Download Review: Time – Varying Fields

Electromagnetic Fields
Review: Time – Varying Fields
In the dynamics case, we can distinguish between two regimes:
Low Frequency (Slowly-Varying Fields) – The displacement
current is negligible in the Maxwell’s equations, since
G
G
∂D ( t )
<< J ( t )
∂t
High Frequency (Fast-Varying Fields) – The general set of
Maxwell’s equations must be considered, with no approximations.
© Amanogawa, 2006 – Digital Maestro Series
24
Electromagnetic Fields
In the low frequency regime we use the complete set of Maxwell’s
equations, but the displacement current is omitted
G
G
∂B
∇× E= −
dt
G G
∇× H = J
G
∇⋅ D= ρ
G
∇⋅ B = 0
G
G
D=ε E
G
G
B= µ H
G
G G G
F = q ( E + v × B)
© Amanogawa, 2006 – Digital Maestro Series
25
Electromagnetic Fields
The concept of low frequency and slowly-varying phenomena is
relative to the situation at hand. Any disturbance (time-variation) of
the electromagnetic field propagates at the speed of light. If a
length L is the maximum dimension of the system under study, the
maximum propagation time for a disturbance is
Maximum Propagation Time
L
td =
vp
Maximum Length
Phase velocity of light
We can assume slow-varying fields if the currents are practically
constant during this time period.
For sinusoidal currents, with a period of oscillation T , we have
Wavelength
Period
Frequency
1 λ
T= =
>> t d
f vp
© Amanogawa, 2006 – Digital Maestro Series
and
L << λ
26
Electromagnetic Fields
The electric potential is now by itself insufficient to completely
describe the time-varying electric field, because there is also a
direct dependence on the magnetic field variations. By recalling the
definition of magnetic vector potential, we can derive a relationship
between electric field and electric potential
Time-Varying Fields
G
G
G
∂ B( t )
∂
∇ × E(t) = −
= − ∇ × A( t )
∂t
∂t
G
∂ A( t ) 
G
⇒ ∇ ×  E(t) +
=0

∂t 

G
G
∂A ( t )
E(t) +
= −∇φ ( t )
∂t
© Amanogawa, 2006 – Digital Maestro Series
Statics
G
∇× E=0
G
E = −∇φ
27
Electromagnetic Fields
We can also obtain an integral relation between electric field and
magnetic flux, by integrating the curl of the electric field over a
surface
G
G
G
G
G
∂B ( t )
∂
∫ ∫S ∇ × E ( t ) ⋅ dS = ∫ ∫S − ∂ t dS = − ∂ t ∫ ∫S B ( t ) ⋅ dS
Stoke’s Theorem
Magnetic Flux
Φ(t)
G G
∂ Φ(t)
v∫ E ⋅ d l = − ∂ t
© Amanogawa, 2006 – Digital Maestro Series
28
Electromagnetic Fields
In the electrostatic case, we do not need to distinguish between
voltage and potential difference. The voltage between two points is
always defined as
G
bG
Vba = − E ⋅ d l = − e. m. f .
a
∫
but in terms of potential φ we have
Time-Varying Fields
G
b
∂A G
Vba ( t ) = ∫  ∇φ +  ⋅ d l
a
∂t
G
∂ bG
= φ b − φ a + ∫ A( t ) ⋅ d l
∂t a
© Amanogawa, 2006 – Digital Maestro Series
Statics
G
bG
Vba = − E ⋅ d l
a
∫
= φb − φa
29
Electromagnetic Fields
Note that for time-varying fields the line integral of the magnetic
vector potential between two given points depends on the actual
path of integration. In general:
∫
G
G
G
bG
A ( t ) ⋅ d l ≠ A( b, t ) − A ( a, t )
a
Consider now the integral of the electric field along a closed path:
Time-varying fields
G
G
v∫ E ( t ) ⋅ d l ≠ 0
Statics
G
G
v∫ E ( t ) ⋅ d l = 0
The closed path could be a metallic wire which confines the current
due to moving electric charge.
© Amanogawa, 2006 – Digital Maestro Series
30
Electromagnetic Fields
The line integral of the electric field gives the work necessary to
move a unit charge along the path of integration, under the
influence of time-varying electric and magnetic fields.
For a closed wire loop at rest, the work necessary to move a unit
charge once around the loop is
W = v∫
Force
Charge
G
G
G
G
G
⋅ d l = v∫ E ( t ) ⋅ d l = ∫ ∇ × ( E ( t ) ) ⋅ d S
G
G
G
∂ B( t ) G
∂
=∫ −
⋅ d S = − ∫ B( t ) ⋅ d S
S
∂t
∂t S
∂
=−
Φ(t)
∂t
Magnetic Flux
© Amanogawa, 2006 – Digital Maestro Series
31
Electromagnetic Fields
As a more general case, consider a wire loop in motion. The
complete Lorentz force must be considered:
W = e. m. f . = v∫
Force
Charge
G
G
G
G
G
⋅ d l = v∫ ( E ( t ) + v ( t ) × B ( t ) ) ⋅ d l
G
G
G
G
= ∫ (∇ × ( E ( t ) + v ( t ) × B ( t ))) ⋅ d S
G
G
G
 ∂ B( t )
 G
d G
G
= ∫−
+ ∇ × ( v ( t ) × B ( t ))  ⋅ d S = − ∫ B ( t ) ⋅ d S
∂t
dt


G
dB ( t )
dt
Flux
Φ(t)
If the velocity of motion is constant, note that
G G
∇ × ( v × B ( t )) =
G
G
G
G G G G
G G
G
v∇ ⋅ B − B∇ ⋅ v + ( B ⋅ ∇ ) v − ( v ⋅ ∇ ) B = − ( v ⋅ ∇ ) B
0
0
© Amanogawa, 2006 – Digital Maestro Series
0
32