Transcript Inductancex - MrsCDsAPPhysics

Ch. 30 Inductance
AP Physics

According to
Faraday’s law, an emf
is induced in a
stationary circuit
whenever the
magnetic flux varies
with time. If this flux
variation is caused by
a varying current in a
second circuit, the
induced emf can be
expressed in terms of
varying current, i.
Mutual Inductance

A changing current in
one circuit causes a
changing magnetic
flux and induced emf
in a neighboring
circuit that is
proportional to the
rate of change in
current. The
proportionality
constant is called the
mutual inductance.
Mutual Inductance

As the current
changes in the
primary coil, the
magnetic flux
changes in the
secondary coil
which induces an
emf in the
secondary coil.
 2   N2
d  B2
dt
Mutual Inductance

Mutual inductance
of the coil is defined
to be M  N 2  B
2
21
i1
N 2  B2  M 21i1
N2
d  B2
dt
 M 21
 2   M 21
di1
dt
di1
dt
Mutual Inductance
If the coils are in a vacuum, then M21
depends on the geometry of the coils. If
a magnetic material is present, then M21
depends on the properties of the magnetic
material.
 Note: We will assume that the flux is
directly proportional to the current; that
is, M21 = M12 always!!

Mutual Inductance

In symbols,
M 

Therefore,

and
N 2  B2
i1

N1 B1
di1
2  M
dt
di2
1  M
dt
Mutual Inductance
i2

The negative sign show that the direction
of the induced emf in each coil is opposite
to the rate of change of current in the
other coil.

The measurement unit for mutual
inductance is: 1 Wb/A = 1 V-s/A = 1 Ω-s
= 1 Henry (H)
Mutual Inductance

A long solenoid with
length l and crosssectional area A and
N1 turns of wire is
surrounded by a coil
of N2 turns. Find
the mutual
inductance.
Tesla Coil

Suppose the length of the solenoid in the
previous slide is 0.50 m, A = 10 sq. cm.,
N1 = 10 turns, and N2 = 1000, calculate
the mutual inductance.
Sample Problem #1

Suppose i1 = (2.0 A/s) t, at t = 3.0 s,
what is the average magnetic flux through
each turn of the solenoid caused the
current in the larger coil of the Tesla coil.

What is the induced emf in the solenoid?
Sample Problem #2

When a current is present in any circuit,
the current sets up a magnetic field that
links with the same circuit and changes
when the current changes. Any circuit
that carries a varying current has an
induced emf in it resulting from the
variation in its own magnetic field. Such
an emf is called a self-induced emf.
This is also called back emf.
Self-Inductance

Consider a coil with N turns, carrying a
current i. As a result of this current, a
magnetic flux passes through each turn.
Self-inductance is defined to be
NB
L
i
Li  N  B

From Faraday’s law,
Self-Inductance
dB
di
L
N
dt
dt
di
  L
dt
An inductor or choke is an electrical
component of a circuit that is designed to
have a particular inductance.
 The schematic symbol is:
 The cause of the induced emf and the
field is the changing current in the
inductor, and the emf always act to
oppose this change according to Lenz’s
law; that is,
di
VL    L

dt
Inductor

Case 1. Current is increasing,
0, ε < 0
di
dt >
0, VL >
i increasing

Case 2. Current is constant

Case 3. Current is decreasing
How do the values change in cases
2 & 3?

Find the inductance of a uniformly wound
solenoid having N turns and length l.
Assume that l is much longer than the
radius of the windings and that the core of
the solenoid is air.
Sample Problem #3
Calculate the inductance of an air-core
solenoid containing 300 turns if the length
of the solenoid is 25.0 cm and its crosssectional area is 4.00 sq. cm.
 Calculate the self-induced emf in the
solenoid if the current through it is
decreasing at the rate of 50.0 A/s.

Sample Problem #4

Establishing a current within an inductor
requires an input of energy, and an
inductor carrying a current has energy
stored in it. A changing current in an
inductor causes an emf between their
terminals. The source supplies the
current must maintain a corresponding
potential difference between its terminals.
Magnetic Field Energy


Let the current at some time be i and its
increasing rate of change is di/dt. Then VL =
Ldi/dt and P = Vli = Lidi/dt. Since P = dU/dt,
then dU/dt = Lidi/dt.
I
Therefore,
U  L idi

0


The total energy is U  1 LI 2
2
The energy in an inductor is stored in the
magnetic field within the coil.
Magnetic Field Energy

The energy stored in an inductor is 1.00
kWh. If the current is 2.00 A, what is the
inductance?
Sample Problem #5




The inductance of the inductor results ina back
emf, an inductor in a circuit opposes
changes in the current through that circuit.
If the battery voltage in the circuit increased so
that the current rises, the inductor opposes this
change, and the rise is not instantaneous.
If the battery voltage is decreased, the
presences of the inductor results in a slow drop
in the current rather than an immediate drop.
Thus, the inductor causes the circuit to be
‘sluggish’ as it reacts to changes in the current.
RL Circuits

A switch controls the current in a circuit
that has a large inductance. Is a spark
more likely to be produced at the switch
when the switch is being closed or when it
is being opened, or doesn’t it matter?
RL Circuits

Consider the RL Circuit below:

Sketch a graph of current versus time as the
current increases in the circuit when the
switch is closed at t = 0
Apply Kirchhoff’s loop rule to the circuit, and
then solve for the current as a function of
time.

RL Circuit—Current Growth
Defined to be: τ = L/R
Suppose that the resistance is 6.00 Ω and
the inductance is 30.0 mH. Find the time
constant.
 Calculate the current
at t = 2.00 ms. The
voltage of the battery is
12.0 V.
 Compare the potential
difference across the resistor with that across
the inductor at t = 2.00 ms.


Time Constant-Sample Problem #6

Given the RL circuit,
when the current has reached its steady
state, the switch is moved from point A to
point B.
Sketch a current vs time. Graph.
 Apply Kirchhoff’s loop rule, and solve for the
current as a function of time.
RL Circuit—Current Decay
RL Circuit-Current Growth & Decay

In the circuit below, once the current
reached its steady state value the switch
is moved from point A to point B. The
resistor has a resistance of 6.00 ohms
and the inductor has an inductance of
30.0 mH. What fraction of the stored
energy has been dissipated after 2.3 time
constants?
Sample Problem #7






http://www.falstad.com/circuit/
After the applet opens, click on ‘Circuits,’ go
to ‘Basics.’ Select inductor.
How does the current change when the
battery is in the circuit?
How does the voltage drop across the resistor
(yellow) compare with the voltage drop
across the inductor (green)?
How does the current change when the
battery is not in the circuit?
How does the voltage drop across the resistor
(yellow) compare with the voltage drop
across the inductor (green)?
Circuit Simulator
When a capacitor is connected to an
inductor, as shown below, the combination
is an LC circuit.
 If the capacitor is initially charged and the
switch is closed, what will happen?
Assume that there is no resistance.

LC Circuit
When the capacitor is charged to a
potential difference of V, the initial charge
is Q = CV, and the energy stored in the
electric field of the capacitor is U = Q2/2C.
 At this time, the switch is opened and
there is no current flowing in the circuit.
Therefore, there is no energy stored in
the magnetic field of the inductor.

LC-Circuit
At t = 0, the switch is closed. The rate at
which the charge leaves the plates of the
capacitor is equal to the current in the
circuit; that is, i = dq/dt.
 The energy in the electric field of the
capacitor decreases and the energy in the
magnetic field of the inductor increases so
that Umax = q2/2C + Li2/2, since there is
no resistance in the circuit.

LC Circuit

When you apply Kirchhoff’s loop rule to
di
the LC circuit, you get: q
C
L
dt
0
q
d 2q
L 2 0
C
dt
q  Qmax cos(t   )
LC Circuit
dq
i
 Qmax sin(t   )
dt
mechanical spring
current:
q
t
½mv2
inductor's energy:
½Li2
½kx2
capacitor's energy:
velocity:
kinetic energy:
LC circuit
potential energy:
dv
 k x
dt
Eq. of motion:
m
frequency:
1
2
k
m
Kirchhoff's law:
frequency:
Oscillation Systems
L
½q2/C
di
1

q
dt
C
1
2
1
LC

The 9.00 pF capacitor in a LC circuit is
initially charged using a source of 12.0 V.
It is disconnected from its source and
then connected to a 2.81 mH inductor.
◦ What is the frequency of oscillation of the
circuit?
◦ What are the maximum values of charge on
the capacitor and current in the circuit?
◦ What is the total energy stored in the circuit?
Sample Problem #8
http://www.falstad.com/circuit/
 Go to ‘Circuits’ and click on ‘Basic’
 Select the LRC circuit
 What happens with the voltages and the
currents over time?

LRC Circuit