Transcript PHYS_2326_042309

Direct current (dc) generators
Split ring (commutator) does the job of reversing polarity every half cycle
Motional emf – conductor moving in a constant magnetic field
 B  Blx
FB  qvB will move charges
until compensated by the electric
field of end accumulations
qvB  qE  qV /l
V  Bvl

dx
  Bl
  Blv
dt
Generators as Energy Converters
I  Blv / R
Presistor  I 2 R  ( Blv ) 2 / R
Who does the work?
Generator does not produce electric energy
out of nowhere – it is supplied by whatever
entity that keeps the rod moving. All it does
is to convert it to a different form, namely to
electric energy (current)
We! - By moving the bar:
Papplied  Fv  IBlv  ( Blv )2 / R
Energy conserved
After initial push,
velocity w ill relax
decelerate d by the
magnetic force :
Rotating bar :
v ( r )  r
Small element :


d  Bv dr
Total emf :
l
B l 2
  Br dr 
2
0
2
m
dv
( Bl )
  IBl  
v
dt
R
v  v0 exp( t /  )
  mR /( Bl ) 2
Motion does not
necessarily
mean changing
magnetic flux!
Significance of the minus sign – Lenz’s Law
Induced current has such direction that its
own flux opposes the change of the external
magnetic flux
Magnetic field of the induced current wants
to decrease the total flux
Magnetic field of the induced current wants
to increase the total flux
Correspondingly, magnetic forces oppose the
motion – consistently with conservation of
energy!
Lenz’s Law – the direction of any magnetic induction
effect as to oppose the cause of the effect
Lenz’s Law – a direct consequence of the energy conservation principle
Finding the direction of the induced current
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.