sources of magnetic field what are some sources of magnetic field? moving point charges: current...
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Sources of Magnetic Field
What are some sources of Magnetic Field?
Moving Point Charges:
30
20
4
ˆ
4
r
rvqr
rvqB
Current Elements
(Biot-Savart Law):
30
20
4
ˆ4
rrId
rrId
Bd
r
points from source to field point
(CH 27, #23)
CBA
BABAC
,sin|||||| Direction
from RHR I
27
70
A/N10
A/mT104
Sources of Magnetic Field
Magnets
(magnetic dipole moment ):
N
S
B
E
Electric dipole
Magnetic dipole moment
N
S
B
30 24 x
B
Magnetic field along axis, distance x away from center:
Exercise: Compass and B-Field
Take your compass and place it flat on your table.
The compass needle will lie along the direction of the total local magnetic field.
Hold your cylindrical bar magnet so its axis is perpendicular to the compass needle and it is in the plane of the compass (see figure).
Hold magnet far from the compass to start. Slowly bring it closer to the compass and observe the needle deflection.
Biot-Savart and RHR
The Biot-Savart Law is
A small piece of a current carrying wire lying along the y-axis is indicated on the sketches.
P2
(x1,0)
(0,y)
● P1 (0,y1)
30
20
4
ˆ
4 rrId
rrId
Bd
P3
(x3,0)
a) What is the direction of the magnetic field at the point P1?
b) What is the direction of the magnetic field at the point P2?
c) What is the direction of the magnetic field at the point P3?
Id
More Biot-Savart
A long straight current carrying wire, length 2L, is oriented along the y-axis as shown. A small piece of the wire at (0,y) is indicated. A field point P is on the x-axis at a distance xP.
What is ?
P
(xp,0)
(0,y)
Id
-L
L
r
30
20
4
ˆ
4 rrId
rrId
Bd
What is ?
dI
Example: Biot-Savart Applied to Long Straight Wire
P
(xp,0)
(0,y)
Id
-L
L
30
20
4
ˆ
4 rrId
rrId
Bd
jIdyId ˆ)(
jyixr Pˆˆ
2/3223
22
)( yxr
yxr
P
P
kdyIxyIdykxIdy
jjyIdyijxIdyjyixjIdyrId
pp
pP
ˆ))(()0)()(()̂)()((
ˆˆ))((ˆˆ))(()̂ˆ(ˆ)(
Note: points into page as expectedk̂
30
20
44 r
rId
r
r̂IdBd
kyx
dyIx
P
P ˆ)(4 2/322
0
L
L/
PP
)yx(
dyk̂)Ix(BdB
232204
Limits of integration from sketch; smallest y-value for wire is –L, highest y-value for wire is L.
2222/322 )( ubb
u
ub
du
222222222
2222/322
2
)(
Lxx
L
Lxx
L
Lxx
L
yxx
y
yx
dy
pPpPpP
Ly
LypP
L
L P
kLxx
IL
Lxx
LkIxB
PP
PP
P
ˆ2
2ˆ)(4
220
2220
Magnetic Field from Straight Wire
On bisector:
“Infinite” wire: take limit as
2202 LRR
ILBB
L
RI
B
20
Direction: Magnetic field makes closed circles around wire.
RHR applied to straight wires:
Point thumb in current direction; fingers curl in direction of magnetic field.
R
Exercise: Magnetic Field from Wires
1) Take your compass and place it flat on your desk or notebook.
2) The compass needle will lie along the direction of the total local magnetic field.
3) Take out your batteries/pack and an alligator clip lead (or several).
4) Lay the a length of wire on top of the compass so that the wire is parallel to the needle (don’t complete circuit yet.
5) Arrange so that the current in the wire will flow from south to north (north should point towards the front of the room). Don’t complete circuit yet.
6) Use RHR to PREDICT the direction the needle should deflect.
7) Check your prediction.
8) Repeat, but this time with wire beneath compass.
9) Repeat, but this time with wire perpendicular to compass needle. Try both above and below the compass.
ConcepTest #15:
Two long straight wires each carry the same current and are the same distance from the origin. For the wire on the y-axis, the current goes into the page, and for the wire on the x-axis, the current comes out of the page, as shown.
Consider the following directions. Hold up as many cards as you need to specify the direction.
What is the direction of the net magnetic field at the origin?
1. Up 4. Left
2. Down 5. Into the page
3. Right 6. Out of the page
(back of card) No Direction
Use formula to get magnitude of B
Use RHR to get direction of B
DRAW vectors
Add VECTORS!
ConcepTest #16:
Wire 1 carries current I going up as shown. Wire 2 is a distance d away from wire 1, and carries current I going down as shown.
Consider the following directions. Hold up as many cards as you need to specify the direction.
1. Up 4. Left
2. Down 5. Into the page
3. Right 6. Out of the page
(back of card) No Direction
1 2
d
a) What is the direction of the magnetic field at the location of wire 2 due to the current in wire 1?
b) What is the direction of the magnetic force acting on wire 2 due to the magnetic field from wire 1?
BIdFd
20 ˆ4 r
rIdBd
Idr
Id
RHR applied to arcs or loops of current at center of arc or loop:
Curl fingers in current direction; thumb points in magnetic field direction.
Example: Biot-Savart Applied to Circular Arc (at center of arc)
Direction: Magnetic field perpendicular to plane of loop.
RHR applied to arcs or loops:
Curl fingers in current direction, thumb points in direction of magnetic field.
2
0
4 RId
dBB
f
i
dRI
RIRd
440
20
Rdd
Biot-Savart Applied to Circular Arc (at center of arc)
Circular Loop at center of loop
RI
RI
B
2
4400
RI
B
40
RI
B20
2/322
20
)(
24 Rx
RIB
Example: Biot-Savart Applied to Circular Loops (on axis of loop)
Direction: Magnetic field along axis of loop.
RHR applied to loops:
Curl fingers in current direction, thumb points in direction of magnetic field.
AIRI 2
Note resemblance to field from magnetic moment!
2/322
20
)(2
4 RxRI
Baxis
B
Equivalence of Current Loops and Bar Magnets: MAGNETIC MOMENTS
N
S
B
30 24 x
Baxis
Magnetic Field of Special Current Configurations
Axis of Loop of radius R, distance x from center:
Direction: fingers along current; thumb along magnetic moment
2/322
20
)(
24 Rx
RIB
Center of Loop of radius R, so that x = 0:
Direction: fingers along current; thumb in field direction
RI
RI
B2
24
00
Center of Arc of radius R, angle :
Direction: fingers along current; thumb in field direction
22400
RI
RI
B
Axis of bar magnet with moment , distance x from center:
Direction: magnetic moment from South pole to North pole
30 24 x
B
Axis of long solenoid, away from edges, with n turns per meter:
Direction: fingers along current; thumb in field direction
nIB 0
Long (“infinite”) Straight Wire, distance R away from wire:
Direction: fingers along current; thumb in field direction
RI
B2
40