1
ECE 3030
Electromagnetic Fields and Waves
Fall 2009 Lecture 11
2009/9/21
Force, Energy, Power, and Work
Energy Stored in Capacitors
and Inductors
Virtual Work
1
Rana and Swartz 06/5/25
Electromagnetic Fields and Waves
– Fall 2009
Lecture 11:
Instructor:
Dr. Wesley E. Swartz
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ε
B
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F
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Problem Assignments and Prelim 1
•
Demo 4 Problems:
–
Problem 5.13
– Nanodots, long, but straight forward.
–
Problem 6.6
– Nturn coil with
neat
integration steps.
•
Workshop 4 Problems:
–
Problem 5.11
– Use of diverse spherical boundary conditions.
–
Problem 6.5
– Applying BiotSavart Law.
No Homework 4
2
Rana and Swartz 06/5/25
Electromagnetic Fields and Waves
– Fall 2009
Lecture 11:
•
No Homework 4.
•
Prelim 1, Tuesday, September 29, 7:30pm:
–
Covers Lectures 110, Text Chapters 16, Problem sets 14.
–
Expect problems like in Demos, Workshops, and Homeworks,
but with twists.
–
The concepts of Problem 4.13 (Workshop 3) may be useful.
Lorentz Force on Charges
•
The force on a charge
q
in an Efield is:
•
Example
: consider two charges
q
1 and
q
2:
E
q
F
=
q
E
1
q
2
q
d
2
1
ˆ
→
n
3
Rana and Swartz 06/5/25
Electromagnetic Fields and Waves
– Fall 2009
Lecture 11:
–
The force that
q
1 exerts upon
q
2 can be obtained by multiplying the Efield
that
q
1 produces at the location of
q
2 by the charge
q
2 :
–
The force that
q
2 exerts upon
q
1 is just equal and opposite to what
q
1
exerts upon
q
2
(Newton’s third law):
2
1
2
o
1
2
1
2
2
1
n
ˆ
d
4
q
q
E
q
F
→
→
=
=
π
1
2
2
2
1
2
1
2
2
1
1
2
ˆ
4
ˆ
4
→
→
→
=
−
=
n
d
q
q
n
d
q
q
F
o
o
Potential Energy and Force
•
The force on a charge
q
in an Efield is:
•
The potential energy
U
of a charge in the Efield is:
x
x
qE
F
=
q
E
()
x
qE
x
q
U
x
−
=
=
φ
x
x
qE
dx
dU
−
=
Then note
4
Rana and Swartz 06/5/25
Electromagnetic Fields and Waves
– Fall 2009
Lecture 11:
•
So the force on the charge can also be written as:
•
This formula is much more general than it appears (see the slides
down the road a bit …)
dx
dU
F
x
−
=
Electric force is the derivative
of potential energy.
Energy, Force, and Work in
Electromagnetics
•
Lets generalize the relationship between energy and force.
•
Suppose one has a closed electromagnetic system with
electromagnetic energy
U
.
–
If the system does some mechanical work
Fdx
then its electromagnetic
energy must decrease by the same amount:
Fdx
dU
=
5
Rana and Swartz 06/5/25
Electromagnetic Fields and Waves
– Fall 2009
Lecture 11:
•
Here “
F
” is some force that results in the change of some system
length parameter by “
dx
”.
Electromagnetic system
−
or
dx
dU
F
−
=
Force Between Charged Plates (1)
•
Consider two charged plates carrying charges per unit area of
σ
and
–
σ
, respectively.
area =
A
+
+
+
+
+
+
+
+








σ
−
–
Need to find the total force on the
right plate exerted by the left plate
(which is also equal an opposite to
the force on the left plate exerted by
the right plate).
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 RANA
 Energy, Force, Electromagnet, Magnetic Field, Rana, Electromagnetic Fields

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