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Slide 1
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CHAPTER 20
- INDUCED VOLTAGE
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AND INDU=
CTANCE
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<=
/td>
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SECTION 20.1 INDUCED EMF & MAGNETIC FLU=
X
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SECTION
20.2 FARADAY’S LAW<=
br>
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SECTION 20.4=
LENZ’S LAW
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![](3D"v3_slide0001_image00=) |
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Φ =3D
Bn A where
Bn is the component of B normal to the<=
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the s=
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Slide 1
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Slide 1
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In this case, B=
n
=3D Bz so:
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If Bz changes
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![](3D"v3_slide0011_image004=) |
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From Maxwell’s =
Equation IV
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![](3D"v3_slide0011_image005.=) |
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and=
Stoke”s
Theorem from Calculus it can be shown that an electric field, El
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closed path
will be induced by ΔB / Δt
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Slide 1
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<=
/td>
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(5)=
is Faraday’s
Law in terms of
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ind=
uced E.
The induced E in (5)
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wil=
l produce a
voltage difference, V
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(ε in text=
) around the
path.
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![](3D"v3_slide0012_image006.gif") |
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The magnitude o=
f V is given
by:
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![](3D"v3_slide0012_image007.gif") |
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![](3D"v3_slide0012_image008.gif") |
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(6)=
is Faraday”s
Law in terms of
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induced voltage=
(EMF). It
says:
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NOT=
E: Text uses
emf, έ instead of V
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See eq. 20.2 p =
667
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![](3D"v3_slide001=) |
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<=
/td>
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The=
voltage
induced around a closed path
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due=
to changing
magnetic flux equals the
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neg=
ative of the rate
of change of the flux
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through the bou=
nded surface
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![](3D"v3_slide0012_image010.gif") |
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| <=
/td>
| <=
/td>
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(6)=
is used to
find V
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(5)=
is used to
find the direction of
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the induced E
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Slide 1
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![](3D"v3_slide0002_image011.gif") |
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| <=
/td>
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whe=
re V (or ε) is
the voltage induced by =
the
changing
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magnetic flux
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=
There Are
Three Means of Changing <=
u>Φ; Return to the closed path in the
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x,y plane with =
Bn =3D
Bz
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Φ =3D (B cos θ) A =3D
Bz A
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Slide 1
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Each term repre=
sents a
different means of producing a change in Φ
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<=
/td>
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I -
changing B ( time
dependent B)
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II =
- rotating
surface ( time dependent θ)
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III =
– time
dependent A (motional EMF)
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Let’s deal with=
term I.
V =3D ε =3D -ΔΦ / Δt =3D
-(ΔBz / Δt) (A)
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Slide 1
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![](3D"v3_slide0004_image013.gif") |
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V =3D ε =3D
-(Δ Bz / Δt) (A)
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Consider 2
Cases:
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(i)=
Suppose Bz > 0 and Δ Bz / Δ t=
> 0; then:
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![](3D"v3_slide0004_image014.=) |
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/td>
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<=
/td>
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NOT=
E: this E
would tend
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tp =
produce an induced
I’ which
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wou=
ld cause a negative
induced
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B’=
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![](3D"v3_slide0004_image015.=) |
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/td>
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/td>
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Bz’ is
caused by the
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ind=
uced I’. It is
not
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the original B
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/td>
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/td>
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ALS=
O NOTE: The
induced
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Bz’
tends to oppose the
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change in flux
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Slide 1
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![](3D"v3_slide0005_image016.gif") |
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![](3D"v3_slide0005_image017.gif") |
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V =3D ε =3D
-(Δ Bz / Δ t) (A)
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This induced E would
produce a +
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ind=
uced BZ’ (through the induced I’).
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/td>
| =
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IN =
BOTH CASES,
THE INDUCED E
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WOU=
LD TEND TO
PRODUCE AN
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IND=
UCED CURRENT,
I’ THAT WOULD
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CAU=
SE AN INDUCED
MAGNETIC
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FIE=
LD B’ THAT
WOULD OPPOSE THE
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CHANGE IN B.
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Slide 1
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IN<=
span
style=3D'mso-spacerun:yes'> GENERAL
LENZ’ LAW STATES: The induced electric field, E wo=
uld
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ten=
d to produce a
current that would offset the change in magnetic flux,
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Φ through the bou=
nded
surface.
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Len=
z’ Law is
valid for any application of Faraday’s Law, not just for the time-
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dependent magne=
tic field
case.
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One
Interpretation of Lenz’ Law: A magnetic field of magnitude B contains
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ene=
rgy
proportional to B2. To increase that field requires an input=
of
energy
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which is used i=
n offsetting
the induced electric field, E
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Slide 1
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II -
generators, alternators
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III -
one type of motional EMF
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If N turns of w=
ire compose
the bounding path, then:
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![](3D"v3_slide0008_image020.=) |
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If Bn is the
component of B normal to the surface then:
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/td>
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A bar magnet is inserted inside =
a 20
turn coil. B inside is approximately uniform
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and=
of magnitude,
B =3D 10-2 T. the radius of each turn is 8 cm. The magnet is=
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rem=
oved so that B
drops to zero in 0.1 s. What is the magnitude of the
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average voltage=
induced
across the coil ?
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![](3D"v3_slide0009_image022.g=) |
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A =3D π r<=
sup>2 =3D π
(8x10-2 m)2
=3D 0.02 m2<=
/font>
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V =3D
20 (10-2 T / 0.1 s) (0.02 m2)
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Slide 1
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![](3D"v3_slide0013_image025.g=) |
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/td>
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H=
OMEWORK –
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If I pull the magnet out from the=
South
Pole, in which direction is the
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induced E=
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Slide 1
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| SECTION 20.6 – SELF INDUCTANCE
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A conducting coil of area, A carri=
es a
current, I. The magnetic
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Flu=
x, Φ
through the coil results from Bn, the
normal component of
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B which =
results from
I.
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Define the self-inductance, L of =
the
coil as:
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(9)=
shows that L
is the magnetic flux through the coil divided by the
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current that pr=
oduces that
flux.
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Slide 1
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From (9):=
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(10)=
Φ =3D
L I
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Now if I change=
s with time,
Bn and Φ change with time. From (10)
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![](3D"v3_slide0014_image027=) |
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So,=
in any case
the voltage, VL across
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an inductor in =
a circuit
is:
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From Faraday’s
Law and (11):
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![](3D"v3_slide0014_image028.gif") |
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![](3D"v3_slide0014_image029.gif") |
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next, for a coi=
l with N
turns:
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![](3D"v3_slide0014_image030.gif") |
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Slide 1
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(13) shows:
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(i)=
if I
increases ( ΔI / Δt > 0 ) then VL < 0 and th=
ere
is a voltage drop in the
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dir=
ection of I
(just like a resistor)
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![](3D"v3_slide0015_image032.gif") |
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(ii=
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(ΔI / Δt < 0), then VL > 0 and there is a voltage
rise in the
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direction of I =
(just like a
voltage source)
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![](3D"v3_slide0015_image033.gif") |
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(ii) VL results
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Slide 1
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/td>
| =
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<=
/td>
| <=
/td>
| <=
/td>
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The polarity of voltages induced =
across
a coil according to Lenz’ Law would
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see=
m to opposite
the ΔV’s just discussed. However the voltages obtained from=
|
(13=
) can be
thought of as the voltage that must be applied by a source outside
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the=
inductor in
order to overcome the electric field induced according to=
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Faraday’s Law.
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Also, to increase the current in=
a
coil involves increasing the magnetic field.
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The=
re is energy
stored in a magnetic field and that energy must be supplied
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fro=
m an outside
source in the form of a voltage times charge.
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EXAMPLE –
A 25 mH inductor is connected to a capacitor in an LC circuit.
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At =
one instant,
the current in the circuit is changing at the rate of 500 A/s. Find
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the magnitude o=
f the
voltage across the inductor at the instant described.
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Slide 1
CHAPTER
20 - INDUCED VOLTAGE
<=
/div>
=
; &n=
bsp;
AND INDUCTANCE
SECTION 20.1 INDUCED EMF & MAGNETIC
FLUX
SECTION
20.2 FARADAY’S LAW
=
;
SECTION
20.4 LENZ’S LAW
A
uniform B makes an angle, θ with the z axis. B produces net magnet=
ic
flux,
Φ, through a closed path in the x,y pl=
ane.
the path bounds a surface of area, A.
![](3D"slide0001_image036.gif")
![](3D"slide0001_image037.gif")
![](3D"slide0001_image038.gif")
B
Bz
θ
x
y
z
(1)Φ&nbs=
p;
=3D Bn A =
;
where Bn is the component of B normal to the
=
; &n=
bsp;
the surface
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Slide 1
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Slide 1
![](3D"slide0011_image039.gif")
![](3D"slide0011_image040.gif")
![](3D"slide0011_image041.gif")
B
Bz
x
y
z
In
this case, Bn =3D Bz<=
/span> so:
(2) =
;
Φ&nbs=
p;
=3D Bz A
(3)
Φ&nbs=
p;
=3D B (cosθ) A
θ
If Bz<=
/span> changes with time and A is constan=
t, (2)
shows
![](3D"slide0011_im=)
From
Maxwell’s Equation IV :
=
and
Stoke”s Theorem from Calculus it can be shown that an electric field=
, El<=
/span>
tangent
to the closed path will be induced by ΔB / Δt
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Slide 1
![](3D"slide0012_image046.gif")
![](3D"slide0012_image047.gif")
![](3D"slide0012_image048.gif")
x
y
z
A
Bz
ΔB/Δt
El
![](3D"slide0012_image049.gif")
![](3D"slide0012_im=)
=
(5) is
Faraday’s Law in terms of
induced
E. The induced E in (5)
will
produce a voltage difference, V
(ε in text) around the path.
The
magnitude of V is given by:
=
(6) is
Faraday”s Law in terms of
induced
voltage (EMF). It says:
=
The
voltage induced around a closed path
due
to changing magnetic flux equals the
negative
of the rate of change of the flux
through
the bounded surface
(6)
is used to find V
(5)
is used to find the direction of
the induc=
ed E
=
NOTE:
Text uses emf, έ instead of
V
See eq. 20.2 p 667
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Slide 1
![](3D"slide0002_ima=)
=
=
where V
(or ε)&nb=
sp;
is the voltage induced by the changing&#=
13;
magnetic flux
=
There
Are Three Means of Changing <=
/span>Φ=
; Return to the closed path in
the
x,y plane with Bn =3D
Bz
![](3D"slide0002_image056.gif")
![](3D"slide0002_image057.gif")
![](3D"slide0002_image=)
x
y
z
Bz
![](3D"slide0002_image059.gif")
θ
B
E
dl
Φ =3D=
(B cos θ) A =3D Bz A
![](3D"slide0002_image060.gif")
A
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Slide 1
![](3D"slide0003_image062.gif")
![](3D"slide0003_image063.gif")
![](3D"slide0003_image=)
=
x
y
z
Bz
![](3D"slide0003_image065.gif")
θ
B
E
dl
![](3D"slide0003_ima=)
Each
term represents a different means of producing a change in Φ
=
I -
changing B ( time
dependent B)
II
- rotating surface ( time dependent θ)
III – time dependent A (motional EMF)
Let&#=
8217;s
deal with term I. V =3D ^=
9; =3D
-ΔΦ / Δt =3D
-(ΔBz / Δt) (A)
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Slide 1
V =3D ε =3D<=
span
style=3D'mso-spacerun:yes'> -(Δ B=
z<=
/span> / Δt) (A)
![](3D"slide0004_ima=)
=
Consider 2
Cases:
(i)
Suppose Bz > 0 and Δ=
Bz<=
/span> / Δ t<=
span
style=3D'mso-spacerun:yes'> > 0; then:
and
if El is uniform ov=
er the
path, this means:
=
; &n=
bsp;
El < 0
![](3D"slide0004_image072.gif")
![](3D"slide0004_image073.gif")
![](3D"slide0004_image074.gif")
x
y
z
Bz
E
dl
A
![](3D"slide0004_image075.gif")
dBz / dt
N=
OTE:
this E would tend
tp
produce an induced I’ which
would
cause a negative induced
B’
![](3D"slide0004_image076.gif")
![](3D"slide0004_image077.gif")
B=
z’’
E
I’
z
Bz<=
/span>’
is caused by the
induced
I’. It is not
the
original B
ALSO
NOTE: The induced
Bz<=
/span>’
tends to oppose the
change
in flux
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Slide 1
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