Transcript Document

Induction
Faraday’s Law
Induction
• We will start the discussion of Faraday’s
law with the description of an experiment.
• A conducting loop is connected to a
sensitive ammeter. Since there is no battery
in the circuit there is no current.
Induction
Induction
• Whenever there is a change in the number of
field lines passing through a loop of wire a
voltage (emf) is induced (generated).
• More formally:
• The magnitude of the emf induced in a
conducting loop is equal to the rate at which
the magnetic flux through the loop changes
with time.
Induction
• Faraday’s law expresses this phenomena,
d B
 
dt
• Where the magnetic flux through the loop is
given by the closed integral,
 
 B   B  dA
Induction
• For a coil with N turns Faraday’s law
becomes,
d B
  N
dt
Induction
• In general the induced emf tends to oppose
the change in flux producing it.
• This opposition is indicated by the negative
sign in equation for Faraday’s law.
Induction
•
The general means of changing the flux
are;
1. Change the magnitude of the magnetic
field within the coil.
2. Change the area of the coil or the area
cutting the field.
3. Change the angle between B and dA.
Induction
Lenz’s Law
Induction
• The direction of the induced current in a
loop is determined from Lenz’s law.
• The law states that: An induced current has
a direction such that the magnetic field due
to the current opposes the change in the
magnetic flux that induces the current.
• Consider.
• The direction of increasing B is
to the left.
• The direction opposing this is to
the right.
• Using the screw rule point the
thumb in the direction opposing
the change.
• The fingers give the direction of
the induced current.
Induction
Induction
• Let us look at the following case as an
example of induction.
• Let us look at what happens as a conduction
moves through a magnetic field.
• There is a change in the area of flux cut
hence an induced i and an emf.
 
d B
• Remember    N
where  B   B  dA
dt
Induction
• Consider a conductor (length L) sliding
along a rail with a velocity v in an uniform
magnetic field B.
x
x
x
x
x
x
v
x
x
x
x
x
x
Induction
• As the conductor moves through the field
electrons are push upward (Fleming’s left
hand rule) making the top –ve and the
x
bottom +ve.
-ve
x
x
x
x
x
x
v
+ve
x
x
x
x
x
x
• Fleming’s left hand:
• 1st finger: magnetic field, 2nd current and the
thumb direction of movement.
• We can also use the Lorentz law.
Induction
• This induces an electrostatic field and emf
across the conductor (induced emf) which acts
as a source.
• The direction of the induced current
(conventional current) is clockwise. x
I
-ve
x
x
x
x
v
x
x
+ve
x
x
x
x
x
x
Induction
• The flux cut as the conductor moves through
 
the field is,  B  B.dA  B dA  BA
  B  B(Lx)


x
I
-ve
x
x
x
x
v
x
x
+ve
x
x
x
x
x
x
Induction
• The flux cut as the conductor moves through
 
the field is,  B  B.dA  B dA  BA
  B  B(Lx)


d B d
dx
• Rate of change of flux, dt  dt B(Lx)  BL dt
x
I
-ve
x
x
x
x
v
x
x
+ve
x
x
x
x
x
x
Induction
d B
 BLv
• Therefore,
dt
• Hence from Faraday’s Law the induced emf
is,   d B  BLv
dt
x
I
-ve
x
x
x
x
v
x
x
+ve
x
x
x
x
x
x
Ampere-Maxwell Law
Ampere-Maxwell Law
 
• Recall Ampere’s Law, B.ds   0ienc

Ampere-Maxwell Law
 
• Recall Ampere’s Law, B.ds   0ienc
• Ampere’s Law can be modified as follows
to incorporate the findings of Maxwell,

 
d E
 B.ds   0ienc   0 0 dt
Ampere-Maxwell Law
 
• Recall Ampere’s Law, B.ds   0ienc
• Ampere’s Law can be modified as follows
to incorporate the findings of Maxwell,

 
d E
 B.ds   0ienc   0 0 dt
• That is, there are two ways for a magnetic
field to be formed:
Ampere-Maxwell Law
1. By a currentN (given by Ampere’s Law
 0ienc   0  in ).
d E
n 1
2. By a change in flux (  0 0
).
dt
Ampere-Maxwell Law
1. By a currentN (given by Ampere’s Law
 0ienc   0  in ).
d E
n 1
2. By a change in flux (  0 0
).
dt
• The later part of the equation governing
the induction of a magnetic field.
 
d E
 B.ds  0 0 dt
Ampere-Maxwell Law
• That is, a magnetic field is induced along a
closed loop by changing the electric flux in
the region encircled by the loop.
Ampere-Maxwell Law
• That is, a magnetic field is induced along a
closed loop by changing the electric flux in
the region encircled by the loop.
• An example of this induction occurs during
the charging of a parallel plate capacitor.
Ampere-Maxwell Law
• Ex: Consider a
parallel-plate capacitor
with circular plates of
radius R which is
being charged. Derive
an expression for B at
radii r for r ≤ R and
r ≥ R.
Ampere-Maxwell Law
• Using the methodology of Ampere’s Law
we draw a closed loop between the plates.
Ampere-Maxwell Law
 
d E
• Recall,  B.ds   0ienc   0 0
dt
• There is no current between the plates,
 
d E
  B.ds   0 0
dt
Ampere-Maxwell Law
 
d E
• Recall,  B.ds   0ienc   0 0
dt
• There is no current between the plates,
 
d E
  B.ds   0 0
dt
• For r  R
d E
B2r    0 0
dt
Ampere-Maxwell Law
 
d E r 
• Note:  E  EA  E r 2
 B2r    0 0
2
dt
2 dE
  0 0  r
dt
0 0 r  dE 
B 
 
2  dt 
Ampere-Maxwell Law
• The equation tells us that B increases
linearly with increasing radial distance r.
Ampere-Maxwell Law
• For r  R
 
d E
 B.ds   0 0 dt
 
• In this case,  E  EA  E R2
B 
0 0 R  dE 
2
2r
 
 dt 
Ampere-Maxwell Law
• B decreases as1/r.
Ampere-Maxwell Law
B
linear
decay 1/r
R
r
Ampere-Maxwell Law
 
d E
 B.ds   0ienc   0 0 dt
• Comparing the the two terms on the right of
the Ampere-Maxwell equation, we see that
that the two terms must have dimensions of
current.
d E
• ie. The dimensions of ienc and  0
must
dt
be the same
Ampere-Maxwell Law
• The later product will be treated as a current
and is called the displacement current id .
d E
id   0
dt
Ampere-Maxwell Law
• The later product will be treated as a current
and is called the displacement current id .
d E
id   0
dt
• Therefore Ampere-Maxwell’s Law can be
rewritten as,
 
 B.ds   0id ,enc   0ienc
Ampere-Maxwell Law
• The direction of the magnetic field is found
by assuming the direction of the
displacement current is that of the current.
• Then use the screw rule.
Ampere-Maxwell Law
id
Ampere-Maxwell Law
• Rewriting the results of the circular
capacitor using the displacement current,
 0id
B
2
2r
rR
 0 id
B
2r
rR
Ampere-Maxwell Law
• Remember the displacement current is not a
flow of electrons.
Faraday’s Law
Induced Electric fields
• From the previous discussion of Faraday’s
law we recognise that,
: a conducting ring placed in magnetic field
of changing strength will induce an emf
which in turn will induce a current.
• The induced emf
and current are
illustrated to the
right.
• Since a magnetic
field can’t directly
produce a current it
must be due to an
electric field.
• What we find is
that the changing
magnetic flux
through the ring
produces an electric
field.
The electric field provides
the work need to move
charge around the ring.
• The work done in
moving a charge q0
around a ring is:
 
 
W   F .ds  q0  E  ds
• The work done in
moving a charge q0
around a ring is:
 
 
W   F .ds  q0  E  ds
• So that,
 
   E  ds
• Therefore Faraday’s Law can be
reformulated as,
 
d B
 E  ds   dt
• The electric field
exists independent
of the conductor!
• Permeates all of the
space within the
region of changing
magnetic field.
The red lines indicate
the electric field lines.
• Consider a ring of radius 8.5cm in a
magnetic field which changes as 0.13T/s.
Find the expression for the induced electric
field at a radius of 5.2cm from the centre.
 
d
Recall:  E  ds   dt
•
• LHS:


E

d
s
  Eds E  ds

 E 2r 
• RHS:

   B  dA BA  B r 2
 
• Thus: E 2r   r 2  dB
r dB
 E  3.4mV m
E
dt
2 dt
THE END