Lecture -- Energy in Electrostatic Fields · Electrostatic Energy Density Slide 21 Observe what we...
Transcript of Lecture -- Energy in Electrostatic Fields · Electrostatic Energy Density Slide 21 Observe what we...
9/21/2019
1
Electromagnetics:
Electromagnetic Field Theory
Energy in Electrostatic Fields
Outline
• Energy in terms of potential
• Energy in terms of the field
•Power and energy in conductors
Slide 2
1
2
9/21/2019
2
Energy in Termsof Potential
Slide 3
Recall Potential Difference
Slide 4
Recall the relation between potential difference, work, and charge.
Ad
B
E
AB B A
WV V V
Q
Therefore, the work it takes to move charge Q from A to B is
ABW QV
3
4
9/21/2019
3
Energy in an Ensemble of Charges
Slide 5
1Q
2Q
3Q
An ensemble of charges contains energy because the charges are putting a force on each other and so they have the potential to do work.
We will calculate how much energy the ensemble contains by calculating how much energy it took to assemble it.
Point Charge #1
Slide 6
1Q
No other charges are present, so placing Q1 at P1 takes no work.
1 0W
1P
5
6
9/21/2019
4
Point Charge #2
Slide 7
1Q
Placing Q2 at P2 takes work because charge Q1 is present.
2 2 21W Q V
2Q
2P
Point Charge #3
Slide 8
1Q
Placing Q3 at P3 takes work because charges Q1 and Q2 are present.
3 3 31 3 32W QV QV
2Q
3P 3Q
7
8
9/21/2019
5
Total Work So Far
Slide 9
1Q
The total work placing all three charges is
1 2 3W W W W
2Q
3Q
2 21 3 31 320 Q V Q V V
Assembly in Reverse Order
Slide 10
1Q
If we had placed the charges in the reverse order,
3 2 1W W W W
2Q
3Q
2 23 1 12 130 Q V Q V V
9
10
9/21/2019
6
Add Both Approaches
Slide 11
Equation obtained by placing Q1, then Q2, and then Q3.
2 23 1 12 130W Q V Q V V
2 21 3 31 320W Q V Q V V
Equation obtained by placing Q3, then Q2, and then Q1.
2 21 3 31 32
2 23 1 12 13
2 0
0
W Q V Q V V
Q V Q V V
Add the two equations above.
1 12 13 2 21 23 3 31 322W Q V V Q V V Q V V
1V 2V 3V
1 1 2 2 3 32 W QV Q V QV
Total potentials
Final Expression
Slide 12
1 1 2 2 3 32 W QV Q V QV
1 1 2 2 3 3
1 1 1
2 2 2W QV Q V QV Solve for W.
It is straightforward to generalize this for any number of charges.
1
1 joules
2
N
i ii
W QV
11
12
9/21/2019
7
Energy in Charge Distributions
Slide 13
Point Charge
Q
Charge
CQ
Total Charge
Total1
N
ii
Q Q
Total Energy
1
1
2
N
i ii
W QV
Line Charge
C m
Line Charge Density
TotalQ d L
Total Charge
1
2L
W Vd
Total Energy
Sheet Charge
2s C m
s
Surface Charge Density
Total s s
S
Q ds S Total Charge
s
s
1
2W Vds
Total Energy
Volume Charge
3v C m
v
Volume Charge Density
Total v v
v
Q dv V Total Charge
v
v
1
2W Vdv
Total Energy
Energy in Termsof the Field
Slide 14
13
14
9/21/2019
8
Derivation (1 of 5)
Slide 15
The energy in a volume charge is
v
v
1
2W Vdv
Recall from Maxwell’s equations that .v D
v
1
2W D Vdv
Recall the product rule for divergence fA f A A f
VD V D D V
D V VD D V
Derivation (2 of 5)
Slide 16
Apply the product rule for our equation for work.
v
1
2W D Vdv
v
1
2VD D V dv
v v
1 1
2 2VD dv D V dv
15
16
9/21/2019
9
Derivation (3 of 5)
Slide 17
Recall the divergence theorem
v v
1 1
2 2W VD dv D V dv
S V
F ds F dv
Apply this to our equation for work.
1
2 S
VD ds
v
1 1
2 2S
W VD ds D V dv
Derivation (4 of 5)
Slide 18
Look more closely at the surface integral.
v
1 1
2 2S
W VD ds D V dv
1V
r
2
1D
r
2ds r
22
1 1 1Overall r
r rr
We are free to choose whatever surface S we wish.
As we enlarge the surface out to infinity, the surface integral becomes negligible relative to the volume integral.
v
1 1
2 2S
W VD ds D V dv
17
18
9/21/2019
10
Derivation (5 of 5)
Slide 19
Our equation for work is now
v
1
2W D V dv
Associate the negative sign with ∇𝑉.
v
1
2W D V dv
This is the electric field intensity 𝐸.
v
1
2W D E dv
This is the general equation for energy stored in the electrostatic field.
It is valid for anisotropic and inhomogeneous media.
Electrostatic Energy in LHI Media
Slide 20
The more common expression for energy in the electrostatic field is for the special case of linear, homogeneous, and isotropic (LHI) media.
In isotropic media we have 𝐷 𝜀𝐸.
v
1
2W D E dv
v
1
2E E dv
2
v
1
2W E dv
Simpler equation that is only valid in LHI media.
19
20
9/21/2019
11
Electrostatic Energy Density
Slide 21
Observe what we have been integrating to get total energy.
2
v
1
2W E dv
v
1
2W D E dv
These expressions must be energy density w.
We can now think of calculating total energy by integrating the energy density w.
v
W wdv 2
1 General case
21
LHI media 2
D Ew
E
Power & Energyin Conductors
Slide 22
21
22
9/21/2019
12
Joule’s Law
Slide 23
Joule’s law states that
v
P E J dv
This is equivalent to P = VI in circuit theory.
From this, we can extract the energy density in a conductor.
w E J
Applying Ohm’s law for electromagnetics 𝐽 𝜎𝐸 gives
w E J
E E
2E
2
v
P E dv
Most common form.
23