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Physics 2112
Unit 18
Today’s Concepts:
A) Induction
B) RL Circuits
Electricity & Magnetism Lecture 18, Slide 1
Where we are…..
Just finished
introducing
magnetism
Will now apply
magnetism to AC
circuits
Unit 17, Slide 2
Remember Example 17.6 (solenoid)??
1. Changing current in outer
loop
2. Caused changing magnetic
field in inner loop
3. Induced changing current in
inner loop.
What if you only had one loop?
Would changing current in that induce anything?
Unit 17, Slide 3
Self Inductance
Define:
B
L
I
Depends only
on the
geometry of
the coils
d B
d ( LI )
dI
 

 L
dt
dt
dt
Voltage drop
in a coil
…caused by
changes in the
current in
that same coil
Electricity & Magnetism Lecture 18, Slide 4
Example 18.1 (Inductance of Solenoid)
L
What is the inductance of a solenoid?
Electricity & Magnetism Lecture 18, Slide 5
Checkpoint 1
Two solenoids are made with the same cross sectional area and total
number of turns. Inductor B is twice as long as inductor A
LB  0n r z
2
2
Compare the inductance of the two solenoids
A)
B)
C)
D)
E)
LA  4 L B
LA  2 L B
LA  LB
LA  (1/2) LB
LA  (1/4) LB
Electricity & Magnetism Lecture 18, Slide 6
What the minus sign means…..
A
dI
  L
dt
IAB
Increasing
current
A
DV < 0
VA > VB
Acts like a
resistor
DVAB
Electricity & Magnetism Lecture 18, Slide 7
What the minus sign means…..
A
dI
  L
dt
IAB
DVAB
decreasing
current
B
DV > 0
VA < VB
Acts like a
battery
However you try to change
the current through an
inductor, the inductor
resists that change.
Electricity & Magnetism Lecture 18, Slide 8
What this really means:
emf induced across L tries to keep I constant.
dI
L  L dt
L
current I
Inductors prevent discontinuous current changes!
It’s like inertia!
Units of inductance are Henrys (H) [Tm2/A]
Electricity & Magnetism Lecture 18, Slide 9
Time Constant
V
t /
I  (1  e )
R
I0
L
R
VBATT
L

R
Electricity & Magnetism Lecture 18, Slide 10
Example 18.2 (Current in Solenoid)
A solenoid has 6500 loops in a length
of 10cm and a radius of 6cm. It is
attached to a 120W resistor and a 12V
battery.
S
L
R=100W
12V= VBATT
L
V
t /
I  (1  e )  
R
R
What is the current through the
resistor 0.01sec after the switch is
closed?
What is the current through the
resistor 2.0 sec after the switch is
closed?
Would the current have been at
0.01sec if the inductor had an internal
resistance of 20W?
Electricity & Magnetism Lecture 18, Slide 11
How to think about RL circuits:
When no current is flowing initially:
VL
I0
L
I = V/R
R
L
L

R
R
I
VBATT
At t = 0:
I0
VL  VBATT
VR  0
(L is like a giant resistor)
VBATT
At t >> L/R:
VL  0
VR  VBATT
I  VBATT/R
(L is like a short circuit)
Electricity & Magnetism Lecture 18, Slide 12
CheckPoint 2A
In the circuit, the switch has
been open for a long time, and
the current is zero everywhere.
I
At time t  0 the switch is
closed.
What is the current I through the vertical resistor immediately
I
after the switch is closed? (+ is in the direction of the
arrow)
A) I  V/R
B) I  V/2R
C) I  0
D) I  V/2R
E) I  V/R
Electricity & Magnetism Lecture 18, Slide 13
CheckPoint 2B
After a long time, the switch is
opened, abruptly
disconnecting the battery from
the circuit.
What is the current I through the vertical resistor immediately
after the switch is opened?
(+ is in the direction of the arrow)
A) I  V/R
B) I  V/2R
C) I  0
D) I  V/2R
E) I  V/R
Electricity & Magnetism Lecture 18, Slide 14
How to Think about RL Circuits Episode 2:
VBATT
When steady current is flowing initially:
VL
I=0
R
L
R
L
R
I  V/R
At t  0:
I  VBATT/R
VR  IR
VL  VR
L

R

L
R
At t >> L/R:
I0
VL  0
VR  0
Electricity & Magnetism Lecture 18, Slide 15
Why is there Exponential Behavior?

VL
dI
dt
L
+
+
I
VL
V  IR
R
L

R


dI
L + IR  0
dt
I (t )  I 0e tR / L  I 0e t /
L
R
L
where  
R
Electricity & Magnetism Lecture 18, Slide 16
What are Inductors and Capacitors Good For?
” Can you have capacitors and inductors in the same circuit?
“why inductors are important as opposed to capacitors.
why use one instead of the other?”
Inside your i-clicker
Electricity & Magnetism Lecture 18, Slide 17
Quick comment…
I
L
VL
R
VBATT
L

R
Lecture:
Prelecture:
Did we mess up?
No: The resistance is simply twice as big in one case.
Electricity & Magnetism Lecture 18, Slide 18
CheckPoint 3A
After long time at 0, moved to 1
After long time at 0, moved to 2
After switch moved, which case has
larger time constant?
A) Case 1
B) Case 2
C) The same
Electricity & Magnetism Lecture 18, Slide 19
CheckPoint 3B
After long time at 0, moved to 1
After long time at 0, moved to 2
Immediately after switch moved,
in which case is the voltage
across the inductor larger?
A) Case 1
B) Case 2
C) The same
Electricity & Magnetism Lecture 18, Slide 20
CheckPoint 3C
After long time at 0, moved to 1
After long time at 0, moved to 2
After switch moved for finite time,
in which case is the current
through the inductor larger?
A) Case 1
B) Case 2
C) The same
Electricity & Magnetism Lecture 18, Slide 21
Example 18.3 (3R and L circuit)
R1
The switch in the circuit shown
has been open for a long time.
At t  0, the switch is closed.
R2
V
L
R3
What is dIL/dt, the time rate of change of the current through the
inductor immediately after switch is closed?
 Conceptual Analysis
Once switch is closed, currents will flow through this 2-loop circuit.
KVR and KCR can be used to determine currents as a function of time.
 Strategic Analysis
Determine currents immediately after switch is closed.
Determine voltage across inductor immediately after switch is closed.
Determine dIL/dt immediately after switch is closed.
Electricity & Magnetism Lecture 18, Slide 22