Physics 202, Lecture 17
Today’s Topics
„
Inductance (Ch 32)
„ Reminder of Faraday’s and Lenz’s Laws
„ Mutual Inductance
„ Self Inductance (Inductor L)
„
Energy Stored in magnetic Field
„
LR Circuit
Review: Faraday’s Law of Induction
‰ Faraday’s Law in plain words: When the magnetic
flux through an area is changed, an emf is produced
along the closed path enclosing the area.
conventional
Quantitatively:
direction of ε
B
ε = − dΦ B
dt
Note the - sign
Review: Lenz’s Law
θ
A
Φ B = ∫ B • dA
Examples of Lenz’s Law
‰ Lenz’s law in plain words: the induced emf always
tends to work against the original cause of flux change
Cause of dΦB/dt
“Current” due to Induced ε will:
Increasing B
produce B in opposite dir.
Decreasing B
produce B in same dir.
Relative motion
subject to a force in opposite
direction of relative motions
Note: “Current” will not be actually produced if no circuit
1
Mutual Inductance
Magnetically Coupled Coils (Transformers)
‰ For coupled coils:
ε 1 N1
=
ε 2 N2
ε2 = - M12 dI1/dt
ε1 = - M21 dI2/dt
Can prove (not in class):
M12=M21=M
Æ M: mutual inductance
(unit: Henry)
N1
N2
ε2 = - M dI1/dt
ε1 = - M dI2/dt
Question: Why use iron core?
Self Inductance
‰ When the current in a conducting device changes,
an induced emf is produced in the opposite direction
of the source current. Æ self inductance.
Exercise: Calculate Inductance of a Solenoid
‰ (Text example 32.1)
show that for an ideal solenoid:
L=
(see board)
‰ The magnetic flux due to self inductance is
proportional to I: ΦB= LI
ÎThe induced emf is proportional to dI/dt:
εL = - L dI/dt
L: Inductance, unit: Henry (H)
μ0 N 2 A
Area: A
l
l
# of turns: N
B =μ0nI
(ideal case, recall Apere’s Law)
2
Charging and Discharging
LR Circuit
Inductors
I=0
Dis-Charging
Before
I
Energy in an Inductor
‰ The work needed to increase the current in an inductor
from zero to some value I
Transient
Stabilized
I=i(t)
Before
‰ When an inductor of inductance L is carrying a current
changing at a rate dI/dt, the power supplied is
I=Imax=ε/R
I=i(t)
Charging
‰ Inductance is intrinsic to a conductive circuit.
‰ Two factors that determine the inductance:
ƒ Geometric configuration of the circuit
ƒ Filling of magnetic material.
‰ Specifically configured inductance devices (inductors) are very
useful in electronic and electrical applications:
I=0
After
Transient
Energy in a Magnetic Field
‰ U= ½
ƒ Solenoid: B =μ0 N/l I and L= μ0 N2A/l
Æ U= ½ B2/μ0 (Al)
LI2
¾ The energy is in the form of B field:
ƒ energy
gy density:
y uB = ½ B2/μ
μ0
(recall: uE= ½ ε0E2)
Solenoid as example
‰ Compare:
Inductor: energy stored U= ½ LI2 Î ½ B2/μ0
Capacitor: energy stored U= ½ C(ΔV)2 Î ½ ε0E2
Resistor: no energy stored, (all energy consumed )
3
Basic Circuit Components
Component
Symbol
Behavior in circuit
Ideal battery, emf
ΔV=V+-V- =ε
Resistor
ΔV= -IR
ε r
Realistic Battery
LR Circuit
Æ
‰ Current as a function of time after switching on: I(t)
I (t ) =
(Ideal) wire
Capacitor
ΔV=0 (ÆR=0, L=0, C=0)
ΔV=V- - V+ = - q/C, dq/dt =I
Inductor
ΔV= - LdI/dt
(Ideal) Switch
L=0, C=0, R=0 (on), R=∞ (off)
ε
R
(1 − e
−
t
L/R
)
switching on at t=0
ε − L dI (t ) − RI (t ) = 0
dt
Transformer
Diodes,
Transistors,…
Future Topics
Note: the time constant is τ=L/R
Quiz: What is the current when t=∞ ?
Turn on LR Circuit: Algebra Details
Homework:
“Switching off”
LR Circuit: Time Constant
turning on
turning off
Apply Kirchhoff loop rule
I=
t
−
V
I = 0 (1− e L / R )
R
t
−
V0
(1− e L / R )
R
I = I 0e
−
t
L /R
Time Constant of the LR circuit: τ=L/R
4