6.
MAXWELLS EQUATIONS IN TIME-VARYING FIELDS
Applets
http://www.oerrecommender.org/visits/119103 http://webphysics.davidson.edu/physlet_resources/ bu_semester2/ Make a video http://www.phy.ntnu.edu.tw/ntnujava/index.php?to pic=687.0 Very Cool Applets http://micro.magnet.fsu.edu/electromag/java/fara day2/
So far.
Static Electromagnetic .
No
change in time (static)
We now look at cases where currents and charges vary in time H & E fields change accordingly
Examples:
light, x-rays, infrared waves, gamma rays, radio waves, etc
We refer to these waves as time-varying electromagnetic waves
A
set of new equations are required!
Maxwells Equations
Same for Dynamic & Static
Static :Zero
Static :Zero
Static :J
Static :I
In this chapter, we will examine Faradays and Ampres laws
A little History
Oersted demonstrated the relation between electricity and magnetism
Current
Fm
impacts (excerpts force on) a compass needle
is due to the magnetic field When current in Z then needle moves to Phi direction
http://micro.magnet.fsu.edu/electromag/java/com pass/index.html
Induced magnetic field can influence the direction of the compass needle. When we connect the circuit, the conducting wire wrapped around the compass is energized creating a magnetic field that counteracts the effects of the Earth's magnetic field and changes the direction of the compass needle.
A Little History
Faraday (in London) hypothesized that magnetic field should induce current!
Henry
in Albany independently trying to prove this!
They showed that magnetic fields can produce electric current
The
key to this induction process is CHANGE http://micro.magnet.fsu.edu/electromag/java/faraday 2/index.html
Another applet: http://phet.colorado.edu/sims/faradayslaw/faradays-law_en.html
galvanometer needle moves
Faradays Law
Electromotive force (voltage) induced by time-varying magnetic flux:
Magnetic Flux (Wb)
We can generate the current through the loop By moving the loop or changing direction of current
http://phet.colorado.edu/sims/faradays-law/faradays-law_en.html When the meter detects current voltage has been induced http://phet.colorado.edu/sims/faradays-law/faradays-law_en.html electromotive force has been created this process is called emf induction
Three types of EMF
Stationary loop; B changing
Moving loop; B fixed
Stationary Loop in Time-Varying B
Stationary Loop in Time-Varying B
Assuming S is stationary and B is varying: Transformer EMF Two types of B fields are generated
Changing
No change of surface
B(t) Induced B (Bind)
Applet: http://micro.magnet.fsu.edu/electromag/java/lenzl aw/index.html
Lenzs Law
Increasing B(t) change of magnetic flux I generates;
I(t)
Bind Bind with be opposite of B(t) change Direction of Bind Direction of I(t)
Lenzs Law
Increasing B(t) change of magnetic flux I generates;
I(t) Bind Bind with be opposite of B(t) change Direction of Bind Direction of I(t)
If I(t) clockwise I moving from + to
V2> V1 emf is negative
Faradays Law
magnetic field induces an E field whose CURL is equal to the negative of the time derivative of B
Note A0
Example
Note the direction of B
cont.
Note B0 Finding the Magnetic Flux:
Find ds! Find Polarity of Vtr at t=0 Find Vtr:
Find Vrt!
What is V1 V2 based on the Given polarity?
Note B0 Finding the Magnetic Flux:
Find Polarity of Vtr at t=0 Find Vtr:
Ideal Transformer
Due to the primary coil
Primary and sec. coils Separated by the magnetic core (permittivity is infinity) Magnetic flux is confined in the core
Motional EMF
In the existence of constant (static) magnetic field the wire is moving
Fm
is generated in charges Thus, Em = Fm/q = UXB Em is motional emf
http://webphysics.davidson.edu/physlet_resources/ bu_semester2/ (motional EMF)
Motional EMF
Magnetic force on charge q moving with velocity u in a magnetic field B: This magnetic force is equivalent to the electrical force that would be exerted on the particle by the electric field Em given by This, in turn, induces a voltage difference between ends 1 and 2, with end 2 being at the higher potential. The induced voltage is called a motional emf
Induced emf
Example : Sliding Bar
Note that B increases with x
The length of the loop is related to u by x0 = ut. Hence
Note A1
EM Generator
EM Motor/ Generator Reciprocity
Current passing through the loop
Load
Angular Velocity
Electrical energy being converted to mechanical turning the loop
Motor: Electrical to mechanical energy conversion
Generator: Mechanical to electrical energy conversion
EM Motor/ Generator Reciprocity
Load The loop is turning due to external force B = Bo in Z direction
Motor: Electrical to mechanical energy conversion
Generator: Mechanical to electrical energy conversion
EM Motor/ Generator Reciprocity
Load
Motor: Electrical to mechanical energy conversion
Generator: Mechanical to electrical energy conversion
Applet
http://www.walter-fendt.de/ph14e/electricmotor.htm http://www.walter-fendt.de/ph14e/generator_e.htm Good tutorial: http://micro.magnet.fsu.edu/electromag/electricity/g enerators/index.html
Other Applications
Relay Generator
Relay
Solenoid
EM Generator EMF
n is normal to the surface
As the loop rotates with an angular velocity about its own axis, segment 12 moves with velocity u given by
AC Current - Commutator
Also: Segment 3-4 moves with velocity u. Hence:
Note A2
A=W .l
EM Generator EMF Alternative Approach
Using Magnetic Flux and Faradays Law
AC Current - Commutator
Note A3
Faradays Law
The compass in the second coil deflects momentarily and returns immediately to its original position The deflection of the compass is an indication that an electromotive force was induced causing current to flow momentarily in the second coil. The closing and opening of the switch cause the magnetic field in the ring to change to expand and collapse respectively. Faraday discovered that changes in a magnetic field could induce an electromotive force and current in a nearby circuit. The generation of an electromotive force and current by a changing magnetic field is called electromagnetic induction. http://micro.magnet.fsu.edu/electromag/java/faraday/
Displacement Current
= Jc + Jd For arbitrary open surface Conduction current : (transporting charges) Displacement current: (does not transport)
This term is conduction current IC
This term must represent a current Total Current =
Application of Stokess theorem gives:
Displacement Current
= Total Current = Ic + Id
Define the displacement current as: The displacement current does not involve real charges; it is an equivalent current that depends on
Note: If dE/dt = 0 Id = 0
Remember: Conduction Current
Conduction current density:
Perfect Dielectric: Conductivity = 0 Jc = 0 Perfect Conductor: Conductivity = INF E = Jc/ = 0
Materials: Conductors & Dielectrics Conductors: Loose electrons Conduction current can be created due to E field Dielectrics: electrons are tightly bound to the atom no current when E is applied
Conductivity depends on impurity and temperature!
For metals: T inversely proportional to Conductivity!
From Chapter 4
Capacitor Circuit
Given: Wires are perfect conductors and capacitor insulator material is perfect dielectric. For Surface S1: Vo cos wt E field?? For Surface S2:
Note A3b
I2 = I2c + I2d I2c = 0 (perfect dielectric)
I1 = I1c + I1d
(D = 0 in perfect conductor)
Conclusion: I1 = I2
Capacitor Circuit
Given: Wires are perfect conductors and capacitor insulator material is perfect dielectric. For Surface S1: Vo cos wt For Surface S2:
I2 = I2c + I2d I2c = 0 (perfect dielectric)
I1 = I1c + I1d
Note:
(D = 0 in perfect conductor)
Remember:
21
= Ey. d Ey = V/d
C Conclusion: I1 = I2
Applet
http://www.circuit-magic.com/capacitor.htm
Charge Current Continuity Equation
Current I out of a volume is equal to rate of decrease of charge Q contained in that volume:
Changing over time
Total Current At junction Is zero Kirchhoffs current law: Algebraic sum of currents following out of a junction is zero Used Divergence Theorem
ds
Charge Current Continuity Equation
Current I out of a volume is equal to rate of decrease of charge Q contained in that volume:
Changing over time
Total Current At junction Is zero Kirchhoffs current law: Algebraic sum of currents following out of a junction is zero Used Divergence Theorem
ds
Maxwells Equations General Set
Maxwells Equations Free Space Set
We assume there are no charges in free space and thus, =0
Time-varying E and H cannot exist independently! If dE/dt non-zero dD/dt is non-zero Curl of H is nonzero H is non-zero
If H is a function of time E must exist!
Maxwell Equations Electrostatics and Magnetostatics
Boundary Conditions
General Form // Time-varying
Example: Displacement Current
Does E exist? Why? Does Id exist? Are Ic and Id related?
Using Ohms law; this is not a perfect conductor Jc = E If E exists D exists (assuming it is time-varying, which is because Ic is time-varying!) Id exists E is related to Jc ; Jd is defined as change of Electric flux density (E) in time They MUST be related!
Find the displacement current.
Note A4
Example
Find V12 U = 5 z (m/s)
Note A5
