Download Review (Faraday`s law, magnetic field, Gauss`s law

Electricity and Magnetism
Review
Faraday’s Law
Lana Sheridan
De Anza College
Dec 3, 2015
Overview
• Faraday’s law
• Lenz’s law
• magnetic field from a moving charge
• Gauss’s law
Definition
Reminder:(30.18)
Magnetic
Fluxof magnetic flux
S
field B that makes an
s case is
S
B
(30.19)
a, then u 5 908 and the
the plane as in Figure
maximum value).
a weber (Wb); 1 Wb 5
Magnetic flux
The magnetic flux
e plane is a
magnetic
r to the plane.
S
dA
u
S
dA
Figure 30.19 The magnetic
flux through an area element dA
S
S
is B ? d A 5 B dA cos u, where
Sa magnetic field through
of
d A is a vector perpendicular to
the surface. X
ΦB =
a surface A is
B · (∆A)
Units: Tm2
If the surface is a flat plane and B is uniform, that just reduces to:
ΦB = B · A
n a magnet is moved
When the magnet is
When the magnet is held
d a loop of wire
moved away from the
Changing
emf there is no
ected
to a sensitive flux andstationary,
loop, the ammeter sh
induced current in the
eter, the ammeter
that the induced curr
theof wire there is no
that a current
is opposite
that shown
When ais magnet is atloop,
rest even
near when
a loop
potential
inside
the loop.
ed in the
loop.
part a .
difference
across the magnet
ends ofisthe
wire.
I
I
N
S
N
b
S
N
c
S
When the magnet is
toward a loop of wire
stationary, there is no
connected
to
a
sensitive
Changing flux and emf
induced current in th
ammeter, the ammeter
loop,
eventhe
when the
shows
a current
is
When the north pole of
the that
magnet
is moved
towards the
loop,
magnet
is
inside
the
induced
in
the
loop.
magnetic flux increases.
I
N
.1 A simple experiment
at a current is induced
hen a magnet is moved
away from the loop.
a
A current flows clockwise in the loop.
S
N
b
S
the magnet is held
moved away from the
ary,
there
is
no
Changing flux andloop,
emfthe ammeter shows
d current in the
that the induced current
ven when
the the north pole of
is opposite
that shown
in away from the loop,
When
the magnet
is moved
a
t is inside
the
loop.
part
.
the magnetic flux decreases.
I
N
S
N
S
c
A current flows counterclockwise in the loop.
Faraday’s Law
Faraday’s Law
If a conducting loop experiences a changing magnetic flux through
the area of the loop, an emf EF is induced in the loop that is
directly proportional to the rate of change of the flux, ΦB with
time.
Faraday’s Law for a conducting loop:
E=−
∆ΦB
∆t
Faraday’s Law
Faraday’s Law for a coil of N turns:
EF = −N
∆ΦB
∆t
if ΦB is the flux through a single loop.
Changing Magnetic Flux
The magnetic flux might change for any of several reasons:
• the magnitude of B can change with time,
• the area A enclosed by the loop can change with time, or
• the angle θ between the field and the normal to the loop can
change with time.
Additional examples, video, and practice available at
Lenz’s Law
Lenz’s Law
30-4 Lenz’s Law
An induced current has a direction such that the
Faraday
proposed
histhe
law of induc
magnetic fieldSoon
dueafter
to the
current
opposes
devised a rule for determining the direction of an
change in the magnetic flux that induces the
current.
An induced current has a direction such that the ma
opposes the change in the magnetic flux that induces t
S
The magnet's motion
creates a magnetic
dipole that opposes
the motion.
N
N
µ
Furthermore, the direction of an induced emf is th
a feel for Lenz’s law, let us apply it in two different
where the north pole of a magnet is being moved to
1. Opposition
to Pole
Movement.
The approach
Basically, Lenz’s
law let’s us
interpret
the minus
Fig. 30-4 increases the magnetic flux through
sign in the equation we write to represent
current in the loop. From Fig. 29-21, we know t
Faraday’s Law. netic dipole with a south pole and a north po
:
moment !
is directed from south to north.
∆Φ
i
B
increase
being
E:= − caused by the approaching mag
thus !
) must ∆t
face toward the approaching no
S
30-4). Then the curled – straight right-hand rul
Fig. 30-4 Lenz’s law at work. As the
the current induced in the loop must be counte
magnet is 1moved toward the loop, a current
If we
Figure from Halliday, Resnick, Walker, 9th
ed. next pull the magnet away from th
:
B
:
30-5b.Thus, Bind and B are now in the same direction.
In Figs. 30-5c and d, the south pole of the magnet approaches and retreats from the loop, respectively.
(a)
(b)
Lenz’s Law: Page 795 in Textbook
Increasing the external
field B induces a current
with a field Bind that
opposes the change.
The induced
current creates
this field, trying
to offset the
change.
The fingers are
in the current's
direction; the
thumb is in the
induced field's
direction.
Decreasing the external
field B induces a current
with a field Bind that
opposes the change.
Increasing the external
field B induces a current
with a field Bind that
opposes the change.
(c)
Decreasing the external
field B induces a current
with a field Bind that
opposes the change.
B ind B
B ind
B
i
i
i
i
B
B
B ind
B ind
B ind
B ind
B
i
B
B ind
B ind
B
B ind
i
B
(a)
i
i
i
B
B
B ind B
i
i
i
B
B ind
(b)
B ind
(c)
(d)
:
A
)
x
s
o
e
-
uniform magnetic field that exists
throughout a conducting loop, with the difield B(t)
perpendicular
to the
The rection
graph givesof
thethe
magnitude
of a uniform magnetic
field
that plane
exists throughout
a conducting
directionof
of the
of the loop.
Rankloop,
the with
fivethe
regions
field perpendicular to the plane of the loop. Rank the five regions
the graph according to the magnitude of
of the graph according to the magnitude of the emf induced in the
emffirst.
induced in the loop, greatest first.
loop,the
greatest
Faraday’s Law Question
B
a
b
c
d
e
t
Faraday’s Law
30-4 LENZ’S LAW
795
wnward,
CHECKPOINT
2
The figure
shows three situations
in which identical circular
crease in
:
The figure
shows
three
situations
in whichfields
identical
circular
conconducting
loops
are
in
uniform
magnetic
that
are either
Bind diducting
loops
are
in
uniform
magnetic
fields
that
are
either
inincreasing (Inc) or decreasing (Dec) in magnitude at identical
then the
creasing (Inc) or decreasing (Dec) in magnitude at identical
rates. In each, the dashed line coincides with a diameter. Rank the
ward flux
rates. In each, the dashed line coincides with a diameter. Rank
situations
accordingaccording
to the magnitude
of the current
induced
ig. 29-21
the situations
to the magnitude
of the current
in-in the
loops, duced
greatest
first.
5a.
in the
loops, greatest first.
oses the
ean that
he magInc
Inc
Dec
rom the
but it is
Inc
Dec
Inc
ward inn in Fig.
.
(a)
(b)
(c)
gnet ap-
Magnetic fields from moving charges and currents
We are now moving into chapter 29.
Anything with a magnet moment creates a magnetic field.
This is a logical consequence of Newton’s third law.
Magnetic fields from moving charges
A moving charge will interact with other magnetic poles.
This is because it has a magnetic field of its own.
The field for a moving charge is given by the Biot-Savart Law:
B=
µ0 q v × r̂
4π r 2
Magnetic fields from moving charges
B=
1
Figure from rakeshkapoor.us.
µ0 q v × r̂
4π r 2
Magnetic fields from currents
B=
µ0 q v × r̂
4π r 2
We can deduce from this what the magnetic field do to the current
in a small piece of wire is.
Current is made up of moving charges!
qv = q
q
∆s
=
∆s = I∆s
∆t
∆t
We can replace q v in the equation above.
This element of current creates a
Magnetic fields from currents
magnetic field at P, into the page.
ids
ds
A current-length element
a differential magnetic
int P. The green ! (the
w) at the dot for point P
:
dB is directed into the
i
θ
ˆr
r
P
d B (into
page)
Current
distribution
This is another version of the Biot-Savart Law:
Bseg =
µ0 I ∆s × r̂
4π r 2
where Bseg is the magnetic field from a small segment of wire, of
length ∆s.
ummation
Magnetic fields
fromfield
currents
The magnetic
vector
because of
at any point is tangent to
Magnetic field
around a wire segment, viewed end-on:
is a scalar,
a circle.
being the
Wire with current
into the page
a currentB
(29-1)
that points
y constant,
B
(29-2)
f the cross
The magnetic field lines produced by a current in a long straight wire
Fig. 29-2
HALLIDAY REVISED
Magnetic fields from currents
to determine
direction of the field lines (right-hand rule):
GNETICHow
FIELDS
DUE TOthe
CURRENTS
es the dio a curFig. 29-2,
:
eld B at
perpennd dion of the
b) If the
to the
hed rathe page,
i
B
B
i
(a )
The thumb is
current's dire
The fingers re
the field vecto
direction, whi
tangent to a c
(b )
Here is a simple right-hand rule for finding the direction of the mag
set up by a current-length element, such as a section of a long wire:
Right-hand rule: Grasp the element in your right hand with your extended th
Magnetic field from a long straight wire
The Biot-Savart Law,
Bseg =
µ0 I ∆s × r̂
4π r 2
implies what the magnetic field is at a perpendicular distance R
from an infinitely long straight wire:
B=
µ0 I
2πR
(The proof requires some calculus.)
Gauss’s Law for Magnetic Fields
Gauss’s Law for magnetic fields.:
I
B · dA = 0
Where the integral is taken over a closed surface A. (This is like a
sum over the flux through many small areas.)
We can interpret it as an assertion that magnetic monopoles do
not exist.
The magnetic field has no sources or sinks.
Gauss’s Law for Magnetic Fields
I
32-3 INDUCED MAGNETIC
FIELDS
B · dA = 0
ore complicated than
e does not enclose the
f Fig. 32-4 encloses no
ux through it is zero.
only the north pole of
el S. However, a south
ace because magnetic
like one piece of the
I encloses a magnetic
ttom faces and curved
s B of the uniform and
A and B are arbitrary
s of the magnetic flux
B
Surface II
N
S
Surface I
PA R T 3
863
cates level of problem difficulty
ILW
Interactive solution is at
Law
for
Magnetism
Ch32 # 2
Flying
Circus
of Physics and Question,
at flyingcircusofphysics.com
on Gauss’s
available in The
The figure shows a closed surface. Along the flat top face, which
has a radius of 2.0 cm, a perpendicular magnetic field B of
for Magnetic Fields
page. The total elect
magnitude 0.30 T is directed outward. Along the flat bottom face,
hrough
each of five
faces
a die
(singular
E " (3.00
a magnetic
flux of
0.70 of
mWb
is directed
outward.given
What by
are !
the
onds. What is the m
" #N
whereand
N (" 1 to 5) is the num(a)Wb,
magnitude
(b)
direction
(inward
or
outward)
of
the
magnetic
flux through
field
that is the
induced
The flux is positive (outward) for N even
curved
part
of
the
surface?
cm
and
(b)
5.00
cm?
or N odd. What is the flux through the sixth
a closed surface. Along
has a radius of 2.0 cm, a
:
field B of magnitude
ard. Along the flat botux of 0.70 mWb is diare the (a) magnitude
rd or outward) of the
the curved part of the
B
••8
Nonunifor
29 shows a circular
cm in which an elec
the plane of the pa
concentric circle of
(0.600 V & m/s)(r/R)t
magnitude of the ind
cm and (b) 5.00 cm?
••9
Uniform ele
Summary
• Faraday’s law
• Lenz’s law
• magnetic field from a moving charge
• Guass’s law
Homework
Study!