Subjects
ÎInduced
emf
‹Faraday’s
law (law #3 of electricity and magnetism)
‹Lenz’s law
‹Motional emf
‹Generator
ÎInductors
ÎRL
and Inductance
circuits
ÎMagnetic
energy
ÎTransformers
ÎSection
book!
30-6 (Induced Electric Field): read the
‹Important
concept. Needed in Chapters 32 & 33
PHY2049: Chapter 30
1
Inductors
ÎCome
in many shapes and sizes. Most common
are small solenoids and toroids. Used in electronic
circuits.
PHY2049: Chapter 30
2
Self-induced emf
ÎWhat
is it?
‹If
a current through a coil of wire changes,
B produced by it changes. This causes
magnetic flux in the coil to change,
leading to induced emf — self-induced
emf.
‹Self-induced
emf can destroy a poorlydesigned current supply for an
electromagnet. Also, never touch the
cables carrying the current to an
electromagnet. (See a demo later.)
B
i
PHY2049: Chapter 30
3
Inductance
ÎMeasure
of the strength of self-induced emf with
respect to current change
ÎDefinition:
radius
ÎWhy
Inductance L of a coil with a uniform
NΦ B
L≡
i
is this a good definition?
ε
di
dt
L
Faraday’s law: ε
‹ This is actually the general definition of L
‹ Negative sign means: self-induced emf opposes change
in current (just Lenz’s law!)
‹
dΦ B
= −N
dt
= −L
ÎUnits
‹
‹
‹
Wb/A = T m2/A = henry = H
Inductors used in electronics: µH
Large electromagnets in labs: H
PHY2049: Chapter 30
4
CHECKPOINT 5
ÎThe
figure shows the direction of a self-induced
emf of a coil. Which of the following can describe
the current through the coil?
‹(a)
constant and rightward
‹(b)
constant and leftward
‹(c)
increasing and rightward
‹(d)
increasing and leftward
Lentz’s law
PHY2049: Chapter 30
5
Example
Î
L of a long solenoid
NΦ B
L≡
i
‹
Definition
‹
Field produced by long solenoid
B = µ0 ni
L = µ0 n Al
2
To increase L: increase n (many turns/length), A
(large cross section), l (long solenoid)
‹
Î
Two units for µ0
‹
H/m and T m/A
PHY2049: Chapter 30
6
CHECKPOINT 6
ÎThe
three inductors are identical, as are all the
resistors and batteries. When the switch is closed at
t=0, which circuit has the largest current through the
battery?
‹(a)
1
‹(b) 2
‹(c) 3
‹(d) 1 and 2
‹(e) 1 and 3
(1)
(2)
(3)
If i through L jumps from 0 to a non-zero value at
t=0, then di/dt=∞. emf produced by L would be
∞, which is unphysical. di/dt is finite and i=0.
PHY2049: Chapter 30
7
(continued)
ÎThe
same three circuits, with identical L, R, and
batteries. Long time after the switch is closed, which
circuit has the largest current through the battery?
‹(a)
1
‹(b)
2
‹(c)
3
‹(d)
1 and 2
‹(e)
1 and 3
(1)
(2)
(3)
At t→∞, di/dt=0. So emf produced by L is 0.
Inductor is then just a piece of wire.
PHY2049: Chapter 30
8
RL Circuit
ÎReaching
steady state takes time
‹Self-induced emf of inductor opposes current change
from 0
‹Current takes time to reach full value
ÎClose
switch at t=0
‹Initial current: i = 0
‹Initial emf of L: ℇL = ℇ of battery
t
∞
‹Final current: i = ℇ/R
‹Final emf of L: ℇL = 0
(since di/dt=0. L behaves just as piece of wire.)
ÎAt
PHY2049: Chapter 30
9
(continued)
ÎWhat
happens in between? Use loop rule:
E – i R – L di /dt = 0
ÎSolve
the differential equation
ε
ÎGeneral
R
di
+i =
L L
dt
ε
i = + Ke
solution is
‹(Check
−t /( L / R )
and see!)
R
‹K = −E / R (necessary to make i = 0 at t = 0)
ÎCurrent
i and self-induced emf of L
ε
i = (1 − e
R
‹Compare
−t /( L / R )
)
|
ε |= ε − iR = εe
L
− t /( L / R )
with q and i in RC circuit
PHY2049: Chapter 30
10
Current and V vs Time
ε
i = (1 − e
R
V = ε (1 − e
)
)
−t /( L / R )
− t /( L / R )
across resistor
|
ε |= εe
L
− t /( L / R )
self-induced emf of inductor
t/(L/R)
PHY2049: Chapter 30
11
Question 8
ÎConsider
three circuits wired as shown, containing
the same resistance R and battery but different
inductance L. Which of the three graphs shows the
potential difference VR across the circuit with the
smallest L, after the switch is closed?
a
‹b
‹c
‹
PHY2049: Chapter 30
12
Time Constant
ÎThe
three circuits have identical L, R, and batteries.
Which circuit has the shortest time constant?
‹(a)
1
‹(b)
2
‹(c)
3
‹(d)
1 and 2
‹(e)
1 and 3
(1)
(2)
(3)
To find tau for circuits 2 or 3, use loop rule to
write a differential equation. No need to solve it.
PHY2049: Chapter 30
13
RL Circuit 2 (Disconnect battery)
Îdi/dt
= –∞ will cause
‹Sparks!
εL = – L di/dt = ∞
Can electrocute you
☹
ÎUse
a make-before-break switch to provide a
current path
‹Loop
rule: – i R – L di /dt = 0
‹Solve
‹General
di
R
+i = 0
dt
L
− t /( L / R )
solution is
i = Ke
‹K=ℇ/R to make i=ℇ/R at t=0
ε
i= e
R
−t /( L / R )
ε
V = iR = e − t /( L / R )
PHY2049: Chapter 30
14
Current and V vs Time
ε
i= e
ε
−t /( L / R )
R
V = e −t /( L / R )
ε
V = e
i=
ε (1 − e
R
−t /( L / R )
)
− t /( L / R )
across resistor
|
ε |= εe
L
− t /( L / R )
self-induced emf of inductor
ε
V = e − t /( L / R )
t/(L/R)
PHY2049: Chapter 30
15