Download PHYS_2326_042809

The final will be Thursday, May 7 @ 8:00 AM. It will be 40% comprehensive and
60% what we have covered since the last exam. It will be open book/note.
We will have two review sessions next week:
One on Tuesday at the regular class time
One on Wednesday at at time to be determined (the regular class time?)
Induced Electric Fields
No matter wha t , the total force on a charge is
F  q (E  v  B )
To have current in the loop, F  0
We did explain currents in moving conductors
(" motional emf" ) with FB  qv  B
BUT! Faraday' s experiment s show that currents
are induced when v  0 but B  B(t )
What is it that drives charges then? Electric field E induced by changing B !

emf
is nothing but the work done to move
a unit charge around the loop once, which is
the line integral around the loop
 E  ds
Electric field around a solenoid with alternating current
Current : I(t)  Imax cos(t)
Magnetic field inside the solenoid :
B(t)  0 nI(t) (outside B  0)
Flux through the surface bounded by the path
B (t)  B(t)  R 2
Electric field circulation around the path
 E  ds  E  2r  
:
:
dB
 0 nImax R 2 sin( t)
dt
Outside : E(r,t) 
0 nImax R 2
sin( t)
2r
 nI r
Inside ( R  r) : E(r,t)  0 max sin( t)
2

B
What Maxwell equation w orks is   E  
t
Using Stokes' theorem :
B
 E  ds  S (  E)  n dA   S t  n dA


 B
    B  n dA   
!!!
t  S
t

The electric field E generated by changing B
is very different from the electrosta tic E :
now it is time - dependent E(t ) and nonconserv ative
Do we need a real circuit to have this field? - NO!
We cannot change magnitude of the velocity of a charged particle
in a static magnetic field B
BUT
We can do it in a time-varying magnetic field B(t) – the resulting
electric field E(t) will do the job
And that’s indeed how particles are accelerated in betatrons!
Space Weather Causes Currents in Electric Power Grids
Electric currents in Earth's atmosphere can induce
currents the planet's crust and oceans. During space
weather disturbances, currents associated with the
aurora as large as a million-amperes flow through the
ionosphere at high latitudes. These currents are not
steady but are fluctuating constantly in space and
time - produce fluctuating magnetic fields that are felt
at the Earth's surface - cause currents called GICs
(ground induced currents) to flow in large-scale
conductors, both natural (like the rocks in Earth's
crust or salty ocean water) and man-made structures
(like pipelines, transoceanic cables, and power lines).
Some rocks carry current better than others. Igneous rocks do not conduct electricity very
well so the currents tend to take the path of least resistance and flow through man-made
conductors that are present on the surface (like pipelines or cables). Regions of North
America have significant amounts of igneous rock and thus are particularly susceptible to the
effects of GICs on man-made systems. Currents flowing in the ocean contribute to GICs by
entering along coastlines. GICs can enter the complex grid of transmission lines that deliver
power through their grounding points. The GICs are DC flows. Under extreme space weather
conditions, these GICs can cause serious problems for the operation of the power distribution
networks by disrupting the operation of transformers that step voltages up and down
throughout the network.
Eddy Currents
When magnetic field is on, currents (eddy currents) are induced in
conductors so that the pendulum slows down or stops
Displacement Current
 B  dl  0 I encl
q  C 
0 A
d
( Ed )   0 EA   0 E
dE
dq
ic 
 0
dt
dt
dE
id   0
 displacement current
dt
Inadequacy of Ampere' s Law for time - varying currents :
 B  ds   I
0
becomes contradict ory
once applied to non - steady currents
Its generaliza tion to one of the Maxwell equations
is a great example of a purely the oretical analysis
of the consistenc y of theory culminatin g in a result
with far - reaching consequenc es
 E
t
Not only currents but changing electric fields too
Maxwell' s generaliza tion :  B  ds   0 I   0 0
give rise to circulatin g magnetic fields! !
c B 
2
j
0
E
 c B  
 0 t
2
j
“displacement
current” of the
electric field
flux as opposed
to conduction
current
0 
1
 0c 2
The Reality of Displacement Current
iD  ic
r2
 B  dl  2 rB  0 R 2 iD
 r
B  0 2 ic  inside the capacitor
2 R
0
B
ic  outside capacitor
2 r
Field in the region outside of the capacitor exists
as if the wire were continuous within the capacitor