Lecture 6
Railguns and Maxwell´s equations
Inductors and Energy in AC circuits
Experimentalphysik II in Englischer Sprache
24-6-11
1
Contents
• 6.1 Electromagnetic Induction
– Faraday´s law of induction
– Lenz’s law
– Motional EMF and railguns
d B
 
dt
• 6.2 The need for “displacement current”
– Maxwell´s equations
– Integral and differential form
• 6.3 Inductance
–
–
–
–
–
Mutual Inductance
Self inductance and inductors (L)
Energy stored in magnetic fields
The R-L circuit
The L-C(-R) oscillator
2
6.1 Electromagnetic Induction
• Faraday’s law (1830’s)
– The induced emf in a closed loop equals the negative of the
time rate of change of magnetic flux through the loop
 
M. Faraday (1791-1867)
d B
dt
J. Henry (1797-1878)
Faraday and Henry noticed that time
varying magnetic fields result in the
generation of an induced current
 Arising from an induced EMF ()
 Due to an induced electric field E
3
To think about these effects consider the experimental
arrangement shown on the left consisting of a flexible current
loop in a constant magnetic field caused by an electromagnet…
What happens when I=0, i.e. B=0 in vicinity of coil ?
If B=0 then there is never any induced current.
What happens when we turn on the electromagnet ?
t=t0
As we turn on the magnet, B increases slowly to a
steady value. As B varies then a current is induced in
the coil but that current slowly dies away as dB/dt0
B(t)
I(t)
time
What happens when we “squeeze” the coil such and decrease it´s cross sectional area ?
B=const
I=0
time
4
To think about these effects consider the experimental
arrangement shown on the left consisting of a flexible current
loop in a constant magnetic field caused by an electromagnet…
What happens when I=0, i.e. B=0 in vicinity of coil ?
If B=0 then there is never any induced current.
What happens when we turn on the electromagnet ?
t=t0
As we turn on the magnet, B increases slowly to a
steady value. As B varies then a current is induced in
the coil but that current slowly dies away as dB/dt0
B(t)
I(t)
time
B=const
I=0
time
What about if we rotate the coil around it´s axis ?
DIRECTION ?
5
6.1.2 Lenz’s Law
An induced current has a direction such that the magnetic field due to it opposes the
change in the magnetic flux that induced the current in the first place...
Heinrich Lenz
1804-1865
WHY ?
6
6.1.3 Rules for using Faraday´s law
To solve problems involving Faradays law you need to do the
following:
1.
SET UP PROBLEM
– First identify why the flux threading a loop is changing
(orientation, B(t), conductor moving etc…)
Remember: it is not the flux itself (B) but dB /dt that is important!
– Choose a direction of the area vector (A or dA) with a direction perpendicular
to the plane and identify which direction you will call positive
2.
– Calculate the magnetic flux using
 B   B.d A   BdA cos 
– Calculate the induced EMF using
   d B dt
EXECUTE SOLUTION
– If the resistance of the circuit is know calculate the current I using Ohms law
3.
CHECK SOLUTION
– Make sure that the results have the proper units and double check that you
have properly implemented the sign rules for the magnetic flux and EMF
EXAMPLE : We are going to use Faradays law to calculate the EMF induced in a metal loop that
is being heated in a time varying magnetic field…
The radius of the wire increases with time with a constant velocity v
The magnetic flux density is uniform and increases with time according to B(t)=B0(1+kt)
x
x
x
x
x
x
x
x
x
x
x
 
 
x
x
x
r=vt
x
x
x
x
x
 B 
.d A
v  B .d    
 t 


MOTIONAL
EMF
8
Reminder : Motional EMF, conservation of
energy and the Rail Gun
 
• If we have B constant in time then we have just seen that    v  B .d  , this is
known as the motional EMF
 
• Since F  q v  B we have that q   F .d  , i.e. the motional EMF just
represents the work that can be done on a charge by the mechanical motion…
• This brings up a rather interesting (and dangerous) example – the “rail gun”
WARNING - ANY OF YOU TRYING THIS AT HOME DO SO AT YOUR OWN RISK !!!
A U-shaped fixed wire, bridged by a moving conductor will give rise
to an EMF that produces a current in the loop
From Faraday´s law it is easy to show that
  vBL
The kinetic energy is “converted” to an EMF that can drive a
current around the circuit .
Of course, one can also drive a current around the circuit using a
battery …… this will cause the bar to “move”
9
This is the principle of a rail gun – a device for converting EM energy to KE
Device consists of two long parallel conductors in
which a current flows in opposite directions
µ0 I
B
2r
Circuit is completed by a conducting “projectile”
See http://en.wikipedia.org/wiki/Railgun
Lorentz force acts on the conductor in the sense shown on the figure
 This Lorentz force causes the projectile to accelerate rapidly
 dv 
F  iLB  m 
 dt 
Does it accelerate for ever ?
No, the expanding conducting loop area causes a large –dB/dt that generates a
back EMF back that opposes the source of I
The projectile would reach terminal velocity when |back|~|source|, but in reality
Ohmic voltage drops in the wires caused by the huge currents limit the maximum
muzzle velocities attainable
F µ0i1i2

The maximum current (Forces) limited by the forces between the rails L
2d
10
•
•
A few frightening RailGUN facts and figures
–
Hold world record for fastest object accelerated of a
significant mass
–
Firing of a 1 g pellet at 16 km/s by Sandia National
Research Laboratories in New Mexico
–
Maxwell Laboratories' 32 Megajoule gun fires a 1.6
kg projectile at 3.3 km/s (9 MJ of kinetic energy!)
Applications
–
Defence (obvious one for US government!)
–
Research by NASA for hypervelocity impact
simulations which will allow shields to be developed
which will protect orbiting aircraft from high velocity
debris surrounding the earth.
–
NASA is also studying the possibility of a launcher
which would deliver payloads into orbit at a fraction
of the cost of a rocket launch.
–
Nuclear fusion: studies of the possibility to use Rail
guns in Fusion fuel pellet injectors for experimental
nuclear fusion reactors
www.powerlabs.org/railgun.htm
11
6.1.4 Induced electric fields
•
When a conductor moves in a magnetic field, we can understand the induced EMF on
the basis of the magnetic forces on the conductor
•
However, an EMF is also induced when there is a changing flux through a stationary
conductor that is not in a magnetic field as shown on the figure below…
B  µ0 NI
 
 B  BA  µ0 NIA
d B
 dI 
 µ0 NA 
dt
 dt 
induced EMF in the loop due to time
varying current in solenoid
If the total resistance of the loop is R, then the induced current in the loop (I’) is I´=/R
What drives the charges around the loop in this situation ?
Cannot be magnetic since: (a) conductor is not moving in a magnetic field and
(b) the conductor isn´t even in a magnetic field!
We are forced to conclude that it must be an induced electric field in the
conductor caused by the changing magnetic flux threading the ring…
12
• This is a little problematic !
– We are used to thinking about electric fields as being caused by electric charges
(Gauss´s law)
– Now we are saying that a changing magnetic field somehow acts as a “source” of
electric field
• Furthermore, it’s a weird kind of electric field
– When a charge q goes around the loop, the work done on it by the induced E-field
must be W=q.
– This means that the electric field in the loop is no longer conservative, because the
line integral around the closed path is not zero, it is equal to the emf
 E.d   0
 E.d   
d B
 E.d    dt
STATIONARY
INTEGRATION
PATH
This form of Faraday´s law is true when the path integrated around is stationary
This type of electric field here is known as a nonelectrostatic field
For the loop
 E.d   2rE  
d B
dt
E
1 d B
2r dt
nonelectrostatic field
in a circular loop
13
 
2)
A time varying magnetic field induces an electric field in a
stationary conductor and induces an EMF
d B
dt
d B
 E.d    dt
This changing magnetic field produces a type of electric field that we cannot produce with any static distribution of
charges  nonelectrostatic field
It is non-conservative and, thus, we cannot define a potential for this type of electric field
other than this, this “new” E-field is exactly the same as any other “regular” E-field
These nonelectrostatic fields have many applications
MAGNETIC
STORAGE MEDIA
HYBRID DRIVE CARS
Http://computer.howstuffworks.com/hard-disk.htm
www.toyota.com/vehicles/future/hybridx.html
6.2 The electric displacement current - D
•
•
We have seen that a time varying magnetic field gives rise to an induced electric field…
It turns out that a varying electric field gives rise to an induced magnetic field…
J.C. Maxwell made this link in 1864 and revealed the
beautiful symmetry of electromagnetism to the world !
It must be incomplete !!
To see what leap of imagination Maxwell made to link time varying electric fields
and magnetic fields, let´s return to Ampere´s law as discussed last time…
 B.d   µ i
0 encl
Consider the process of charging the capacitor shown in the figure
ic = conduction current, to distinguish it from the new type of
current we are about to encounter the displacement current id
Apply Ampere´s law to the circular path shown on the figure
 B.d   µ i
0 c
Real current flows
Inconsistent ?
 B.d   0
No current (only E field ?)
15
Maxwell to the rescue !
Although there is no conduction current in the
region between the parallel plates of the capacitor
there is a time dependent electric field as the
capacitor charges up….
As the capacitor charges the electric field E and the electric flux e varies as a function versus time.
We can investigate their rates of change in terms of the charge (q) and current (dq/dt) on the plates of
the capacitor
q  Cv
at any instant in time
q  Cv 
A
d
.Ed
Now
or
C
 0 r A
d
q  AE   e
v
E
d
and
At any instant in time, the charge on the
capacitor plates = epsilon x electric flux
between the plates
dq
 d e 
 

dt
 dt 
The current flowing onto the plates equals the
conduction current in the wire on the left
ic 
Since the current must be conserved in a loop of
a circuit, this conduction current must flow into
the field between the plates…
 d e 
id   

 dt 
We are saying that the changing flux
through the curved surface is
“somehow” equivalent to the current
DISPLACEMENT CURRENT
16
 B.d   µ i
0
c
 id 
The introduction of the concept of the displacement
current (id) leads to a generalized version of Ampere´s law
that removes our apparent paradox …
The displacement current density (jd) then follows:
jd 
id
 dE  DISPLACEMENT CURRENT
 

A
dt

 DENSITY
OK, so the concept of displacement current removes our apparent paradox with Ampere´s law
(Right now you are probably thinking this is “pulled this out of thin air” ! )
Q) Is this a real current, can you measure it ?? (or just cheating ?? )
A) It is real and you can measure its effects
e.g. if the displacement current flowing into the capacitor field is real then it should produce a B-field (from Ampere´s law)
 dE  Current enclosed by
 i 
jd   
iarear  jD .r 2  r 2  D 2 

 dt  this circle, radius r
 R 
 B.d   µ i
0
c
 id 
 r 
B  µ0id 
2 
 2R 
r2
2rB  µ0iD 2
R
This is exactly what is measured, iD is a
very useful concept that repairs all the
equations of electromagnetism
What about if r>R ?
17
6.2.2 Maxwell´s equations in integral form
In non-electric / magnetic materials
The following equations are always true !!!
 E.d A 
Q
0
 B.d A  0
Gauss´s law for E-fields
Gauss´s law for B-fields
 E.d   
 B
t
 e 

 B.d   µ0  ic   0 t 
Faraday´s law
Ampere´s law
We are now at a very important stage in our studies of electromagnetism, we can
summarise everything we have learnt in four beautifully symmetric equations
18
There is a remarkable symmetry about Maxwell´s equations
 E.d A  0
If there are no free charges, e.g. in a vacuum
 B.d A  0
1d
B
.
d


 B.d  c0 2 0 dt  E.d A
With the definitions of electric / magnetic flux
d
 E.d    dt  B.d A
These equations can be stated in another (differential) form using the divergence and Stokes´theorem of vector calculus
 A.d S   .A.dV
S
Divergence theorem
V
Stokes´theorem
   A.d    A.d r

d
19
6.2.3 Maxwell´s equations in differential form

r
.E 
0
Gauss´s law for E-fields
B
 E  
t
.B  0
Gauss´s law for B-fields
  B  µ0 J  µ0 0
Faraday´s law
E
t
Ampere´s law
See handout on web for how to convert between integral and differential forms
20
Let’s take a 10 minute break!
6.3.1 Mutual Inductance
•
If you take a piece of copper wire and wrap it into a coil, put this into a
circuit, does it behave any differently to a straight wire ?
–
•
It does whenever the current is changing in time since inductive effects produce an EMF in
the coil from Faraday´s law
In L5 we looked at the magnetic force between two wires carrying a steady
(DC) current…
–
When the currents are varying in time then another “inductive interaction” is also
present…
F II µ0

L 2r
To see this, consider a pair of coils close to each other as depicted below:
i1-time dependent
current in coil 1
B2 is proportional to i1
N 2 B 2  M 21i1
M21 is the mutual inductance of the 2 coils
when i1 changes, B2 also changes and results in an induced EMF (2) in coil 2
The varying current in coil 1
induces an EMF in coil 2 
 di1 

dt
 
 2   M 21
M
 d B2 

 dt 
 2  N2 
In a similar way, if the current
in coil 2 (i2) changes with time
N2
d B 2
di
 M 21 1
dt
dt
 di 
1   M 12  2 
 dt 
N 2 B 2 N1 B1 Mutual Inductance

Unit = 1 Wb /A= 1Vs / A = 1 HENRY
i1
i2
EXAMPLE : A Tesla coil consists of two concentric coils, the inner one
with many (N1) turns per unit length and the outer one with fewer (N2)
turns per length surrounds it at it center.
Find an expression for the mutual inductance M
SOLUTION: We need to use M 

N 2 B 2 N1 B1

to determine M
i1
i2
We need to know either (a) the flux B2 through each turn of the outer coil due to
the current i1 in the inner solenoid or (b) the flux B1 through each turn of the
solenoid due to the current in the outer coil i1
The inner solenoid carrying a current i1 produces a magnetic field B1 along its axis  B1  µ0 n1i1 
The magnetic flux through the cross section equals B1A,
this must also be the flux threading the outer coil 
The mutual inductance is then given by  M  N 2 B 2 
 B 2  AB1 
µ0 N1i1

µ0 N1i1 A

µ0 N1 N 2 A
, proportional to N1N2

Suppose =0.5m, A=10cm2, N1=1000 and N2 =10 turns 
4 10
M
7



Wb / A.m 100010 1103 m 2
 25µH
0.5m
23
6.3.2 Self-Inductance and inductors
•
mutual inductance = interaction of two independent circuits
•
An important related effect occurs even when we consider only a single isolated circuit.
i changes  B (and B) changes  a self induced EMF is generated
From Lenz´s law, this EMF tries to oppose the change of current
N B Self-inductance or inductance of the circuit
Unit = HENRY = 1Wb / A
i
d B
di
di Self-induced



N

L



L
If i now changes  B changes 
dt
dt
dt EMF
L
Inductors are one of the three essential elements in simple electronic circuits
(resistor, capacitor and inductor), their circuit symbol is a kind of spiral and
they are used to inhibit current fluctuations in unstable circuits…
How do they behave in circuits ?
•
To see how inductors behave in a circuit we are going to develop a Kirchhoff's loop rule that
also applies to an inductor in a circuit…
–
Reminder : The Kirchhoff loop rule states that as we go around a conducting loop, measuring potential differences
across successive circuit elements, the algebraic sum around any closed loop must be zero since the electric field is
produced by charges (Ec) and must, therefore, be conservative.
–
If an inductor is in the circuit the situation changes since the magnetically induced electric field (En) in the coils of
an inductor is not conservative as we saw earlier in the lecture.
If the coils of the inductor have negligible resistance then a vanishingly small field is
required to make the charges move through the coils
  L
di
  E n .d 
dt
Integrating clockwise
around loop
|E|n is non-zero only within the inductor L
En  Ec  0
En , Ec  0
b
 E n .d    L
a
b
Since E n  E c  0 inside the inductor, we can rewrite this  E c .d   L
a
This integral is just the potential difference Vab between point a and b in the circuit
di
Vab  Va  Vb  L
dt
Even though the action of an inductor involves the
non-conservative (non-electrostatic) force there still
exists a potential difference between the terminals!
25
di
dt
di
dt
6.3.3 Energy stored in an inductor
•
A capacitor stores energy in the E-field … What about an inductor ?
–
We can calculate the energy required to change the current in an inductor from zero to i by noting that, as the
current increases from zero, it causes an EMF between the terminals and a corresponding potential difference
Vab between the terminals of the source (point-a at a higher potential than point-b)
Power dissipated
P  Vabi  Li
di
dt
Energy transferred to the inductor dU in a short time dt is then dU  Pdt
dU  Pdt  Li
di
dt
dt
1
U  Li 2
2
ENERGY STORED
in an INDUCTOR
Increasing the
current from 0i
Unlike a resistor – this energy is not dissipated in the
environment to generate heat, rather it is stored in the
magnetic field. When the current decreases this energy is
released again…
What happens when you pull the plug on a
large inductor supporting a current ?
26
• One can develop general relations to describe the energy stored in a magnetic field
similar to those developed for electric fields
µ0 NiA
2r
N B µ0 N 2 A
Self-inductance  L 

i
2r
Flux density inside the coil  B  BA 
1 2 µ0i 2 N 2 A
Energy stored is  U  Li 
2
4r
Volume inside the coil V  2rA
}
U 1
i2 N 2
u   µ0
V 2 2r 2
Ni
We can express this in terms of the magnetic field B  µ0
, so
2r
1 2 ENERGY
u
B DENSITY in
2µ0
B-field
Compare with the energy density stored in an electric field (L4) 
ENERGY
1
2
u   0 E DENSITY in
2
E-field
27
How inductors behave in circuits
R-L circuits and L-C, L-R-C oscillators
28
6.3.4 Inductors in circuits
•
One thing is already clear; when one has an inductor in a circuit it is difficult
to have rapid changes of current (due to self induced EMF)
Vab  L
We can use this equation, together with Kirchhoff's loop rules to analyse circuits
containing inductors
•
Example: The R-L circuit
Suppose both switches are initially open and we then close switch S1 at time t=0
Current cannot change instantaneously from 0 Imax since dI/dt
Instead the current and voltages across the elements will increase slowly with time
At any instant in time vab  iR
and vbc  L
di
dt
Apply Kirchhoff's loop rule, starting at the negative
terminal and moving around the loop counter-clockwise
  iR  L
di
0
dt
Note : at t=0, i=0 so the initial dI/dt is
As t , di/dt=0 so
0
  iR
L
di   iR  iR

 
dt
L
L L

 di 

 
 dt t 0 L
  iR
29
di
dt
di
To obtain the explicit time dependence notice that:
i 

R
R
dt
L
 i  
R Rt
ln 
  
L
R 

t
 1 

di   R dt
0  i   
L 0
R

i
it   

 R 
1  exp   t  
R
 L 
The current flowing in the circuit initially increases rapidly and then
approaches the value /R asymptotically
The quantity t=L/R is a measure of how quickly the current reaches the
steady state value, it is known as the time constant of the circuit
Discharging an L-R circuit
Now suppose that S1 has been closed for a while. If we reset out stopwatch to
redefine the initial time, we close the switch S2 at time t=0
Kirchhoff's loop rule again but omitting the EMF of the battery yields
di
iR

dt
L
 R 
it   I 0 exp   t 
 L 
30
EXAMPLE: Inductors can “save” your devices
EXAMPLE : We have a sensitive electronic device with a resistance of R=175W and want to connect it to a
source via a switch.
The device is designed to operate with a steady current of 36mA, but to avoid damage to the device, the current can rise to
no more than 4.9mA in the first 58µs after S1 is closed. To protect our device we want to connect it in series with an inductor
(a) What EMF should the source have, (b) What Inductance is required and (c) What is the time constant
Sensitive device
6.3.5 The L-C series circuit
• The combination of a capacitor and inductor in series is rather more exciting in
it´s electrical response – it is capable of functioning as an oscillator
The energy transfers from the electric field of the capacitor to the magnetic field of the inductor
•
We can analyse this behaviour using Kirchhoff's loop rule on the circuit
Clockwise
from lower
right around
di q
L  0
dt C
d 2q  1 

q  0
2
dt
 LC 
di d 2 q
 2
dt dt
dq
i
dt
d 2x  k 
  x  0
dt 2  m 
Exactly the same equation of motion as
for a Simple Harmonic Oscillator
Inductance analogous
To mass
1/C analogous
To spring constant
q – analogous to
displacement x
LC -circuit
i(t)
1

LC
qt   A´cost   
time
it  
dqt 
 A´sint   
dt
SHO

k
m
xt   A cost   
vt  
dxt 
 A cost   
dt
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EXAMPLE: Building an oscillator
A 300V dc supply is used to charge a 25µF capacitor. After it is fully charged, it is disconnected from the
power supply and connected across a 10mH inductor. Neglecting the resistance in the circuit find:
(a) The frequency and period of oscillation in the circuit
(b) The capacitor charge and circuit current 1.2ms after the inductor and capacitor are connected
t=0
6.3.6 Damping – the L-R-C series circuit
•
In the last discussion of an L-C circuit, we assumed that the circuit had zero resistance
 This is clearly an idealisation - in reality we have to consider a series L-R-C circuit
It turns out that the resistance acts to “damp” the oscillation in the L-C circuit
 it introduces as “frictional term” into the equation of motion that dissipates energy as heat
To see this consider the circuit
shown left and apply the
Kirchoff´s loop rule (a d)
dq
i
dt
2
di d q

dt dt 2
 iR  L
di q
 0
dt C
d 2 q R dq  1 


q  0
2
dt
L dt  LC 
“velocity” dependent
term= damping”
The form of the solutions of this “equation of motion” depend on the relative size of R and L/C
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In the “under-damped” case
Have the form
(R2<4L/C)
2
d
q R dq  1 
the solutions of


q  0
2
dt
L dt  LC 

  R    1
R2
qt   Q exp    t  cos
 2 t  

  2 L    LC 4 L


 1

  R  
dqt 
R2
1
R2
R2
  R   1

 Q exp    t    cos
 2 t   
 2 sin
 2 t   
 LC 4 L

 LC 4 L

dt
2L
LC 4 L
  2 L  



   
oscillation “resonant”
frequency changes
Current
´
1
R2

LC 4 L2
time
Amplitude decays with a time constant
t
2L
R
We´ll look at the practical relevance of the L-R-C circuit next time when we
drive it using an AC voltage – FORCED “ELECTRICAL” OSCILLATIONS
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