a net flux through each box. Determine if there is charge inside each
box. Explain how you know.
(a)
(b)
(c)
(pictures from Ruth Chabay and Bruce Sherwood, Electric and Magnetic Interactions, John, Wiley, and Sons Inc., NY,
1995)
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PHY2054 – Gauss’s Law
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2) (from Ruth Chabay and Bruce Sherwood, Electric and Magnetic Interactions, John, Wiley, and Sons Inc., NY, 1995)
Here is a disk-shaped region of radius 2cm, on which there is a uniform electric field of
magnitude 300 N/C at an angle of 30 degrees to the plane of the disk. Calculate the
electric flux on the disk, and include the correct units.
3) (from Ruth Chabay and Bruce Sherwood, Electric and Magnetic Interactions, John, Wiley, and Sons Inc., NY, 1995)
Here is a box on whose surfaces the electric field is measured to be horizontal and to the
right. On the left face (3cm by 2cm) the magnitude of the electric field is 400 N/C, and on
the right face the magnitude of the electric field is 1200 N/C.
a) Calculate the electric flux on every face of the box.
b) Calculate the total flux of the box.
c) Calculate the total amount of charge that is inside the box.
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