Transcript Self-Inductance

Self-Inductance
AP Physics C
Montwood High School
R. Casao
Resistance, Capacitance, & Inductance
V
• Ohm’s law defines resistance as: R 
I
• Resistors do not store energy; they transform
U  0.5  C  V
electrical energy into
thermal energy at a rate
2
of:
V
2
E
P  I R 
2
R
• Capacitance is the ability to hold charge:
Q
C
V
• Capacitors store electric energy in the electric
field between the plates when fully charged:
• Inductance can be described as the ability to
“hold” current.
• Inductors store energy in the magnetic field
inside the inductor once the current flows
through it.
Terminology:
• EMF and current are associated with batteries
or other primary voltage sources.
• Induced EMF and induced current are
associated with changing magnetic flux.
Inductance, the definition
When a current flows through a coil, there
is magnetic field established. If we take
the solenoid assumption for the coil:
B  0  n  I
I
E
When this magnetic field flux changes, it
induces an emf, EL, called self-induction:
d  N  A B
d  N  A  0  n  I 
d B
EMFL  


dt
dt
dt
dI
dI
EMFL   0  n 2 V    L 
dt
dt
dI
EMFL   L 
dt
+
2
L



n
V
For a solenoid:
0
EL
–
Where
n= # of turns per unit
length.
N = # of turns in length l.
A = cross section area
V = Volume for length l.
Inductance L, is a constant related only to the coil.
The self-induced emf εL is generated by (changing) current in the coil.
According to Lenz’s Law, the emf generated inside this coil is always
opposing the change of the current which is delivered by the original emf ε.
• Consider an isolated circuit consisting of
a switch, a resistor, and a source of
EMF.
• When the switch is
closed, the current
doesn’t immediately
jump from zero to
its maximum value,
Im ax
EMF

R
• The law of electromagnetic induction
(Faraday’s law) prevents this from
happening.
• As the current
increases with time,
the magnetic flux
through the loop
due to this current
also increases with
time.
• The increasing flux induces an EMF in the
circuit that opposes the change in the net
magnetic flux through the loop.
• By Lenz’s law, the induced electric field in the
wires must be opposite to the
direction of the
conventional current
and the opposing EMF
and the induced current
that results establishes a
magnetic field that opposes
the change in the source
magnetic field.
• The direction of the induced EMF is opposite
the direction of the source EMF; this results in
a gradual rather than instantaneous increase
in the source current to its final equilibrium
value.
• This effect is called self-induction since the
changing flux through the circuit arises from
the circuit itself.
• The EMF, EMFL, that is set up in this case is
called a self-induced EMF or back EMF.
• Faraday’s law tells us that the induced EMF is
the negative time rate of change of the
magnetic flux.
– The magnetic flux is proportional to the magnetic
field, which is proportional to the current in the
circuit.
– Therefore, the self-induced EMF is always
proportional to the time rate of change of the
current.
– For a closely spaced coil of N turns of fixed
geometry (a toroidal coil or a solenoid):
d m
dI
EMF  N 
 L 
dt
dt
– where L is a proportionality constant, called the
inductance of the device, that depends on the
geometric features of the circuit and other physical
characteristics.
– Remember: the minus sign is a reflection of Lenz’s
law; it says that the self-induced EMF in a circuit
opposes any change in the current in that circuit.
• The inductance of a coil containing N turns is:
N  m
L
I
– where it is assumed that the same flux passes
through each turn.
• Inductance can also be rewritten as the ratio:
EMF
L
dI
dt
- This is usually taken to be the defining equation for
the inductance of any coil, regardless of its shape,
size, or material characteristics.
• Just as resistance is a measure of the
opposition to current, inductance is a measure
of the opposition to the change in current.
• Inductance unit: henry (H);
V s
1H=1
A
• The inductance of a device depends on its
geometry, much like the capacitance of a
capacitor depended on the geometry of its
plates.
• Consider a coil wound on a cylindrical iron
core and that the source current in the coil
either increases or decreases with time.
• When the source current is in the direction
shown (figure a), a magnetic field directed
from right to left is set up inside the coil.
• As the source current changes with time, the
magnetic flux through the coil also changes
and induces an EMF in the coil.
• From Lenz’s law, the polarity of this induced
EMF must be such that it opposes the change
in the magnetic field from the source current.
• If the source current is increasing, the
polarity of the induced EMF is as shown in
figure b.
• If the source current is decreasing, the
polarity of the induced EMF is as shown in
figure c.
• Many problems will ask you to identify the point on either
side of an inductor that has the higher potential.
• To determine which point is at the higher potential, curl
the fingers of your right hand in the direction of the
induced current within the inductor; the right thumb
points in the direction of the point which is at the higher
potential.
• The point on either side of an inductor that is at the higher
potential is the point that is in the direction of the Binduced
vector.
Inductance of a Solenoid
• Find the inductance of a uniformly wound
solenoid with N turns and length l that is long
compared with the radius and that the core of
the solenoid is air.
• The magnetic field of the solenoid is:
N
B  o   I
l
• The magnetic flux through each turn is:
m
N
 B  A  o   I  A
l
– where A is the cross-sectional area of the solenoid.
• Inductance:
N  m
L
I
N
N  o   I  A
2


N
A
l
o


I
l
• This shows that L depends on geometric factors and is
proportional to the square of the number of turns.
• Since n = N/l, and N = n·l:
L
2
o  N  A
l
2
o  n  l   A
2

L  o  n  volume
l
2
 o  n  l  A
where A  l = volume
Calculating Inductance and EMF
• Calculate the inductance of a solenoid
containing 300 turns if the length of the
solenoid is 25 cm and its cross-sectional area
is 0.0004 m2.
N  m
L
I
N
N  o   I  A
2
o  N  A
l


I
l
T  m  300 
2

 0.0004 m
A
0.25 m
2
L  4   x 10
7
L  0.000181 H  0.181mH
• Calculate the self-induced (back) EMF in the
solenoid described if the current through it is
decreasing at a rate of 50 A/s.
– dI/dt is negative because the rate of change is
decreasing.
dI
A
EMF  L 
 0.000181 H  50
dt
s
EMF  0.00905 V  9.05 mV
• The potential difference across a resistor depends on
the current.
– Resistor with current i flowing from a to b;
potential drops from a to b.
• The potential difference across an inductor depends
on the rate of change of the current.
– Inductor with constant current i flowing from a to
b; no potential difference.
– Inductor with increasing current I flowing from a
to b; potential drops from a to b.
– Inductor with decreasing current I flowing from a
to b; potential increases from a to b.
Inductors in Series and in Parallel
• Inductors, like resistors and capacitors,
can be placed in series.
– Total inductance can be increased by
placing inductors in series.
LT = L1 + L2 + L3 + . . . + LN
Inductors in Series and in Parallel
• Inductors, like resistors and capacitors,
can be placed in parallel.
– Total inductance can be decreased by
placing inductors in parallel.
1
1
1
1




LT L1 L2 L3

1
1
LT  L
1
1

LN
1
 L2  L3  ...  LN
1

1