Transcript ppt - plutonium

Physics II:
Electricity & Magnetism
Binomial Expansions,
Riemann Sums, Sections 21.6
to 21.11
Thursday
(Day 11)
Binomial Expansion
Warm-Up
Thurs, Feb 5
 Calculate the force acting on Q2 at distance of 0.50 m.
 Place your homework on my desk:
 “Foundational Mathematics’ Skills of Physics” Packet (Page 5)
 Electrostatics Lab #2: Lab Report
 Have you complete WebAssign Problems: 21.1 - 21.4?
 For future assignments - check online at www.plutonium-239.com
Warm-Up Review
Calculate the force acting on Q2 at
distance of 0.50 m.
Essential Question(s)
 WHAT PRIOR FOUNDATIONAL MATHEMATICS’
SKILLS ARE NECESSARY IN PHYSICS II?
Vocabulary
 Static Electricity
 Electric Charge
 Positive / Negative
 Attraction / Repulsion
 Charging / Discharging
 Friction
 Induction
 Conduction
 Law of Conservation of
Electric Charge
 Non-polar Molecules
 Polar Molecules
 Ion
 Ionic Compounds
 Force
 Derivative
 Integration (Integrals)
 Test Charge
 Electric Field
 Field Lines
 Electric Dipole
 Dipole Moment
Foundational Mathematics
Skills in Physics Timeline
Day
Pg(s)
Day
Pg(s)
Day
Pg(s)
Day
Pg(s)
1
1
2
6
3
11
16
16
21
2
13
14
7
4
12
17
17
8
3
22
23
8
5
13
18
18
9
4
24
†12
9
6
14
19
19
10
5
15
10
7
15
20
20
11
WHAT PRIOR FOUNDATIONAL MATHEMATICS’ SKILLS ARE NECESSARY IN PHYSICS II?
Agenda
 Review “Foundational Mathematics’ Skills of Physics”
Packet (Page 5) with answer guide.
 Begin The Four Circles Graphic Organizer
 DERIVATIVE PROOF USING BINOMIAL EXPANSION
 Derivative practice
 FRIDAY:
 INTEGRAL PROOF USING RIEMANN SUMS
 Integral Practice
 MONDAY:
 Discuss Electric Fields & Gravitational Field
 Apply Electric Fields
 Continue with The Four Circles Graphic Organizer
Topic #1: Determine the slope at
point A for f(x)=xn
y = 1/2 x
WHAT PRIOR FOUNDATIONAL MATHEMATICS’ SKILLS ARE NECESSARY IN PHYSICS II?
Topic #1: Determine the slope at
point A for f(x)=xn
y = 1/4 x2
WHAT PRIOR FOUNDATIONAL MATHEMATICS’ SKILLS ARE NECESSARY IN PHYSICS II?
Topic #1: Determine the slope at
point A for f(x)=xn
Slope “is defined as” the rise over the run.
f (b)  f (a)
slope  lim
ba
ba
Trick #1: Factor out (b-a)
f (b)  f (a)
bn  an
(b  a)("something")
slope  lim
 lim
 lim
ba
ba b  a
ba
ba
b  a 
The something is determined by using binomial expansion.
Binomial Expansion is defined as:
b n  a n  (b  a)(b n1a 0  b n2 a1  b n3a 2  b 2 a n3  b1a n2  b 0 a n1 )
WHAT PRIOR FOUNDATIONAL MATHEMATICS’ SKILLS ARE NECESSARY IN PHYSICS II?
Aside #1: Binomial Expansion
Practice
Using binomial expansion:
b n  a n  (b  a)(b n1a 0  b n2 a1  b n3a 2  b 2 a n3  b1a n2  b 0 a n1 )
Expand b1-a1:
WHAT PRIOR FOUNDATIONAL MATHEMATICS’ SKILLS ARE NECESSARY IN PHYSICS II?
Aside #2: Binomial Expansion
Practice
Using binomial expansion:
b n  a n  (b  a)(b n1a 0  b n2 a1  b n3a 2  b 2 a n3  b1a n2  b 0 a n1 )
Expand b2-a2:
WHAT PRIOR FOUNDATIONAL MATHEMATICS’ SKILLS ARE NECESSARY IN PHYSICS II?
Aside #3: Binomial Expansion
Practice
Using binomial expansion:
b n  a n  (b  a)(b n1a 0  b n2 a1  b n3a 2  b 2 a n3  b1a n2  b 0 a n1 )
Expand b3-a3:
WHAT PRIOR FOUNDATIONAL MATHEMATICS’ SKILLS ARE NECESSARY IN PHYSICS II?
Aside #4: Binomial Expansion
Practice
Using binomial expansion:
b n  a n  (b  a)(b n1a 0  b n2 a1  b n3a 2  b 2 a n3  b1a n2  b 0 a n1 )
Expand b4-a4:
WHAT PRIOR FOUNDATIONAL MATHEMATICS’ SKILLS ARE NECESSARY IN PHYSICS II?
Aside #5: Binomial Expansion
Practice
Using binomial expansion:
b n  a n  (b  a)(b n1a 0  b n2 a1  b n3a 2  b 2 a n3  b1a n2  b 0 a n1 )
Expand b5-a5:
WHAT PRIOR FOUNDATIONAL MATHEMATICS’ SKILLS ARE NECESSARY IN PHYSICS II?
Determination of Slope
Recall: The definition for slope of a polynomial is now:
f (b)  f (a)
bn  an
(b  a)("something")
slope  lim
 lim
 lim
ba
ba
ba
ba
ba
b  a 
Calculate the slope for f(x) = x1
The expansion of b1-a1 = (b-a)(1)
b1  a1
(b  a)(1)
slope  lim
 lim
 lim(1)
ba b  a
ba b  a 
ba
WHAT PRIOR FOUNDATIONAL MATHEMATICS’ SKILLS ARE NECESSARY IN PHYSICS II?
Determination of Slope
Recall: The definition for slope of a polynomial is now:
f (b)  f (a)
bn  an
(b  a)("something")
slope  lim
 lim
 lim
ba
ba
ba
ba
ba
b  a 
Calculate the slope for f(x) = x2
The expansion of b2-a2 = (b-a)(b+a)
b2  a2
(b  a)(b  a)
slope  lim
 lim
 lim(b  a)
ba b  a
ba
ba
b  a 
WHAT PRIOR FOUNDATIONAL MATHEMATICS’ SKILLS ARE NECESSARY IN PHYSICS II?
Determination of Slope
Recall: The definition for slope of a polynomial is now:
f (b)  f (a)
bn  an
(b  a)("something")
slope  lim
 lim
 lim
ba
ba
ba
ba
ba
b  a 
Calculate the slope for f(x) = x3
The expansion of b3-a3 = (b-a)(b2+ba+a2)
b3  a3
(b  a)(b 2  ba  a 2 )
slope  lim
 lim
 lim(b 2  ba  a 2 )
ba b  a
ba
ba
b  a 
WHAT PRIOR FOUNDATIONAL MATHEMATICS’ SKILLS ARE NECESSARY IN PHYSICS II?
Determination of Slope
Recall: The definition for slope of a polynomial is now:
f (b)  f (a)
bn  an
(b  a)("something")
slope  lim
 lim
 lim
ba
ba
ba
ba
ba
b  a 
Calculate the slope for f(x) = x4
The expansion of b4-a4 = (b-a)(b3+b2a+ba2+a3)
b4  a4
(b  a)(b 3  b 2 a  ba 2  a 3 )
slope  lim
 lim
ba b  a
ba
b  a 
 lim(b 3  b 2 a  ba 2  a 3 )
ba
WHAT PRIOR FOUNDATIONAL MATHEMATICS’ SKILLS ARE NECESSARY IN PHYSICS II?
Determination of Slope
Recall: The definition for slope of a polynomial is now:
f (b)  f (a)
bn  an
(b  a)("something")
slope  lim
 lim
 lim
ba
ba
ba
ba
ba
b  a 
Calculate the slope for f(x) = x5
The expansion of b5-a5 = (b-a)(b4+b3a+b2a2+ba3 +a4)
b5  a5
(b  a)(b 4  b 3a  b 2 a 2  ba 3  a 4 )
slope  lim
 lim
ba b  a
ba
b  a 
 lim(b 4  b 3a  b 2 a 2  ba 3  a 4 )
ba
WHAT PRIOR FOUNDATIONAL MATHEMATICS’ SKILLS ARE NECESSARY IN PHYSICS II?
Determination of Slope
 Now we can show that:
slope  lim(b n1a 0  b n2 a1  b n3a 2  b n4 a 3  )
ba
 Now this is where it gets fun! It is time to decrease the
distance between points a and b to get a more accurate slope
at point a.
 This is called taking the ‘limit’ as “b” approaches “a.”
 All we have to do is change all “b’s” to “a’s”
slope  (a n1a 0  a n2 a1  a n3a 2  a n4 a 3  )
slope  (a n10  a n21  a n32  a n43  )
slope  (a n1  a n1  a n1  a n1  )
slope  n  a n1
WHAT PRIOR FOUNDATIONAL MATHEMATICS’ SKILLS ARE NECESSARY IN PHYSICS II?
Determination of Slope
 The slope of any power is now:
slope  n  a n1
WHAT PRIOR FOUNDATIONAL MATHEMATICS’ SKILLS ARE NECESSARY IN PHYSICS II?
Method Determination
Comparision
 Checking the slope for f(a) = an where n = 2
 Using the binomial expansion method
slope  lim(b  a)  (a  a)  2a
 Using the slope =
ba
n an-1
method
slope  n  a n1  2  a 21  2a
 Do they agree?
WHAT PRIOR FOUNDATIONAL MATHEMATICS’ SKILLS ARE NECESSARY IN PHYSICS II?
Method Determination
Comparision
 Checking the slope for f(a) = an where n = 1
 Using the binomial expansion method
slope  lim(1)  1
ba
 Using the slope = n an-1 method
slope  n  a n1  1 a11  1
 Do they agree?
WHAT PRIOR FOUNDATIONAL MATHEMATICS’ SKILLS ARE NECESSARY IN PHYSICS II?
Method Determination
Comparision
 Checking the slope for f(a) = an where n = 3
 Using the binomial expansion method
slope  lim(b 2  ba  a 2 )  (a 2  aa  a 2 )  3a 2
ba
 Using the slope = n an-1 method
slope  n  a n1  3 a 31  3a 2
 Do they agree?
WHAT PRIOR FOUNDATIONAL MATHEMATICS’ SKILLS ARE NECESSARY IN PHYSICS II?
Method Determination
Comparision
 Checking the slope for f(a) = an where n = 4
 Using the binomial expansion method
slope  lim (b 3  b 2 a  ba 2  a 3 )
ba
 (a 3  a 2 a  aa 2  a 3 )  4a 3
 Using the slope = n an-1 method
slope  n  a n1  4  a 41  4a 3
 Do they agree?
WHAT PRIOR FOUNDATIONAL MATHEMATICS’ SKILLS ARE NECESSARY IN PHYSICS II?
Method Determination
Comparision
 Checking the slope for f(a) = an where n = 5
 Using the binomial expansion method
slope  lim(b 4  b 3a  b 2 a 2  ba 3  a 4 )
ba
 (a 4  a 3a  a 2 a 2  aa 3  a 4 )  5a 4
 Using the slope = n an-1 method
slope  n  a n1  5  a 51  5a 4
 Do they agree?
WHAT PRIOR FOUNDATIONAL MATHEMATICS’ SKILLS ARE NECESSARY IN PHYSICS II?
Summary of Derivative
 After adding the constant back in and changing a to the
variable x we get
f(x) =C xn
Slope = n C x n-1
 How is the derivative expressed?
dy
slope  f (x) 
dx
WHAT PRIOR FOUNDATIONAL MATHEMATICS’ SKILLS ARE NECESSARY IN PHYSICS II?
Summary of Derivative
n
 How is the derivative for a polynomial, f (x)   Cx ,
represented?
slope   nCx n1
 What about finding the area under a curve (or line)?
Find the integral.
For a polynomial:
1
area   f (x) dx 
Cx n1
n 1
WHAT PRIOR FOUNDATIONAL MATHEMATICS’ SKILLS ARE NECESSARY IN PHYSICS II?
So what is the relationship
...
between finding the slope and
taking the first derivative?
They are the same.
Other derivative representations
dy d  f (x) d
f (x)  f 

  f (x)
dx
dx
dx

WHAT PRIOR FOUNDATIONAL MATHEMATICS’ SKILLS ARE NECESSARY IN PHYSICS II?
Summary
 After comparing the force constants for electrostatics and gravity, identify
which Force is stronger.
 HW (Place in your agenda):
 “Foundational Mathematics’ Skills of Physics” Packet (Page 6)
 Derivative Practice
 Future assignments:
 Electrostatics Lab #3: Lab Report (Due in 3 classes)
How do we use Coulomb’s Law and the principle of superposition to determine the force that acts between point charges?
Friday
(Day 12)
Riemann Sums
Warm-Up
Fri, Feb 6
 Calculate the velocity of the electron moving around the hydrogen
nucleus (r = 0.53 x 10-10 m)
 Place your homework on my desk:
 “Foundational Mathematics’ Skills of Physics” Packet (Page 6)
 Derivative Practice
 For future assignments - check online at www.plutonium-239.com
Warm-Up Review
 Calculate the velocity of the electron moving around the
hydrogen nucleus (r = 0.53 x 10-10 m)
Essential Question(s)
 WHAT PRIOR FOUNDATIONAL MATHEMATICS’
SKILLS ARE NECESSARY IN PHYSICS II?
 HOW DO WE DESCRIBE THE NATURE OF
ELECTROSTATICS AND APPLY IT TO VARIOUS
SITUATIONS?
 How do we describe and apply the concept of electric field?
 How do we describe and apply Coulomb’s Law and the Principle
of Superposition?
Vocabulary









Static Electricity
Electric Charge
Positive / Negative
Attraction / Repulsion
Charging / Discharging
Friction
Induction
Conduction
Law of Conservation of Electric
Charge
 Non-polar Molecules











Polar Molecules
Ion
Ionic Compounds
Force
Derivative
Integration (Integrals)
Test Charge
Electric Field
Field Lines
Electric Dipole
Dipole Moment
Foundational Mathematics
Skills in Physics Timeline
Day
Pg(s)
Day
Pg(s)
Day
Pg(s)
Day
Pg(s)
1
1
2
6
3
11
16
16
21
2
13
14
7
4
12
17
17
8
3
22
23
8
5
13
18
18
9
4
24
†12
9
6
14
19
19
10
5
15
10
7
15
20
20
11
WHAT PRIOR FOUNDATIONAL MATHEMATICS’ SKILLS ARE NECESSARY IN PHYSICS II?
Agenda
 Review “Foundational Mathematics’ Skills of
Physics” Packet (Page 6) with answer guide.
 Review Derivative Practice
 INTEGRAL PROOF USING RIEMANN SUMS
 Integral Practice
 MONDAY:
 Discuss Electric Fields & Gravitational Field
 Apply Electric Fields
 Continue with The Four Circles Graphic Organizer
Topic #1: Determine the slope at
point A for f(x)=xn
y = 1/2 x
WHAT PRIOR FOUNDATIONAL MATHEMATICS’ SKILLS ARE NECESSARY IN PHYSICS II?
Topic #1: Determine the slope at
point A for f(x)=xn
y = 1/4 x2
WHAT PRIOR FOUNDATIONAL MATHEMATICS’ SKILLS ARE NECESSARY IN PHYSICS II?
Summary
 Identify one section that in the Integral Proof using Riemann Sums that was
confusing?
 HW (Place in your agenda):
 “Foundational Mathematics’ Skills of Physics” Packet (Page 7)
 Go through the Riemann sum derivation - determine what you do not
understand.
 Integral Practice
 Future assignments:
 Electrostatics Lab #3: Lab Report (Due in 3 classes)
How do we use Coulomb’s Law and the principle of superposition to determine the force that acts between point charges?
Monday
(Day 13)
Riemann Sums (Day II)
Section 21.6
Section 21.8
Warm-Up
1.
2.
3.


Mon, Feb 9
If I measured the distance of each step I took and summed them all together,
what would I have calculated?
If I was driving in a car on the turnpike at a constant speed and I multiplied
my speed by the time I was traveling, what would I have calculated?
Now make it more complex, what if my speed was slowly changing and I
1. Wrote down my velocity and the amount of time I was traveling at that
velocity;
2. Multiplied those two numbers together;
3. Added those new numbers together;
 What would I have calculated?
Place your homework on my desk:
 “Foundational Mathematics’ Skills of Physics” Packet (Page 7)
 Integral Practice
For future assignments - check online at www.plutonium-239.com
Warm-Up
1.
2.
3.
Mon, Feb 9
If I measured the distance of each step I took and summed them all
together, what would I have calculated?
If I was driving in a car on the turnpike at a constant speed and I
multiplied my speed by the time I was traveling, what would I have
calculated?
Now make it more complex, what if my speed was slowly changing
and I
1. Wrote down my velocity and the amount of time I was traveling
at that velocity;
2. Multiplied those two numbers together;
3. Added those new numbers together;
 What would I have calculated?
Essential Question(s)
 WHAT PRIOR FOUNDATIONAL MATHEMATICS’
SKILLS ARE NECESSARY IN PHYSICS II?
 HOW DO WE DESCRIBE THE NATURE OF
ELECTROSTATICS AND APPLY IT TO VARIOUS
SITUATIONS?
 How do we describe and apply the concept of electric field?
 How do we describe and apply Coulomb’s Law and the Principle
of Superposition?
Vocabulary









Static Electricity
Electric Charge
Positive / Negative
Attraction / Repulsion
Charging / Discharging
Friction
Induction
Conduction
Law of Conservation of Electric
Charge
 Non-polar Molecules











Polar Molecules
Ion
Ionic Compounds
Force
Derivative
Integration (Integrals)
Test Charge
Electric Field
Field Lines
Electric Dipole
Dipole Moment
Foundational Mathematics
Skills in Physics Timeline
Day
Pg(s)
Day
Pg(s)
Day
Pg(s)
Day
Pg(s)
1
1
2
6
3
11
16
16
21
2
13
14
7
4
12
17
17
8
3
22
23
8
5
13
18
18
9
4
24
†12
9
6
14
19
19
10
5
15
10
7
15
20
20
11
WHAT PRIOR FOUNDATIONAL MATHEMATICS’ SKILLS ARE NECESSARY IN PHYSICS II?
Agenda
 Review “Foundational Mathematics’ Skills of Physics”
Packet (Page 7) with answer guide.
 Complete the Integral Proof using Riemann Sums
 Review Integral Practice
 Discuss
 Electric Fields
 Gravitational Field
 Field Lines
 Continue with The Four Circles Graphic Organizer
 Apply Electric Fields
Riemann Sums Proof
ADD RIEMANN SUMS PROOF HERE
Riemann Sums
Riemann Sums Related to Reality
Big Picture Ideas and Relationships
 If F(x) is written as x(t) (aka. Displacement as a function
of time)
 Then slope of the graph, F’(x), can be written as x’(t) (aka
v(t), the velocity as a function of time)
 And F(b) - F(a) is the really the just the xfinal - xinitial.
 This is also equal to the summation of all velocity x time
calculations [f(ci)(xi-xi-1)] or rewritten as [v(ci)(ti-ti-1)]
The graph of y(x)
Referred to as F(x) [or x(t)]
x (ci )  v(ci )
x(t i )  x t i1 
t i  t i1
x(ci )
x ti1 
only a reference point;
not the "height"
t
ti  ti1
x(ti )
t
x(a)
x(b)
The graph of y’(x);
Called F’ (x); [or x’ (t)]
The graph of F’(x) is renamed
f(x); [or x’ (t) is renamed v(t)]
Riemann Sum with only 1
approximation (t: large)
x (ci )  v(ci )
x
x ti1 
x(ci ) only a reference point; not the "height"
x(t i )  x t i1 
t i  t i1
x(a)
x(ti )
ti  ti1
x(b)
t
Riemann Sum with only 2
approximations (t: still large)
x
x(t i )  x t i1 
t i  t i1
x (ci )  v(ci )
x ti1 
x(a)
x(ci )
only a reference point;
not the "height"
ti  ti1
x(b)
x(ti )
t
Riemann Sum with 9
approximations (t: medium)
x(t i )  x t i1 
t i  t i1
x
x (ci )  v(ci )
x ti1 
x(ci )
only a reference point;
x(ti )
t
not the "height"
x(a)
ti  ti1
t
x(b)
t
Riemann Sum with 17
approximations (t: small)
x(t i )  x t i1 
t i  t i1
x
x (ci )  v(ci )
x ti1 
x(ci )
x(ti )
only a reference point;
not the "height"
x(a)
ti  ti1
t
x(b)
t
Riemann Sum with only 33
approximations (t: smaller)
x
t
Riemann Sums
 Confusing Points:
 x(ci) is only a point of reference, not the “height” to which
the t is multiplied to get the area under the curve.
 In fact, it is the area under the v(t) graph that we are trying
to find in order to determine the total displacement.
Riemann Sums
 As t decreases, your approximations become more
accurate.
 Note: Summing up all of the “slope of x vs t times t”
(aka. “velocity x time”) calculations will equal the
total displacement (aka. The final position minus the
starting position).
Section 21.6
How do we describe and apply the concept
of electric field?
How do we define electric fields in terms of the
force on a test charge?
Section 21.6
How do we describe and apply Coulomb’s
Law and the Principle of Superposition?
How do we use Coulomb’s Law to describe the
electric field of a single point charge?
How do we use vector addition to determine
the electric field produced by two or more point
charges?
21.6 The Electric Field
The electric field is the
force on a small charge,
divided by the charge:
21.6 The Electric Field
For a point charge:
21.6 The Electric Field
Force on a point charge in an electric field:
Superposition principle for electric fields:
21.6 The Electric Field
Problem solving in electrostatics: electric
forces and electric fields
1. Draw a diagram; show all charges, with
signs, and electric fields and forces with
directions
2. Calculate forces using Coulomb’s law
3. Add forces vectorially to get result
Section 21.8
How do we describe and apply Coulomb’s
Law and the Principle of Superposition?
How do we compare and contrast Coulomb’s
Law and the Universal Law of Gravitation?
21.8 Field Lines
The electric field can be represented by field
lines. These lines start on a positive charge
and end on a negative charge.
Electric Field created by a
spherically charged object
Electric Field created by a
spherically charged object
21.8 Field Lines
The number of field lines starting (ending)
on a positive (negative) charge is
proportional to the magnitude of the charge.
The electric field is stronger where the field
lines are closer together.
21.8 Field Lines
Electric dipole: two equal charges, opposite in
sign:
21.8 Field Lines
Summary of field lines:
1. Field lines indicate the direction of the
field; the field is tangent to the line.
2. The magnitude of the field is proportional
to the density of the lines.
3. Field lines start on positive charges and
end on negative charges; the number is
proportional to the magnitude of the
charge.
21.8 Field Lines
Summary of field lines:
4. Field lines never cross because the
electric field cannot have two values for
the same point.
EM Field uses color to represent the field strength (ie. Red
is stronger; blue is weaker). Each charge below is ±10q.
Summary
 Using Newton’s Second Law, what the formula for force?
 HW (Place in your agenda):
 “Foundational Mathematics’ Skills of Physics” Packet (Page 16)
 Web Assign 21.5 - 21.7
 Future assignments:
 Electrostatics Lab #3: Lab Report (Due in 2 classes)
How do we use Coulomb’s Law and the principle of superposition to determine the force that acts between point charges?
Tuesday
(Day 14)
Section 21.9
Section 21.10
Warm-Up
Tues, Feb 10
 Each charge on the next slide is ±q. What will happen to the lines if a 3rd
charge of +q is added to the (1) right side and (2) left side?
 Place your homework on my desk:
 “Foundational Mathematics’ Skills of Physics” Packet (Page 16)
 Web Assign 21.5 - 21.7
 For future assignments - check online at www.plutonium-239.com
Field Example #1: Each charge below is ±q. What will
happen to the lines if a 3rd charge of +q is added to the
(1) right side and (2) left side?
Essential Question(s)
 WHAT PRIOR FOUNDATIONAL MATHEMATICS’ SKILLS ARE
NECESSARY IN PHYSICS II?
 HOW DO WE DESCRIBE THE NATURE OF ELECTROSTATICS
AND APPLY IT TO VARIOUS SITUATIONS?
 How do we describe and apply the nature of electric fields in and
around conductors?
 How do we describe and apply the concept of induced charge and
electrostatic shielding?
 How do we describe and apply the concept of electric fields?
Vocabulary









Static Electricity
Electric Charge
Positive / Negative
Attraction / Repulsion
Charging / Discharging
Friction
Induction
Conduction
Law of Conservation of Electric
Charge
 Non-polar Molecules











Polar Molecules
Ion
Ionic Compounds
Force
Derivative
Integration (Integrals)
Test Charge
Electric Field
Field Lines
Electric Dipole
Dipole Moment
Foundational Mathematics
Skills in Physics Timeline
Day
Pg(s)
Day
Pg(s)
Day
Pg(s)
Day
Pg(s)
1
1
2
6
3
11
16
16
21
2
13
14
7
4
12
17
17
8
3
22
23
8
5
13
18
18
9
4
24
†12
9
6
14
19
19
10
5
15
10
7
15
20
20
11
WHAT PRIOR FOUNDATIONAL MATHEMATICS’ SKILLS ARE NECESSARY IN PHYSICS II?
Agenda
 Review “Foundational Mathematics’ Skills of Physics” Packet (Page
16) with answer guide.
 Discuss
 Electric Fields and Conductors
 Motion of a Charged Particle in an Electric Field
 Work on Web Assign
Field Example #2: Each charge below is ±5q. What will
happen to the lines if a 3rd charge of +q is added to the
(1) right side and (2) left side?
Field Example #3: Each charge below is ±10q. What will
happen to the lines if a 3rd charge of +q is added to the
(1) right side and (2) left side?
Section 21.9
 How do we describe and apply the nature of electric fields
in and around conductors?
 How do we explain the mechanics responsible for the absence of
electric field inside of a conductor?
 Why must all of the excess charge reside on the surface of a
conductor?
 How do we prove that all excess charge on a conductor must reside
on its surface and the electric field outside of the conductor must
be perpendicular to the surface?
Section 21.9
How do we describe and apply the concept of
induced charge and electrostatic shielding?
What is the significance of why there can be no electric
field in a charge-free region completely surrounded by
a single conductor?
21.9 Electric Fields and Conductors
The static electric field inside a conductor is
zero – if it were not, the charges would move.
The net charge on a conductor is on its
surface.
Charge ball suspended in
a hollow metal sphere
Observations
The hollow sphere had a charge on
the outside.
The charged ball still had a charge.
Conclusions
The charged ball on the inside
induces an equal charge on the
hollow sphere.
21.9 Electric Fields and Conductors
The electric
field is
perpendicular
to the surface
of a
conductor –
again, if it
were not,
charges
would move.
Charge ball placed into a
hollow metal sphere
Observations
The hollow sphere had a charge on
the outside.
The charged ball no longer had a
charge.
Conclusions
The charge resides on the outside of
a conductor.
Applications of E-fields and conductors:
Faraday Cages

Faraday cages protect you from lightning because there is no electrical field inside the metal cage
(Notice (1) it completely surrounds him and (2) the size of the gaps in the fence (it is not a solid piece
of metal).
Section 21.10
How do we describe and apply the nature of
electric fields in and around conductors?
How do we determine the direction of the force
on a charged particle brought near an uncharged
or grounded conductor?
Section 21.10
How do we describe and apply the concept
of induced charge and electrostatic
shielding?
How do we determine the direction of the force
on a charged particle brought near an uncharged
or grounded conductor?
Section 21.10
How do we describe and apply the concept
of electric field?
How do we calculate the magnitude and
direction of the force on a positive or negative
charge in an electric field?
How do we analyze the motion of a particle of
known mass and charge in a uniform electric
field?
Electron accelerated by an electric field
 An electron is accelerated in the uniform field E
(E=2.0x104N/C) between two parallel charged
plates. The separation of the plates is 1.5 cm. The
electron is accelerated from rest near the negative
plate and passes through a tiny hole in the positive
plate. (a) With what speed does it leave the hole? (b)
Show that the gravitational force can be ignored.
[NOTE: Assume the hole is so small that it does not
affect the uniform field between the plates]
Electron accelerated by an electric field
(a) With what speed does it leave the hole?
F qE

F  ma  a 
m
m
v 2  v02  2ax  v  2ax
F  qE
0
 qE 
v  2   x
 m
1.60 x 10 C2.0 x 10
2
9.1 x 10 kg 
-19
v
v  1.0 x 10 7 m s
31
4
N

C 0.015 m 
Electron accelerated by an electric field
(b) Show that the gravitational force can be ignored.
FE  qE


FE  1.60 x 10-19 C 2.0 x 10 4 N C
FE  3.5 x 1015 N
FG  mg


FG  9.1 x 10-31 kg 9.8 m
s2


FG  8.9 x 10-30 N
Note that FE is 1014 times larger than the FG.
Also note that the electric field due to the electron
does not enter the problem since it cannot exert a
force on itself.
Applications of an electron accelerated by an
E-Field: Mass Spectrometer
 Mass Spectrometers are used to separate isotopes of atoms.
 The charged isotopes (a.k.a. ions) are accelerated to a velocity by the parallel plates
(located from S to S1)
Projectile Motion of a Charged Particle:
Electron moving perpendicular to E
 Suppose an electron is traveling with
a speed, v0 = 1.0x107m/s, enters a
uniform field E at right angles to v0.
Describe the motion by giving the
equation of its path while in the
electric field. Ignore gravity.
F
eE
F  ma  a   
m
m
F  qE  eE
eE 2
eE 2


x
t
y  v0 y t  ayt  
2
qmv0
2m
0
1
2
2
This is the equation of a
parabola (i.e. projectile motion).
2
2


x
x
x
x
2
2
1

t
   2

t


x  v0 x t  2 ax t
 v0  v0
v0 x v0
0
Electrons moving perpendicular to E:
The discovery of the electron: J.J. Thomson’s
Experiment
 J. J. Thomson’s famous experiment that allowed him to discover the electron.
Applications of an electron moving
perpendicular to E: Cathode Ray Tube (CRT)
 Television Sets & Computer Monitors (CRT)
Applications of an electron moving
perpendicular to E: Mass Spectrometer
 Mass Spectrometers are used to separate isotopes of atoms.
 The charged isotopes (a.k.a. ions) are accelerated to a velocity by the parallel plates
(located at the - & + plates)
Applications of an electron moving
perpendicular to E: e/m Apparatus
 e/m Apparatus
Applications of an electron moving
perpendicular to E: e/m Apparatus
 e/m Apparatus
Summary
 Using your kinematic equations, determine the equation that relates y to v0, g,
, and x?
 HW (Place in your agenda):
 “Foundational Mathematics’ Skills of Physics” Packet (Page 17)
 Web Assign 21.12 - 21.14
 Future assignments:
 Electrostatics Lab #3: Lab Report (Due in 1 class)
How do we use Coulomb’s Law and the principle of superposition to determine the force that acts between point charges?
Wednesday
(Day 15)
Work Day
Warm-Up
Wed, Feb 11
 Write down the steps that you would use to explain how to open a door to a
blind person?
 Place your homework on my desk:
 “Foundational Mathematics’ Skills of Physics” Packet (Page 17)
 Web Assign 21.12 - 21.14
 For future assignments - check online at www.plutonium-239.com
Essential Question(s)
 WHAT PRIOR FOUNDATIONAL MATHEMATICS’ SKILLS ARE
NECESSARY IN PHYSICS II?
 HOW DO WE DESCRIBE THE NATURE OF ELECTROSTATICS
AND APPLY IT TO VARIOUS SITUATIONS?
 How do we describe and apply the nature of electric fields in and
around conductors?
 How do we describe and apply the concept of induced charge and
electrostatic shielding?
 How do we describe and apply the concept of electric fields?
Vocabulary









Static Electricity
Electric Charge
Positive / Negative
Attraction / Repulsion
Charging / Discharging
Friction
Induction
Conduction
Law of Conservation of Electric
Charge
 Non-polar Molecules











Polar Molecules
Ion
Ionic Compounds
Force
Derivative
Integration (Integrals)
Test Charge
Electric Field
Field Lines
Electric Dipole
Dipole Moment
Foundational Mathematics
Skills in Physics Timeline
Day
Pg(s)
Day
Pg(s)
Day
Pg(s)
Day
Pg(s)
1
1
2
6
3
11
16
16
21
2
13
14
7
4
12
17
17
8
3
22
23
8
5
13
18
18
9
4
24
†12
9
6
14
19
19
10
5
15
10
7
15
20
20
11
WHAT PRIOR FOUNDATIONAL MATHEMATICS’ SKILLS ARE NECESSARY IN PHYSICS II?
Agenda
 Review “Foundational Mathematics’ Skills of Physics” Packet (Page
17) with answer guide.
 Work Day
 Coulomb’s Law
 Web Assign
 REVISE: Complete Graphic Organizer (up to Sections 21.10)
Summary
 Using Newton’s Second Law, what the formula for force?
 HW (Place in your agenda):
 “Foundational Mathematics’ Skills of Physics” Packet (Page 18)
 Web Assign 21.12 - 21.14
 Future assignments:
How do we use Coulomb’s Law and the principle of superposition to determine the force that acts between point charges?
Thursday
(Day 16)
Section 21.11
Warm-Up
Thurs, Feb 12
 Complete Graphic Organizers for Sections 21-8 & 21-10.
 Place your homework on my desk:
 “Foundational Mathematics’ Skills of Physics” Packet (Page 18)
 Web Assign 21.12 - 21.14
 For future assignments - check online at www.plutonium-239.com
Essential Question(s)
 WHAT PRIOR FOUNDATIONAL MATHEMATICS’ SKILLS ARE
NECESSARY IN PHYSICS II?
 HOW DO WE DESCRIBE THE NATURE OF ELECTROSTATICS
AND APPLY IT TO VARIOUS SITUATIONS?
 How do we compare and contrast the basic properties of an
insulator and a conductor?
 How do we describe and apply the concept of electric field?
Vocabulary









Static Electricity
Electric Charge
Positive / Negative
Attraction / Repulsion
Charging / Discharging
Friction
Induction
Conduction
Law of Conservation of Electric
Charge
 Non-polar Molecules











Polar Molecules
Ion
Ionic Compounds
Force
Derivative
Integration (Integrals)
Test Charge
Electric Field
Field Lines
Electric Dipole
Dipole Moment
Foundational Mathematics
Skills in Physics Timeline
Day
Pg(s)
Day
Pg(s)
Day
Pg(s)
Day
Pg(s)
1
1
2
6
3
11
16
16
21
2
13
14
7
4
12
17
17
8
3
22
23
8
5
13
18
18
9
4
24
†12
9
6
14
19
19
10
5
15
10
7
15
20
20
11
WHAT PRIOR FOUNDATIONAL MATHEMATICS’ SKILLS ARE NECESSARY IN PHYSICS II?
Agenda
 Review “Foundational Mathematics’ Skills of Physics” Packet (Page
18) with answer guide.
 Discuss





Torque, factors that affect torque, r X F
Electric Dipoles
Electric Dipoles in an Electric Field
The electric field produced by a dipole
Calculations: Dipoles in an electric field
Section 21.11
How do we compare and contrast the basic
properties of an insulator and a conductor?
What are characteristics and classification(s) of
electrically . . .
conductive atoms?
insulative atoms?
semi-conductive atoms?
conductive compounds?
insulative compounds?
semi-conductive compounds?
Section 21.11
How do we describe and apply the concept
of electric field?
How do we calculate the net force and torque
on a collection of charges in an electric field?
How do we calculate the net force and torque on a collection of charges in an electric field?
Torque
To make an object start rotating, a force is needed;
the position and direction of the force matter as well.
The perpendicular distance from the axis of rotation
to the line along which the force acts is called the
lever arm.
How do we calculate the net force and torque on a collection of charges in an electric field?
Torque
A longer lever
arm is very
helpful in
rotating objects.
How do we calculate the net force and torque on a collection of charges in an electric field?
Torque
Here, the lever arm for FA is the distance from the
knob to the hinge; the lever arm for FD is zero;
and the lever arm for FC is as shown.
How do we calculate the net force and torque on a collection of charges in an electric field?
Torque
The torque is defined
as:
 = r F
or
 = r F
Torque, 
 Torque is perpendicular to the direction of the rotation.
 Right-hand rule - The direction of the positive torque is
in the direction of increasing angle)
 In general, if we define torque as
 = r x F = r F sin 
 Also, torque can be defined about any point using
net = (ri x Fi)
 where ri is the position vector of the ith particle and Fi is
the net force on the ith particle.
How do we calculate the net force and torque on a collection of charges in an electric field?
Electric Dipoles
•The combination of two equal charges of opposite sign, +Q and -Q , separated by a
distance l, is referred to as an electric dipole. The quantity Ql is called the dipole
moment, p. The dipole moment points from the negative to the positive charge. Many
molecules have a dipole moment and are referred to as polar molecules.
•It is interesting to note that the value of the separated charges may be less than that of a
single electron or proton but cannot be isolated.
How do we calculate the net force and torque on a collection of charges in an electric field?
Electric Dipoles
•
Electric Dipoles: The combination of two equal charges of opposite sign, +Q
and -Q, separated by a distance l.
•
•
•
•
The dipole moment, p: The quantity Ql.
The dipole moment points from the negative to the positive charge.
Many molecules have a dipole moment and are referred to as polar molecules.
It is interesting to note that the value of the separated charges may be less than that
of a single electron or proton, but they cannot be isolated.
Dipole in an External Field
+q
F
F±
F±
F
A dipole, p = Ql, is placed in an electric field E.
First, let us analyze the angle , for torque and
about its bisector at point O.
Point O
sinPoint
0
F±
sinF±
0
180
0
45
2
135
90
1
-q
Note that the choice of the
angle does not change our value
for sin   point O will be used
for all reference angles instead
of the rxF angle to relate the
direction of the dipole moment
to the E-Field.
135
180
O
2
2
2
0
90
45
0
2
2
1
2
2
0
How do we calculate the net force and torque on a collection of charges in an electric field?
Dipole in an External Field
A dipole, p = Ql, is placed in an electric field E.
Next, let us analyze the direction of the torque force
to the change angle , Note: By definition, positive
torque always increases the value of  (I.e. move
the dipole in the counterclockwise direction).
It is also important to note that the applied torque
force will cause the angle  decrease (in the
clockwise direction) instead of increase (counterclockwise) about point O.   about point O is
negative.
How do we calculate the net force and torque on a collection of charges in an electric field?
Dipole in an External Field
A dipole p = Ql is placed in an electric field E.
  r  F  rF sin 
 net   r  F   rF sin 
 net        rF sin   rF sin 
 net
 net
l
l
 QE sin   QE sin 
2
2
 QlE sin   pE sin  p  E
How do we calculate the net force and torque on a collection of charges in an electric field?
Dipole in an External Field
The effect of the torque is to try to turn the dipole
so p is parallel to E. The work done on the dipole
by the electric field to change the angle from 
to  , is

W    d
2
1
Because the direction of the torque is opposite to
the direction of increasing , we write the torque as
  p  E or    pE sin
Then the dipole so p is parallel to E. The work done on the dipole by the electric
field to change the angle from  to  , is
2
W   pE  sin  d  pE cos 12  pE cos 2  cos1 

1
How do we calculate the net force and torque on a collection of charges in an electric field?
Dipole in an External Field
Positive work done by the field decreases the
potential energy, U, of the dipole in the field. If we
choose U = 0 when p is perpendicular to E (that is
choosing  = 90º so cos  = 0), and setting  = 
then


W  pE  cos 2  cos1   pE cos



U  W   pE cos  p  E
How do we calculate the net force and torque on a collection of charges in an electric field?
Torque with respect to the
Dipole’s Orientation
How do we calculate the net force and torque on a collection of charges in an electric field?
Electric Field Produced by
a Dipole
+
–
h
r
To determine the electric field produced by a dipole
in the absence of an external field along the
midpoint or perpendicular bisector of the dipole.
E  E  E
h  r 2  L2 4
1 Q
1
Q

E  E  E 
4 0 h 2 4 0 r 2  L2 4
L
2
Enet  E cos  E cos  2E cos  2E 2
r  L2 4
L
1
Q
L
Enet  E 2

2
2
r  L2 4 4 0 r  L 4 r 2  L2 4

h

Enet 
Enet
1


p


4 0 r 2  L2 4 3 2
1 p

at r >> L
4 0 r 3
How do we calculate the net force and torque on a collection of charges in an electric field?
Electric Field Produced by
a Dipole
+
–
h
r
h
It is interesting to note that at r >> l, the electric
field decreases more rapidly for a dipole (1/r3) than
for a single point charge (1/r2). This is due to the
fact that at large distances the two opposite charges
neutralize each other due to their close proximity
At distances where r >> l, this 1/r3 dependence also
applies for points that are not on the perpendicular
bisector of the dipole.
p
1

4 0 r 3 r 3
1
1 q

E
4 0 r 2 r 2
E
1
For a dipole at r >> l
For a single point charge
How do we calculate the net force and torque on a collection of charges in an electric field?
Table of Dipole Moment
Values
How do we calculate the net force and torque on a collection of charges in an electric field?
Dipoles in an Electric Field
The dipole moment of a water molecule is 6.1 x 10-30 C•m. A water
molecule is placed in a uniform electric field with magnitude
2.0 x 105 N/C.
•
What is the magnitude of the maximum torque that electric
field can exert on the molecule?
90

p1 cos
p2 cos

 max  p  E  pE sin   pE  6.1 x 10 30 C m  2.0 x 10 5 N C  1.2 x 1024 N m
1
•
What is the potential energy when the torque is at its maximum?
•
What is the dipole moment, p1 and p2, for a single O-H bond (where 2 =
 pOH .
104.5°)? pNote:
1  p2Let
p  pnet  p1 cos  p2 cos  2 pOH cos  2 
U  p  E   pE cos   pE cos 90
 pOH

 0

6.1 x 10 30 C m
p


 4.98 x 10 30 C m
2 cos 104.5 2 
2 cos  2 
How do we calculate the net force and torque on a collection of charges in an electric field?
Dipoles in an Electric Field
The dipole moment of a water molecule is 6.1 x 10-30 C•m. A water
molecule is placed in a uniform electric field with magnitude
2.0 x 105 N/C.
•
In what position will the potential energy take on its greatest
value?
The potential energy will be maximized when cos  = –1, so  = 180°, which
means p and E are antiparallel. The potential energy is maximized when the
dipole moment is oriented so that it has to rotate through the largest angle, 180°,
to reach equilibrium at  = 0°.
•
Why is this different than the position where the torque is
maximized?
The torque is maximized when the electric forces are perpendicular to p.
How do we calculate the net force and torque on a collection of charges in an electric field?
Dipoles in an Electric Field
The carbonyl group (C=O) dipole. The distance between the carbon
(+) and oxygen (–) atoms in the carbonyl group which occurs
in many organic molecules is about 1.2 x 10-10 m and the dipole
moment of this group is about 8.0 x 10-30 C•m. A formaldehyde
molecule, CH2O, is placed in a uniform electric field with
magnitude 2.0 x 105 N/C.
•
What the direction of the dipole moment, p?
•
p
What is the magnitude of the maximum torque that electric
field can exert on the molecule?
90


 max  p  E  pE sin   pE  8.0 x 1030 C m  2.0 x 105 N C  1.6 x 1024 N m
1
•
What is the potential energy when the torque is at its
maximum?
U  p  E   pE cos   pE cos 90
  0
How do we calculate the net force and torque on a collection of charges in an electric field?
Dipoles in an Electric Field
The carbonyl group (C=O) dipole. The distance between the
carbon (+) and oxygen (–) atoms in the carbonyl group
which occurs in many organic molecules is about
1.2 x 10-10 m and the dipole moment of this group is about
8.0 x 10-30 C•m. A formaldehyde molecule, CH2O, is
placed in a uniform electric field with magnitude
2.0 x 105 N/C.
•
What is the partial charge (±) of the carbon (+) and oxygen (–) atoms in the
carbonyl group?
8.0 x 10 30 C m
p
20
p  Ql  Q  

6.7
x
10
C
l
1.2 x 10 10 m


•


(a) How much of the quantized charge of an electron/proton is the partial charge
of the carbonyl group to? (b) What is this value in percent?
(a)
n
Q 
Q
6.7 x 10 20 C
(b) Percentage of an electron's charge  n x 100%

0.42

 .42 x 100%  42%
1.602 x 10 19 C
How do we calculate the net force and torque on a collection of charges in an electric field?
Dipoles in an Electric Field
The carbonyl group (C=O) dipole. The distance between the carbon
(+) and oxygen (–) atoms in the carbonyl group which occurs
in many organic molecules is about 1.2 x 10-10 m and the dipole
moment of this group is about 8.0 x 10-30 C•m. A formaldehyde
molecule, CH2O, is placed in a uniform electric field with
magnitude 2.0 x 105 N/C.
•
In what position will the potential energy take on its greatest
value?
The potential energy will be maximized when cos  = –1, so  = 180°, which
means p and E are antiparallel. The potential energy is maximized when the
dipole moment is oriented so that it has to rotate through the largest angle, 180°,
to reach equilibrium at  = 0°.
•
Why is this different than the position where the torque is
maximized?
The torque is maximized when the electric forces are perpendicular to p.
How do we calculate the net force and torque on a collection of charges in an electric field?
Summary
 How does positive torque relate to the change in the angle?
 HW (Place in your agenda):
 “Foundational Mathematics’ Skills of Physics” Packet (Page 19)
 Web Assign 21.15
 Future assignments:
How do we calculate the net force and torque on a collection of charges in an electric field?
Supplementary Notes
Vector Cross Product
 Known as the vector product or cross product
 The cross product of two vectors A and B is
defined as another vector C = A x B whose
magnitude is
C = |A x B| = AB sin 
where  < 180º between A and B and whose
direction is perpendicular to both A and B.
 Right hand rules for cross products
Vector Cross Product
 The cross product of two vectors
 A = Axi + Ayj + Azk
 B = Bxi + Byj + Bzk
 Can be written as
i
j
k
A  B  Ax
Bx
Ay
By
Az
Bz
A x B = (AyBz-AzBy)i + (AzBx-AxBz)j + (AxByAyBx)k

Properties of Vector Cross
Products
A
A
A

xA=0
x B = -B x A
x (B + C) = (A x B) + (A x C)
d
dA
dB
.
A  B   B  A 
dt

dt
dt
NOT FINISHED
START HERE
Review Riemann Sums Proof (see
notes)
Friday
(Day 17)
Warm-Up
Fri, Feb 13
 Complete Graphic Organizer for Section 21-11.
 Place your homework on my desk:
 “Foundational Mathematics’ Skills of Physics” Packet (Page 19)
 Web Assign 21.15
 For future assignments - check online at www.plutonium-239.com
Essential Question(s)
 WHAT PRIOR FOUNDATIONAL MATHEMATICS’ SKILLS ARE
NECESSARY IN PHYSICS II?
 HOW DO WE DESCRIBE THE NATURE OF ELECTROSTATICS
AND APPLY IT TO VARIOUS SITUATIONS?
 How do we apply integration and the Principle of Superposition to
uniformly charged objects?
 How do we identify and apply the fields of highly symmetric
charge distributions?
 How do we describe and apply the electric field created by
uniformly charged objects?
Vocabulary









Static Electricity
Electric Charge
Positive / Negative
Attraction / Repulsion
Charging / Discharging
Friction
Induction
Conduction
Law of Conservation of Electric
Charge
 Non-polar Molecules











Polar Molecules
Ion
Ionic Compounds
Force
Derivative
Integration (Integrals)
Test Charge
Electric Field
Field Lines
Electric Dipole
Dipole Moment
Foundational Mathematics
Skills in Physics Timeline
Day
Pg(s)
Day
Pg(s)
Day
Pg(s)
Day
Pg(s)
1
1
2
6
3
11
16
16
21
2
13
14
7
4
12
17
17
8
3
22
23
8
5
13
18
18
9
4
24
†12
9
6
14
19
19
10
5
15
10
7
15
20
20
11
WHAT PRIOR FOUNDATIONAL MATHEMATICS’ SKILLS ARE NECESSARY IN PHYSICS II?
Agenda
 Review “Foundational Mathematics’ Skills of Physics”
Packet (Page 19) with answer guide.
 Steps to Determine the E-Field Created by a Uniform Charge
Distributions
 Calculate the electric field for continuous charge
distributions for the following:
 Uniformly Charged Ring ( 0  2)
 Uniformly Charged Vertical Wire (–∞+∞)
 Uniformly Charged Vertical Wire (–L/2+L/2)
 Distribute E-Field Derivation Rubrics
Section 21.7
 How do we apply integration and the Principle of Superposition to
uniformly charged objects?
 How do we use integration and the principle of superposition to calculate
the electric field of a straight, uniform charge wire?
 How do we use integration and the principle of superposition to calculate
the electric field of a thin ring of charge on the axis of the ring?
 How do we use integration and the principle of superposition to
calculate the electric field of a semicircle of charge at its center?
 How do we use integration and the principle of superposition to calculate
the electric field of a uniformly charged disk on the axis of the disk?
Section 21.7
 How do we identify and apply the fields of highly
symmetric charge distributions?
 How do we identify situations in which the direction of the electric
field produced by highly symmetric charge distributions can be
deduced from symmetry considerations?
Section 21.7
How do we describe and apply the electric field
created by uniformly charged objects?
How do we describe the electric field of parallel
charged plates?
How do we describe the electric field of a long,
uniformly charged wire?
How do we use superposition to determine the electric
fields of parallel charged plates?
21-7 The Field of a Continuous Distribution
To find the field of a continuous distribution of
charge, treat it as a collection of near-point
charges:
Summing over the
infinitesimal fields:
How do we apply integration and the Principle of Superposition to uniformly charged objects?
How do we identify and apply the fields of highly symmetric charge distributions?
How do we describe and apply the electric field created by uniformly charged objects?
21-7 The Field of a Continuous Distribution
Finally, making the charges infinitesimally
small and integrating rather than summing:
How do we apply integration and the Principle of Superposition to uniformly charged objects?
How do we identify and apply the fields of highly symmetric charge distributions?
How do we describe and apply the electric field created by uniformly charged objects?
21-7 The Field of a Continuous Distribution
Some types of charge distribution are relatively
simple.
Constant linear charge density
:
How do we apply integration and the Principle of Superposition to uniformly charged objects?
How do we identify and apply the fields of highly symmetric charge distributions?
How do we describe and apply the electric field created by uniformly charged objects?
21-7 The Field of a Continuous Distribution
Constant surface charge density
:
How do we apply integration and the Principle of Superposition to uniformly charged objects?
How do we identify and apply the fields of highly symmetric charge distributions?
How do we describe and apply the electric field created by uniformly charged objects?
21-7 The Field of a Continuous Distribution
Constant volume charge density
:
How do we apply integration and the Principle of Superposition to uniformly charged objects?
How do we identify and apply the fields of highly symmetric charge distributions?
How do we describe and apply the electric field created by uniformly charged objects?
Steps to Determine the E-Field Created
by a Uniform Charge Distributions
1.
2.
3.
4.
5.
6.
7.
8.
Graphical:
Draw a picture of the object and 3-D plane.
Label the partial length, area, or volume that is creating the partial E-field.
Determine the distance from the charged object to the location of the desired E-Field and label all
components and lengths.
Mathematical:
Write the formulas for dE and the component of the E-field that contributes to the net E-field (i.e.
does not cancel due to symmetry).
Write the total charge density and solve it for Q.
Write the charge density in relation to the partial charge and solve it for the partial charge (dq).
Set up the integral by determining what key component(s) change.
†Solve the integral and write the answer in a concise manner.
†See the instructor, AP Calculus BC students, or Schaum’s Mathematical Handbook.
Uniformly Charged Ring
( 0  2)
1
dq
1 z
z
dE 
;
dE

dE
cos


dq

dE
z
2
3
4

h
h 4 0 h
0
r
h  r 2  z2
Q

Q
Q

 Q  2 r   r
 
l 2 r r

dq dq
z
 dq  rd


dEz  dE cos 
dl rd
Qtot z
1
Etot
dE

dq
Ez   dEz
3

0
0
4 0
h  rd
dl
Ez 
1
4 0

2
0
z
h
z r
2
Ez 

3  r d

4 0 z 2  r 2

3
2  0  
2


1
z
4 0 z 2  r 2
2
z r
1
z r
1

3
2

4 0 z 2  r 2
2
0

3
d 
Q

4 0 z 2  r 2
 2 r 
2
z r
1
1
3  0
2
2

zQ

4 0 z 2  r 2

3
2
Uniformly Charged
Vertical Wire (–∞+∞)

dq
1 x
x
;

dq
dE

dE
cos

 dE
2
x
3
2
2
4 0 h
h 4 0 h
h x y
Q
dE x  dE cos 
Q


 Q  y

y
l
x
dE
dq dq
 dq   dy


dl dy
dE 
dy
y
-
Ex  
Etot
0
dEx 
1
4 0

Qtot
0
1
x
1
dq

h 3  dy 4 0




x
h

3  dy
x 2 y 2


y
1 
y
Ex 
x  2 2 2  
 2
2
4 0
 x x  y   4 0 x  x  y
1 y
1 Q
Ex 

2 0 x y
2 0 xy
1

1
4 0
x


1
x
2
 y2

3
dy
2

1 
1 

1
1








4

x
2 0 x
0
 
Uniformly Charged
Vertical Wire (–L/2+L/2)
L
2
dq
1 x
x
;

dq
dE

dE
cos

 dE
2
x
3
2
2
dy
4 0 h
h 4 0 h
h x y
y
Q
dE x  dE cos 
Q


 Q  y

y
l
x
dE
dq dq
 dq   dy



L

dl dy
2
Qtot x
L 2
L 2
Etot
1
1
1
1
x
dq


x


dy
Ex   dEx 
3 dy
3


3

0
L
2
L
2
0
2
2
4 0
h  dy 4 0
4 0
h
x y 2
dE 
1
x y
2
L 2


y
1 
y
Ex 
x  2 2 2  
 2
2
4 0
 x x  y  L 2 4 0 x  x  y


1 
L 2
L 2
 
Ex 

2 
2
4 0 x  x 2  L 2 2
x


L
2




1

2
L 2


 L 2
L
2 0 x
1
1
4x 2  L2

Summary
 How did symmetry help to reduce our calculations?
 HW (Place in your agenda):
 “Foundational Mathematics’ Skills of Physics” Packet (Page 20)
 For each of the following, complete 1 derivation with reasons for
each mathematical step and 3 additional derivations: (*Refer to
rubric)
Uniformly Charged Ring ( 0  2)
Uniformly Charged Vertical Wire (–∞+∞)
*Uniformly Charged Vertical Wire (–L/2+L/2)
 Web Assign 21.8 - 21.11
How do we apply integration and the Principle of Superposition to uniformly charged objects?
How do we identify and apply the fields of highly symmetric charge distributions?
How do we describe and apply the electric field created by uniformly charged objects?
Tuesday
(Day 18)
Warm-Up
Tues, Feb 17
 Identify the parts of the derivation that were confusing.
 Place your homework on my desk:
 “Foundational Mathematics’ Skills of Physics” Packet (Page 20)
 For each of the following, 1 derivation with reasons for each mathematical
step and 3 additional derivations: (*Refer to rubric)
 Uniformly Charged Ring ( 0  2)
 Uniformly Charged Vertical Wire (–∞+∞)
 *Uniformly Charged Vertical Wire (–L/2+L/2)
 Web Assign 21.8 - 21.11
 For future assignments - check online at www.plutonium-239.com
Essential Question(s)
 WHAT PRIOR FOUNDATIONAL MATHEMATICS’ SKILLS ARE
NECESSARY IN PHYSICS II?
 HOW DO WE DESCRIBE THE NATURE OF ELECTROSTATICS
AND APPLY IT TO VARIOUS SITUATIONS?
 How do we apply integration and the Principle of Superposition to
uniformly charged objects?
 How do we identify and apply the fields of highly symmetric
charge distributions?
 How do we describe and apply the electric field created by
uniformly charged objects?
Vocabulary









Static Electricity
Electric Charge
Positive / Negative
Attraction / Repulsion
Charging / Discharging
Friction
Induction
Conduction
Law of Conservation of Electric
Charge
 Non-polar Molecules











Polar Molecules
Ion
Ionic Compounds
Force
Derivative
Integration (Integrals)
Test Charge
Electric Field
Field Lines
Electric Dipole
Dipole Moment
Foundational Mathematics
Skills in Physics Timeline
Day
Pg(s)
Day
Pg(s)
Day
Pg(s)
Day
Pg(s)
1
1
2
6
3
11
16
16
21
2
13
14
7
4
12
17
17
8
3
22
23
8
5
13
18
18
9
4
24
†12
9
6
14
19
19
10
5
15
10
7
15
20
20
11
WHAT PRIOR FOUNDATIONAL MATHEMATICS’ SKILLS ARE NECESSARY IN PHYSICS II?
Agenda
 Review “Foundational Mathematics’ Skills of Physics”
Packet (Page 20) with answer guide.
 Calculate the electric field for continuous charge
distributions for the following:
 †Uniformly Charged Horizontal Wire (dd+l)
 Uniformly Charged Disk (0R)
 Complete 4 Derivations + Reasoning for the above
problems
 Complete Web Assign Problems 21.8 - 21.11
Section 21.7
 How do we apply integration and the Principle of
Superposition to uniformly charged objects?
 How do we use integration and the principle of superposition to
calculate the electric field of a straight, uniform charge wire?
 How do we use integration and the principle of superposition to
calculate the electric field of a thin ring of charge on the axis of the
ring?
 How do we use integration and the principle of superposition
to calculate the electric field of a semicircle of charge at its
center?
 How do we use integration and the principle of superposition to
calculate the electric field of a uniformly charged disk on the axis
of the disk?
Section 21.7
 How do we identify and apply the fields of highly
symmetric charge distributions?
 How do we identify situations in which the direction of the electric
field produced by highly symmetric charge distributions can be
deduced from symmetry considerations?
Section 21.7
How do we describe and apply the electric field
created by uniformly charged objects?
How do we describe the electric field of parallel
charged plates?
How do we describe the electric field of a long,
uniformly charged wire?
How do we use superposition to determine the electric
fields of parallel charged plates?
Steps to Determine the E-Field Created
by a Uniform Charge Distributions
1.
2.
3.
4.
5.
6.
7.
8.
Graphical:
Draw a picture of the object and 3-D plane.
Label the partial length, area, or volume that is creating the partial E-field.
Determine the distance from the charged object to the location of the desired E-Field
and label all components and lengths.
Mathematical:
Write the formulas for dE and the component of the E-field that contributes to the net
E-field (i.e. does not cancel due to symmetry).
Write the total charge density and solve it for Q.
Write the charge density in relation to the partial charge and solve it for the partial
charge (dq).
Set up the integral by determining what key component(s) change.
†Solve the integral and write the answer in a concise manner.
†See the instructor, AP Calculus BC students, or Schaum’s Mathematical Handbook.
†Uniformly
Charged
Horizontal Wire (dd+l)
1 1
dq

dq
dE

dE
dE 
;
x
2
2
4 0 x
4 0 h
d
l
Q
 Q  l

l
dE x
dq dq  dq  dx
dx
x


dl
dx
dl 
dl 1
Etot
Qtot 1
1
1
1
dx 

dx
Ex   dEx 
dq 
2
2

2

d
d
0
0
x
4 0
x
4 0
x  dx 4 0
1
d  l  d 
1
1
1
1
1 
 1
1



Ex 
   
  
 


4 0  x  d 4 0  x  dl 4 0  d d  l  4 0  d d  l  
1
dl


1
l

Ex 


4 0
4 0  d d  l 
1
d
 Q 


d
d

l




R
Uniformly Charged Disk
(0R)
z
dq dE  dE cos   dE  1 z dq
dE 
; z
3
2
h
4

h
0
4 0 h
r
2
2
Q
Q
h r z
  2
 Q   r 2
A r

dA
 dA  2 rdr
z
 2 r
dr
dr
dEz  dE cos 
dq
dq
 dq  2 r dr


dA 2 rdr
dE
Qtot z
R
1
Etot
1
z

dq

Ez   dEz
3

3 2 r dr

0
0
0
4 0
h 2 r dr 4 0
h
1
R
Ez 
1
4 0
2 z 
r
R
0
z
2
r
2

3
dr 
2
1

1
1

Ez 
z  2


2
2 0
z
z  R  2 0

z 2 r 2
0




1
1
1
1
 z  2 2  
z 2 2 
2 0
2

z r 0
0

 z r R
z


z

  2

2
2 0
z
z R 



z
1



2
2
z R 

Summary
 How did symmetry help to reduce our calculations?
 HW (Place in your agenda):
 “Foundational Mathematics’ Skills of Physics” Packet (Page 21)
 For each of the following, complete 1 derivation with reasons for each
mathematical step and 3 additional derivations: (*Refer to rubric)
 †Uniformly Charged Horizontal Wire (dd+l)
 Uniformly Charged Disk (0R)
 Web Assign 21.8 - 21.11
How do we apply integration and the Principle of Superposition to uniformly charged objects?
How do we identify and apply the fields of highly symmetric charge distributions?
How do we describe and apply the electric field created by uniformly charged objects?
Wednesday
(Day 19)
Warm-Up
Wed, Feb 18
 Identify the parts of the derivation that were confusing.
 Place your homework on my desk:
 “Foundational Mathematics’ Skills of Physics” Packet (Page 21)
 For each of the following, 1 derivation with reasons for each mathematical
step and 3 additional derivations: (*Refer to rubric)
 †Uniformly Charged Horizontal Wire (dd+l)
 Uniformly Charged Disk (0R)
 Web Assign 21.8 - 21.11
 For future assignments - check online at www.plutonium-239.com
Essential Question(s)
 WHAT PRIOR FOUNDATIONAL MATHEMATICS’ SKILLS ARE
NECESSARY IN PHYSICS II?
 HOW DO WE DESCRIBE THE NATURE OF ELECTROSTATICS
AND APPLY IT TO VARIOUS SITUATIONS?
 How do we apply integration and the Principle of Superposition to
uniformly charged objects?
 How do we identify and apply the fields of highly symmetric
charge distributions?
 How do we describe and apply the electric field created by
uniformly charged objects?
Vocabulary









Static Electricity
Electric Charge
Positive / Negative
Attraction / Repulsion
Charging / Discharging
Friction
Induction
Conduction
Law of Conservation of Electric
Charge
 Non-polar Molecules











Polar Molecules
Ion
Ionic Compounds
Force
Derivative
Integration (Integrals)
Test Charge
Electric Field
Field Lines
Electric Dipole
Dipole Moment
Foundational Mathematics
Skills in Physics Timeline
Day
Pg(s)
Day
Pg(s)
Day
Pg(s)
Day
Pg(s)
1
1
2
6
3
11
16
16
21
2
13
14
7
4
12
17
17
8
3
22
23
8
5
13
18
18
9
4
24
†12
9
6
14
19
19
10
5
15
10
7
15
20
20
11
WHAT PRIOR FOUNDATIONAL MATHEMATICS’ SKILLS ARE NECESSARY IN PHYSICS II?
Agenda
 Review “Foundational Mathematics’ Skills of Physics”
Packet (Page 21) with answer guide.
 Calculate the electric field for continuous charge
distributions for the following:
 Uniformly Charged Disk (0∞)
 Uniformly Charged Hoop (R1R2)
 Uniformly Charged Infinite Plate
 Complete 4 Derivations + Reasoning for the above
problems
 Complete Web Assign Problems 21.8 - 21.11
Section 21.7
 How do we apply integration and the Principle of
Superposition to uniformly charged objects?
 How do we use integration and the principle of superposition to
calculate the electric field of a straight, uniform charge wire?
 How do we use integration and the principle of superposition to
calculate the electric field of a thin ring of charge on the axis of the
ring?
 How do we use integration and the principle of superposition
to calculate the electric field of a semicircle of charge at its
center?
 How do we use integration and the principle of superposition to
calculate the electric field of a uniformly charged disk on the axis
of the disk?
Section 21.7
 How do we identify and apply the fields of highly
symmetric charge distributions?
 How do we identify situations in which the direction of the electric
field produced by highly symmetric charge distributions can be
deduced from symmetry considerations?
Section 21.7
How do we describe and apply the electric field
created by uniformly charged objects?
How do we describe the electric field of parallel
charged plates?
How do we describe the electric field of a long,
uniformly charged wire?
How do we use superposition to determine the electric
fields of parallel charged plates?
Steps to Determine the E-Field Created
by a Uniform Charge Distributions
1.
2.
3.
4.
5.
6.
7.
8.
Graphical:
Draw a picture of the object and 3-D plane.
Label the partial length, area, or volume that is creating the partial E-field.
Determine the distance from the charged object to the location of the desired E-Field and label all
components and lengths.
Mathematical:
Write the formulas for dE and the component of the E-field that contributes to the net E-field (i.e.
does not cancel due to symmetry).
Write the total charge density and solve it for Q.
Write the charge density in relation to the partial charge and solve it for the partial charge (dq).
Set up the integral by determining what key component(s) change.
†Solve the integral and write the answer in a concise manner.
†See the instructor, AP Calculus BC students, or Schaum’s Mathematical Handbook.
R
Ez 
Uniformly Charged Disk
(0∞)
z
dq dE  dE cos   dE  1 z dq
dE 
; z
3
2
h
4

h
0
4 0 h
r
2
2
Q
Q
h r z
  2
 Q   r 2
A r

dA
 dA  2 rdr
z
 2 r
dr
dr
dEz  dE cos 
dq
dq
 dq  2 r dr


dA 2 rdr
dE
Qtot z

1
Etot
1
z

dq

Ez   dEz
3

3 2 r dr

0
0
0
4 0
h 2 r dr 4 0
h
1
2 z 
r
4 0 h 3 0 z 2  r 2

1


3
dr 
2
1


1
1

Ez 
z  2

2
2 0
2 0
z
z  

z 2 r 2
0




1
1
1
1
 z  2 2  
z 2 2 
2 0
2

z r 0
0

 z r 
z  
 z 
2 0
Uniformly Charged Hoop
(R1R2)
z
dq dE  dE cos   dE  1 z dq
dE 
; z
3
2
h
4

h
0
4 0 h
r
2
2
Q
Q
h r z
  2
 Q   r 2
A r

dA
 dA  2 rdr
R1
z
 2 r
dr
dr
dEz  dE cos 
dq
dq
 dq  2 r dr


dA 2 rdr
dE
Qtot z
R2
1
Etot
1
z

dq

Ez   dEz
3

3 2 r dr

0
R
0
4 0
h 2 r dr 4 0 1 h
R2
1
2 z R2
r
Ez 
4 0 h 3 R1 z 2  r 2
1

z 2 r 2
R2

3
2
R1




1
1
1
1
 z  2 2  
z 2 2 
dr 
2 0
z  r  R 2 0

 z  r  R2
1

1
1
1
Ez 
z 2

2 0
z 2  R22
 z  R12

 
z
z



z 2  R22
 2 0  z 2  R12



Uniformly Charged
Infinite Plate
dE
-
dEz  dE cos 
z
1 z
dq

dE

dq
dE 
; dEz  dE cos 
3
2
h 4 0 h
4 0 h
Q
Q
 Q   xy
 
A xy

z
2
2
dA  dx dy
h

r

z

dq
dq

 dq   dx dy
 x 2  y2  z 2

dx dy
dA
2
2
y
r x y
Qtot z
1
Etot
dy

dq
x
E

dE
3

z
z
0

dx
4 0
h  dx dy
0
-





1
1
1
z


z
Ez 

dx
dy
3 dx dy


3






2
2
2
4 0
4 0
h
x y z 2
1
x y z
2
Ez 
1
4 0
 z


2
y2  z 2
dy 
2
1
4 0

2
 2   
 z  2 0
z

No radius?!? What
does that mean?!?
21.8 Field Lines
The electric field between
two closely spaced,
oppositely charged parallel
plates is constant.
21-7 The Field of a Continuous Distribution
From the electric field due to a uniform sheet of charge,
we can calculate what would happen if we put two
oppositely-charged sheets next to each other:
The individual fields:
The superposition:
The
result:
How do we apply integration and the Principle of Superposition to uniformly charged objects?
How do we identify and apply the fields of highly symmetric charge distributions?
How do we describe and apply the electric field created by uniformly charged objects?
Summary
 How does the result of the Uniformly Charged Disk (0∞) & the Uniformly Charged
Infinite Plate compare? Why is this the case when one is circular and the other is a
rectangle?
 HW (Place in your agenda):
 “Foundational Mathematics’ Skills of Physics” Packet (Page 8)
 For each of the following, complete 1 derivation with reasons for each
mathematical step and 3 additional derivations: (*Refer to rubric)
 *Uniformly Charged Disk (0∞)
 *Uniformly Charged Hoop (R1R2)
 Uniformly Charged Infinite Plate
 Web Assign 21.8 - 21.11
How do we apply integration and the Principle of Superposition to uniformly charged objects?
How do we identify and apply the fields of highly symmetric charge distributions?
How do we describe and apply the electric field created by uniformly charged objects?
Thursday
(Day 20)
Warm-Up
Thurs, Feb 19
 Identify the parts of the derivation that were confusing.
 Place your homework on my desk:
 “Foundational Mathematics’ Skills of Physics” Packet (Page 20)
 For each of the following, 1 derivation with reasons for each mathematical
step and 3 additional derivations: (*Refer to rubric)
 *Uniformly Charged Disk (0∞)
 *Uniformly Charged Hoop (R1R2)
 Uniformly Charged Infinite Plate
 Web Assign 21.8 - 21.11
 For future assignments - check online at www.plutonium-239.com
Essential Question(s)
 WHAT PRIOR FOUNDATIONAL MATHEMATICS’ SKILLS ARE
NECESSARY IN PHYSICS II?
 HOW DO WE DESCRIBE THE NATURE OF ELECTROSTATICS
AND APPLY IT TO VARIOUS SITUATIONS?
 How do we describe and apply the nature of electric fields in and
around conductors?
 How do we describe and apply the concept of induced charge and
electrostatic shielding?
 How do we describe and apply the concept of electric fields?
Vocabulary









Static Electricity
Electric Charge
Positive / Negative
Attraction / Repulsion
Charging / Discharging
Friction
Induction
Conduction
Law of Conservation of Electric
Charge
 Non-polar Molecules











Polar Molecules
Ion
Ionic Compounds
Force
Derivative
Integration (Integrals)
Test Charge
Electric Field
Field Lines
Electric Dipole
Dipole Moment
Foundational Mathematics
Skills in Physics Timeline
Day
Pg(s)
Day
Pg(s)
Day
Pg(s)
Day
Pg(s)
1
1
2
6
3
11
16
16
21
2
13
14
7
4
12
17
17
8
3
22
23
8
5
13
18
18
9
4
24
†12
9
6
14
19
19
10
5
15
10
7
15
20
20
11
WHAT PRIOR FOUNDATIONAL MATHEMATICS’ SKILLS ARE NECESSARY IN PHYSICS II?
Agenda
 Review “Foundational Mathematics’ Skills of Physics” Packet (Page
8) with answer guide.
 Work Day
 Web Assign Problems and Final Copy
Summary
 On the 3-2-1 Sheets, write down:
 3 things you already knew about static electricity.
 2 things that you learned about static electricity.
 1 thing you would like to know about static electricity.
 HW (Place in your agenda):
 “Foundational Mathematics’ Skills of Physics” Packet (Page 9)
 Web Assign 21.8 - 21.11
 Web Assign Final Copies
 Future assignments:
 Web Assign Final Copies are due in 2 classes
 ALL “5 Derivations” are due in 2 classes
TOMORROW: CHAPTER 22: ELECTRIC FLUX & GAUSS’S LAW
How do we apply integration and the Principle of Superposition to uniformly charged objects?
How do we identify and apply the fields of highly symmetric charge distributions?
How do we describe and apply the electric field created by uniformly charged objects?
Summary of Chapter 21
• Charge is quantized in units of e
• Objects can be charged by conduction or
induction
• Coulomb’s law:
• Electric field is force per unit charge:
Summary of Chapter 21
• Electric field of a point charge:
• Electric field can be represented by electric
field lines
• Static electric field inside conductor is zero;
surface field is perpendicular to surface
CHAPTER 21
- el fin -
TEMPLATES FOR
Derivations
Uniformly Charged Disk
(0R)
z
dq dE  dE cos   dE  1 z dq
dE 
; z
3
2
h
4

h
0
4 0 h
Q
Q
  2
 Q   r 2
A r
dA
 dA  2 rdr
 2 r
dr
dq
dq
 dq  2 r dr


dA 2 rdr
Qtot z
R
1
Etot
1
z

dq

Ez   dEz
3

3 2 r dr

0
0
0
4 0
h 2 r dr 4 0
h
1
Ez 
2 z R
r
4 0 h 3 0 z 2  r 2
R
1


3
dr 
2
1

1
1

Ez 
z  2


2
2 0
z
z  R  2 0

z 2 r 2
0




1
1
1
1
 z  2 2  
z 2 2 
2 0
2

z r 0
0

 z r R
z


z

  2

2
2 0
z
z R 



z
1



2
2
z R 

Uniformly Charged
Ring (02)
1
dq
1 z
z
dE 
;
dE

dE
cos


dq

dE
z
3
4 0 h 2
h 4 0 h
Q
Q
Q
 Q  2 r  r


l 2 r r
dq dq
 dq  rd


dl rd

Ez  
Etot
0
dEz 
1
4 0

Qtot
0
z
1
dq

h 3  rd 4 0

2
0
z
1 z r 2 
 r d 
d
h3
4 0 h 3 0
1 z
1 zQ
1 z r
2
z r


2

r


2


0


Ez 
 0 4 h 3
4 0 h 3 Q
4 0 h 3
4 0 h 3
0
1
Uniformly Charged
Vertical Wire (–∞+∞)
dE 
Ex  
Etot
0
dEx 
1
4 0

Qtot
0
Ex 




x
h

3  dy
x 2 y 2

1 
y
 
 2
2
  4 0 x  x  y
2 y
1 2Q

4 0 x y
4 0 xy
1
dq
1 x
x
;

dq
dE

dE
cos

 dE
2
x
3
4 0 h
h 4 0 h
Q
Q

 Q  y

y
l
dq dq
 dq   dy


dl dy
x
1
dq

h 3  dy 4 0

y
Ex 
x  2 2 2
4 0
 x x  y
1
1

1
4 0
x


1
x
2
 y2

3
dy
2

1 
1 2

1
1







 4 x
4

x
0
0
 
Uniformly Charged
Vertical Wire (–∞+∞)
dE 
Ex  
Etot
0
dEx 
1
4 0

Qtot
0
Ex 




x
h

3  dy
x 2 y 2

1 
y
 
 2
2
  4 0 x  x  y
2 y
1 2Q

4 0 x y
4 0 xy
1
dq
1 x
x
;

dq
dE

dE
cos

 dE
2
x
3
4 0 h
h 4 0 h
Q
Q

 Q  y

y
l
dq dq
 dq   dy


dl dy
x
1
dq

h 3  dy 4 0

y
Ex 
x  2 2 2
4 0
 x x  y
1
1

1
4 0
x


1
x
2
 y2

3
dy
2

1 
1 2

1
1







 4 x
4

x
0
0
 
Uniformly Charged
Vertical Wire (–L/2+L/2)
dq
1 x
x
;

dq
dE

dE
cos

 dE
2
x
3
4 0 h
h 4 0 h
Q
Q

 Q  y

y
l
dq dq
 dq   dy


dl dy
Qtot x
L 2
L 2
Etot
1
1
1
1
x
dq


x


dy
Ex   dEx 
3 dy
3


3

0
L
2
L
2
0
2
2
4 0
h  dy 4 0
4 0
h
x y 2
dE 
1
x y
2
L 2

2
L 2



y
1 
y
Ex 
x  2 2 2  
 2

2
4 0
 x x  y  L 2 4 0 x  x  y  L 2


1 
L 2
L 2
  1 2L 
Ex 

2 
2
4 0 x  x 2  L 2 2
4 0 x
x


L
2




1
1
4x 2  L2

Uniformly Charged Vertical
Wire
Uniformly Charged
Horizontal Wire