Download Ch 30 - Eunil Won

PHYS152
Lecture 9
Ch 31 Induction and Inductance
Eunil Won
Korea University
Fundamentals of Physics by Eunil Won, Korea University
Induction: two experiments
1. A current appears only if there is relative motion
between the loop and the magnet: the current
disappears when the relative motion between them
ceases
2. Faster motion produces a greater current
3. North pole toward the loop: clockwise current
North pole away from the loop: counter clockwise
current (For South pole, the situation is reversed)
The current produced in the loop is called an induced current
Two conducting loops close to each other but not
touching
1. If we close switch S (to turn on a current in the
right-hand loop), the meter suddenly and briefly
registers a current - an induced current
2. If we open switch S (to turn on a current in the
right-hand loop), the meter also registers a current
(but in the opposite direction)
Fundamentals of Physics by Eunil Won, Korea University
Faraday’s Law of Induction
An emf is induced in the loop at at left in figures
when the number of magnetic field lines that
pass through the loop is changing
Magnetic flux :
ΦB =
!
! · dA
!
B
SI unit of the magnetic flux: 1 weber = 1 Wb = 1 T m2
Faraday’s law becomes:
dΦB
E =−
dt
For the coil of N turns:
dΦB
E = −N
dt
Changing the magnetic flux:
1. Change the magnitude B
2. Change the area of the coil
3. Change the angle between the magnetic field and the
area of the coil
Fundamentals of Physics by Eunil Won, Korea University
Lenz’s Law
An induced current has a direction such that the
magnetic field due to the current opposes the
change in the magnetic flux that induces the
current
Electric Guitar
When the metal string is made to oscillate,
it causes a variation in magnetic flux that
induces a current in the coil
Fundamentals of Physics by Eunil Won, Korea University
Induction and Energy Transfers
You must apply a constant force to pull the loop
(magnetic force with opposite direction exists)
The power need to pull:
P = Fv
The flux through the loop:
ΦB = BA = BLx
d
dx
dΦB
= BLx = BL
= BLv
E=
dt
dt
dt
BLv
i = E/R =
R
F2 and F3 cancel each other
The force opposing you:
F = F1 = iLB sin 900 = iLB
BLv
B 2 L2 v
F = iLB =
· LB =
R
R
Let’s find the rate at which thermal
energy appears in the loop
Fundamentals of Physics by Eunil Won, Korea University
P = i2 R
(
rate of doing work:
P =
!
BLv
R
! ×B
!
F! = iL
)
B 2 L2 v 2
P = Fv =
R
"2
B 2 L2 v 2
R=
R
: equal to the rate at which you are
dong work on the loop
Induced Electric Fields
A changing magnetic field produces an electric field
Reformulation of Faraday’s Law
Consider a particle of charge q0 moving around
the circular path
The work done on it per one revolution by the
induced electric field:
Steady increase of B
: counterclockwise
current is induced
An induced electric
field exists even when
the ring is removed
Or,
!
Eq0
F! · d!s = (q0 E)(2πr)
so we find that
E = 2πrE
More generally, work done on a particle of charge q0 moving along any closed path:
W =
!
F! · d!s = q0
Now, we know that
!
! · d!s
E
dΦB
E =−
dt
so we find that
we get
!
E=
! · d!s
E
dΦB
!
E · d!s = −
dt
Observation: electric field lines due to a changing B field has a closed loop
electric field lines due to an electric charge has no closed loop
Fundamentals of Physics by Eunil Won, Korea University
!
(Faraday’s law)
Inductors and Inductance
capacitor: produces a desired electric field
inductor: produces a desired magnetic field (a long
solenoid is a basic type of an inductor)
N ΦB
Inductance: L =
i
N: number of turns
SI unit: 1 henry = 1 H = T m2/A
Inductance of a solenoid (per unit length)
N ΦB = nlBA
the magnetic field of a
solenoid is:
B = µ0 in
n:
B:
l:
A:
number of turns per unit length
magnitude of the magnetic field
length near the middle of solenoid
cross sectional area
N ΦB
(nl)(BA)
(nl)(µ0 in)(A)
L=
=
=
= µ0 n2 lA
i
i
i
L
= µ0 n2 A
l
Fundamentals of Physics by Eunil Won, Korea University
depends on the geometry of the device
Self-Induction
An induced emf appears in any coil in which the current
is changing : this process is called self-induction
(A self-induced emf will appear in the coil while the
current is changing)
For any inductor,
Li = N ΦB
If the current increases,
the self-induced emf
opposes the increase
Faraday’s law tells us that
d(N ΦB )
d(Li)
di
EL = −
=−
= −L
dt
dt
dt
(self induced emf)
Fundamentals of Physics by Eunil Won, Korea University
If the current decreases,
the self-induced emf
opposes the decrease
RL Circuits
When the switch is on: initially an inductor acts to oppose
changes in the current through it. A long time later, it acts like
ordinary connecting wire
Applying the loop rule,
di
−iR − L + E = 0
dt
or
di
E = Ri + L
dt
This form is mathematically identical to the case with the RC circuit. So we take the solution as:
E
i = (1 − e−Rt/L )
R
or
E
i = (1 − e−t/τL )
R
where
L
τL =
R
(inductive time constant)
When the switch is off:
di
L + iR = 0
dt
E −t/τL
= i0 e−t/τL
i= e
R
Fundamentals of Physics by Eunil Won, Korea University
(decay of current)
Energy Stored in a Magnetic Field
Let’s rewrite the equation appeared in the RL circuit:
dq
Ei = E
dt
Ri2
di
Li
dt
di
E = Ri + L
dt
or
di
Ei = Ri + Li
dt
2
: rate at which the emf device delivers energy to the rest of the circuit
: rate at which energy appears as thermal energy
: must be the energy per unit time stored in the magnetic field of the
inductor
Therefore,
dUB = Li di
di
dUB
= Li
dt
dt
UB : the energy stored in the magnetic field
Integrating this, we get:
q2
UE =
2C
1 2
UB = Li
2
for the energy stored by a capacitor
Energy Density of a Magnetic Field: in a solenoid, the energy density becomes
1
2
Li
2
2
2 2
UB
L i
µ0 n i
µ0 n
uB =
=
=
=
=
Al
Al
l 2A
2
2
Fundamentals of Physics by Eunil Won, Korea University
L
= µ0 n2 A
l
2
!
B
µ0 n
B = µ0 in
"2
B2
=
2µ0
Note: we had
1
UE = !0 E 2
2
Mutual Induction
We use the terminology mutual induction, when we
hate two close-packed coils
If the current in coil 1(2) changes, an emf will be induced in coil 2(1)
We define the mutual inductance of coil 2 with respect to coil 1 as:
M21
N2 Φ21
=
i1
If the current varies with time,
so
dΦ21
di1
= N2
M21
dt
dt
The RHS is the emf appearing in coil 2:
so
M21 i1 = N2 Φ21
di1
E2 = −M21
dt
dΦ21
E2 = −N2
dt
If we interchange the roles of coils 1 and 2
di2
E1 = −M12
dt
It is true (without proof) that
Fundamentals of Physics by Eunil Won, Korea University
M21 = M12 = M
Summary
Magnetic Flux
Faraday’s law of Induction
Emf and induced electric field
Magnetic Energy
Fundamentals of Physics by Eunil Won, Korea University
ΦB =
!
! · dA
!
B
dΦB
E =−
dt
E=
!
! · d!s
E
1 2
UB = Li
2