Download Lecture 28: Force on a wire, torque on a wire loop

Electricity and Magnetism
Force on a Curved Wire
Torque on a Wire Loop
Lana Sheridan
De Anza College
Nov 6, 2015
Last time
• the electron-Volt
• cyclotrons
• synchotrons
• magnetic force on a wire
duce the current into the segment at one end and take it out at the other end.
Warm Up Question
CHECKPOINT
4 through a wire in a uniform magnetic field B.
A current i passes
:
The figure
shows a current
wire
a uniform
field B , as
The magnetic
force iFthrough
theinwire.
The magnetic
field is oriented
so well as
B acts aon
:
the magnetic
FB is
acting
on the wire.
The field
is oriented
so that
the force is maxithat theforce
force
maximum.
In what
direction
is the
field?
mum. In what direction is the field?
y
i
x
z
FB
(A) +y
Sample Problem
(B) −y
Magnetic force on a wire carrying curre
(C) +z
(D) horizontal
−z
A straight,
length of copper wire has a current
! 28 A through it. What are the magnitude and direction
:
duce the current into the segment at one end and take it out at the other end.
Warm Up Question
CHECKPOINT
4 through a wire in a uniform magnetic field B.
A current i passes
:
The figure
shows a current
wire
a uniform
field B , as
The magnetic
force iFthrough
theinwire.
The magnetic
field is oriented
so well as
B acts aon
:
the magnetic
FB is
acting
on the wire.
The field
is oriented
so that
the force is maxithat theforce
force
maximum.
In what
direction
is the
field?
mum. In what direction is the field?
y
i
x
z
FB
(A) +y
(B) −y
Sample Problem
←
Magnetic force on a wire carrying curre
(C) +z
(D) horizontal
−z
A straight,
length of copper wire has a current
! 28 A through it. What are the magnitude and direction
:
Overview
• force on a current carrying wire
• torque on a wire loop in a magnetic field
e wire
The
is perpendicular
Magnetic Force on
a force
Current
Carrying Wire
nt. The
to both the field and the length.
on the
FB
28-28)
nts by
olated
introd.
i
φ
B
L
A wire carrying current i
:
makes an angle f with magnetic field B .
The wire hasFlength
= I L × BL in the field and
:
length vector L (in the direction of the cur-
Fig. 28-16
:
:
:
HALLIDAY REVISED
Problem
A straight, horizontal length of copper wire has a current I = 28 A
through it. The linear density (mass per unit length) of the PA
wireRisT 3
46.6 g/m.
751
TIC FORCE ON A CURRENT-CARRYING WIRE
What are the magnitude and direction of the minimum magnetic
fieldthe
B needed
the wire – that is, to balance the
imagine
wire to suspend
The force is perpendicular
gravitational
force
on
it?
each segment. The
to both the field and the length.
the forces on the
FB
(28-28)
nt of currents by
ng as an isolated
be a way to introhe other end.
i
φ
B
L
A wire carrying current i
:
makes an angle f with magnetic field B .
1
Halliday, Resnick,
Walker,
751. L in the field and
The
wire page
has length
Fig. 28-16
Problem
At equilibrium
F B = Fg
Problem
At equilibrium
F B = Fg
ILB sin θ = mg
B = 1.6 × 10−2 T
Magnetic Force on a Current Carrying Wire
What about the case where the wire is curved, or the B-field is not
uniform?
A wire segment
The magnetic force
on any Wire
Magnetic
S
ape
carrying a Force on a Current
S
SCarrying
S
segment d s is I d s ! B and
B the case where the wire is curved, or the B-field is not
magnetic
field
What
about
is directed out of the page.
uniform?
magnetic
force.
I
S
B
S
ds
Now we must consider each infinitesimal segment of wire:
S
S
of the page for the directions
dF = I ds ×Bof B and d s Sin
ut
Figure
n be considered
as an alternative definition of B . That is,
S that B could depend on s if the field is not uniform.
Remember
netic field B in terms of a measurable force exerted on a
ape carrying a
S
S
S
segment d s is I d s ! B and
magnetic field B
Magnetic Force on
a Current
Carrying
is directed
out of the
page.
magnetic force.
I
S
B
Wire
S
ds
S
Z
ut of the page for the directions
of B and d S
s Sin Figure
F = I ds ×B
n be considered
as an alternative definition of B . That is,
S
netic field B in terms of a measurable
force exerted on a
S of the wire.
The integral is taken over the length
he force is a maximum when B is perpendicular to the ele-
d the left edge, (b) in the plane of the page and toward the
ward
out of the page,
or (d)
downward into
the page?
Example
29.4:
Magnetic
Force
on a Wire
What is the net force on this semicircular wire loop in a uniform
ductorB-field, given that the current in the loop is I?
y
circuit and carries a curtic field is directed along
nd direction of the magon the curved portion.
S
B
I
u
S
ds
R
du
u
cts, we see that the force
S
and the force F 2 on the
S
de than F 1 because the
raight portion?
x
I
Figure 29.20 (Example 29.4) The
magnetic force on the straight portion
of the loop is directed out of the page,
and the magnetic force on the curved
portion is directed into the page.
ying wire in a magnetic
Equation 29.12 to find
S
F1 5 I 3 d S
s 3 B 5 I 3 B dx k^ 5 2IRB k^
S
b
R
d the left edge, (b) in the plane of the page and toward the
ward
out of the page,
or (d)
downward into
the page?
Example
29.4:
Magnetic
Force
on a Wire
What is the net force on this semicircular wire loop in a uniform
ductorB-field, given that the current in the loop is I?
y
circuit and carries a curtic field is directed along
nd direction of the magon the curved portion.
S
B
I
u
S
ds
R
du
u
cts, we see that the force
S
and the force F 2 on the
S
de than F 1 because the
raight portion?
x
I
Figure 29.20 (Example 29.4) The
magnetic force on the straight portion
First:
symmetry.of the
Theloop
wire
is in the x, y -plane,
ying wire
in a use
magnetic
is directed out of the page,
magnetic
force
can
only
point
in
the k-direction.
and
the
magnetic
force
on the curved
Equation 29.12 to find
portion is directed into the page.
S
F1 5 I 3 d S
s 3 B 5 I 3 B dx k^ 5 2IRB k^
S
b
R
B = Bj, any
d the left edge, (b) in the plane of the page and toward the
ward
out of the page,
or (d)
downward into
the page?
Example
29.4:
Magnetic
Force
on a Wire
What is the net force on this semicircular wire loop in a uniform
ductorB-field, given that the current in the loop is I?
circuit and carries a curtic field is directed along
nd direction of the magon the curved portion.
y
S
B
I
u
S
ds
R
du
cts, we see that the force
S
and the force F 2 on the
S
de than F 1 because the
raight portion?
u
x
I
Figure 29.20 (Example 29.4) The
magnetic force on the straight portion
First:
symmetry.of the
Theloop
wire
is in the x, y -plane,
ying wire
in a use
magnetic
is directed out of the page,
magnetic
force
can
only
point
in
the k-direction.
and
the
magnetic
force
on the curved
Equation 29.12 to find
portion is directed into the page.
Mentally, break the wire into two pieces, the bottom, straight piece
b
R
and
top curved
piece.
Sthe S
^
^
F 1 5 I 3 d s 3 B 5 I 3 B dx k 5 2IRB k
S
B = Bj, any
Example 29.4: Magnetic Force on a Wire
Bottom segment:
F = IL × B
Since L = 2Ri, and B = Bj:
F = 2RIB k
Top segment:
Z
F=I
ds ×B
Example 29.4: Magnetic Force on a Wire
Top segment:
Z
F=I
ds ×B
Example 29.4: Magnetic Force on a Wire
Top segment:
Z
F=I
ds ×B
The top segment is semi-circular. A path along it is a circular arc:
θ).
s = Rθ → ds = R dθ(θ̂
Z
F = I
RB(− sin θ) dθ k
Zπ
= −IRB k sin θ dθ
0
= −IRB k [cos θ]π
0
= −2RIB k
Example 29.4: Magnetic Force on a Wire
Bottom segment:
F = 2RIB k
Top segment:
F = −2RIB k
Total:
F=0
This is a general result. The force on any loop of wire in a
uniform magnetic field is zero!
HALLIDAY REVISED
9
16:19 Pageon
751a Loop of Wire with a Current
Torque
HALLIDAY REVISED
Or, how to turn electricity into motion.
PA R T 3
ON AConsider
CURRENT-CARRYING
WIRE field751
two wires in a magnetic
with currents flowing in
Fig. 28-16 A wire carrying current i
:
makes an angle f with magnetic field B .
The wire has length L in the field and
:
length vector L (in the direction of the cur:
:
:
rent). A magnetic force FB ! iL ! B acts
on the wire.
opposite directions.
28-8 MAGNETIC FORCE ON A CURRENT-CARRYING WIRE
e wire
The force is perpendicular
t. The
ght
or the field is not
uniform,
we can
the wire
to both
the field
andimagine
the length.
The force is perpendicula
on
ightthe
segments and apply Eq. 28-26 to each segment. The
to both the field and the l
! i dL ! B,
i
φ
B
FB
(28-28)
φ
:
L
:
i
nts by
:
dF B
φ
B
y
28-28)
d
-
whole is then the vector sum of all the forces on the
F
In the differential limit, weB can write
ultant force on any given arrangement of currents by
i
L
r that arrangement.
L
olated
bear
in mind that there is no such thing as an isolated
intro-of length dL. There must always be a way to introment
28-16 A wire carrying current i
Fig. 28-16 A wire carrying cur
d.segment at Fig.
:
one end and take it out at the other end.
makes
angle f with
magnetic
field B . directions.
makes an angle f with magnetic fi
They
will an
experience
forces
in opposite
The wire has length L in the field
The wire has length L in the field and
:
:
length vector L (in the direction o
length vector L (in the direction of the cur:
:
)
B
FB
e
e
e
The force is perpendicular
to both the field and the length.
Torque on a Loop of Wire with a Current
This is the situation
occurs when a loop of wire is placed in a
752 thatCHAPTER
28 MAGNETIC FIELDS
B-field.
28-9 Torque on
F
Much of the world’s
are the magnetic fo
forces that a magneti
Figure 28-18 sho
i
N
S
loop immersed in a
–F
duce a torque on th
B
many essential detai
Fig. 28-18 The elements of an electric
of a magnetic field o
loop of wire, carrying a
action.a
These opposingmotor.A
forces rectangular
on opposite
sides of the loop creates
current and free to rotate about a fixed axis,
Figure 28-19a sh
torque on the loop.
is placed in a magnetic field. Magnetic
i through uniform m
forces on the wire produce a torque that roits long sides, labeled
tates it.A commutator (not shown) reverses
into the page), but it
the direction of the current every half-revorent into and out of t
i
Torque on a Loop of Wire with a Current
The current on the two sides away from the axle gives an upward
752
CHAPTER
28 right.
MAGNETIC FIELDS
force on the left and downward
on the
28-9 Torque on
F
Much of the world’s
are the magnetic fo
forces that a magneti
Figure 28-18 sho
i
N
S
loop immersed in a
–F
duce a torque on th
B
many essential detai
Fig. 28-18 The elements of an electric
of a magnetic field o
motor.A rectangular loop of wire, carrying a
On the two ends that connect to the axle, the force is action.
zero when
current and free to rotate about a fixed axis,
Figure 28-19a sh
the loop lays flat
parallel
to
the
B-field.
is placed in a magnetic field. Magnetic
i through uniform m
forces on the wire produce a torque that roits equal
long sides,
When the loop tates
rotates,
the forces(not
on shown)
those reverses
two ends are
and labeled
it.A commutator
into
the
page),
but it
the direction of the current every half-revoopposite.
rent into and out of t
lution so that the torque always acts in the
To define the or
same direction.
i
B
(b)
Torque on a Loop
of Wire with a Current
gular loop, of
d carrying a curform magnetic
align the normal
ion of the field. (a)
king in the direcd. (b) A perspechow the right:
ction of n
, which is
ane of the loop. (c)
from side 2.
cated.
F1
n
Side 1
i
θ
b
Side 2
Side 3
Rotation
(c)
B
F3
τF = r × F ;
τnet =
X
i
τnet = r1 × F1 + r2 × F2
τi
3
Torque on a Loop of Wire with a Current
B
gular loop, of
d carrying a curform magnetic
align the normal
ion of the field. (a)
king in the direcd. (b) A perspechow the right:
ction of n
, which is
ane of the loop. (c)
from side 2.
cated.
(b)
F1 = Ia × B = iaB j = −F3
F1
Side 1
i
θ
b
Side 2
Side 3
Rotation
(c)
n
B
F3
τnet = r1 × F1 + r3 × F3
F
Torque
onB a Loop
(b) of Wire with a Current
a
3
19 A rectangular loop, of
and width b and carrying a curocated in a uniform magnetic
orque t acts to align the normal
with the direction of the field. (a)
as seen by looking in the direce magnetic field. (b) A perspece loop showing how the right:
e gives the direction of n
, which is
cular to the plane of the loop. (c)
ew of the loop, from side 2.
rotates as indicated.
F1
Side 1
i
θ
b
Side 2
Side 3
Rotation
(c)
n
B
F3
τnet = r1 × F1 + r2 × F2
b
b
=
(IaB) sin θ +
(IaB) sin θ [cw in diag.]
2
2
Noting that the area of the loop A = ab:
τ = IAB sin θ
as discussed with regard to Figure 29.21 and is zero when the field is
normal
to theon
plane
of the loop
5 0). with a Current
Torque
a Loop
of (u
Wire
S
F2
b–
2
!
S
A
u
u
b– sin u
2
S
B
O
"
S
F4
gure 29.22 An edge view
he loop in Figure 29.21
h the normal to the loop
an angle u with respect to
We can make
magnetic field.
When the normal to the loop
makes an
u with
τ angle
= IAB
sinthe
θ
magnetic field, the moment arm
for the torque
is (b/2)
sin u. by
expression
more
compact
this
where n̂ is normal to the loop plane.
τ = IA × B
defining A = An̂
(i) Rank the magnitudes of the torques acting on the rectangu), and
(c) shown
Figure
29.24
from highest to lowest.
Torque
on edge-on
a Loopin of
Wire
Question
entical Which
and carry
therectangular
same current.
the magnitudes
of the
loops(ii)
hasRank
the largest
magnitude of
of the
cting on
the
rectangular
loops
shown
in
Figure
29.24
from highnet force acting on it?
ick Quiz
loop (seen
es the greatr (c)? Which
atest net
a
b
c
(A) a
ment (B)
of abCoil
(C) cof 25 turns of wire and carries a current of 15.0 mA.
m consists
all the same
ane of(D)
the coil.
1
Jewett, 9th ed.
e moment Serway
of the &
coil.
(i) Rank the magnitudes of the torques acting on the rectangu), and
(c) shown
Figure
29.24
from highest to lowest.
Torque
on edge-on
a Loopin of
Wire
Question
entical Which
and carry
therectangular
same current.
the magnitudes
of the
loops(ii)
hasRank
the largest
magnitude of
of the
cting on
the
rectangular
loops
shown
in
Figure
29.24
from highnet force acting on it?
ick Quiz
loop (seen
es the greatr (c)? Which
atest net
a
b
c
(A) a
ment (B)
of abCoil
(C) cof 25 turns of wire and carries a current of 15.0 mA.
m consists
all the same ←
ane of(D)
the coil.
1
Jewett, 9th ed.
e moment Serway
of the &
coil.
(i) Rank the magnitudes of the torques acting on the rectangu), and
(c) shown
Figure
29.24
from highest to lowest.
Torque
on edge-on
a Loopin of
Wire
Question
entical Rank
and carry
the same current.
(ii) Rank
theonmagnitudes
of
the magnitudes
of the torques
acting
the rectangular
cting on
the
rectangular
loops
shown
in
Figure
29.24
from
highloops from highest to lowest.
ick Quiz
loop (seen
es the greatr (c)? Which
atest net
a
b
c
(A) a, b, c
ment (B)
of ab,Coil
a, c
(C) c,
m consists
of b,
25aturns of wire and carries a current of 15.0 mA.
c, a, b
ane of(D)
the coil.
1
Jewett, 9th ed.
e moment Serway
of the &
coil.
(i) Rank the magnitudes of the torques acting on the rectangu), and
(c) shown
Figure
29.24
from highest to lowest.
Torque
on edge-on
a Loopin of
Wire
Question
entical Rank
and carry
the same current.
(ii) Rank
theonmagnitudes
of
the magnitudes
of the torques
acting
the rectangular
cting on
the
rectangular
loops
shown
in
Figure
29.24
from
highloops from highest to lowest.
ick Quiz
loop (seen
es the greatr (c)? Which
atest net
a
b
c
(A) a, b, c
ment (B)
of ab,Coil
a, c
(C) c,
← of wire and carries a current of 15.0 mA.
m consists
of b,
25aturns
c, a, b
ane of(D)
the coil.
1
Jewett, 9th ed.
e moment Serway
of the &
coil.
Torque on a Coil of Wire with a Current
τ = IA × B
Remarkably, that equation also holds for other shapes of loop as
long as they are flat (in one plane). A is the area of the loop.
For a coil of N loops stacked together, the effect of each loop adds
up:
τ = NIA × B
Electric Motors
This effect can be used to turn electricity into mechanical work.
1
Figure from hyperphysics.phys-arstr.gsu.edu
Electric Motors
Either direct current (DC) or alternating current (AC) can be used
for a motor.
1
Figure from hyperphysics.phys-arstr.gsu.edu
Summary
• force on a curved wire
• torque on a wire loop in a magnetic field
• motors
Homework Serway & Jewett:
• PREVIOUS: Ch 29, Problems: 27
• NEW: Ch 29, Problems: 33, 37, 41, 49, 51