skc biot-savart law
TRANSCRIPT
-
8/13/2019 Skc Biot-Savart Law
1/28
The Biot-Savart Law
AP Physics C
Biot-Savart sounds like Leo Bazaar
-
8/13/2019 Skc Biot-Savart Law
2/28
Biot & Savart produced an equation that gives themagnetic field at some point in space in terms ofthe current that produces the field.
a wire carrying steady current I, themagnetic field dB at some point P hasthe following properties:
The vector dB is to both ds
(direction of I) & to the vector rdirected from the element ds to thepoint P.
The magnetic field wraps in circles
around a wire. The direction is foundusing the right-hand rule.
Thumb of right hand in the directionof the current, fingers curl in thedirection of B. Field at any point is
tangent to curl
dlI
-
8/13/2019 Skc Biot-Savart Law
3/28
1. Which drawing below shows the correctdirection of the magnetic field,B, at the pointP?
A. I.B. II.
C. III.
D. IV.
Direction of Magnetic Field
I II III IV V
i i ii
P P P P P
iB
B
Bintopage
Bintopage
Bintopage
-
8/13/2019 Skc Biot-Savart Law
4/28
The magnitude of dB is inverselyproportional to r2, where r is the distancefrom the element ds to the point P.
The magnitude of dB is proportional to thecurrent I and to the length ds of theelement.
The magnitude of dB is proportional to sin q,where qis the angle between the vectors ds
and r.
Biot-Savart law:
To determine the total magnetic field B at somepoint due to a conductor of specified size, addup contribution from all elements ds that make
up the conductor (integrate)!
2
o
r4
sdsIdB
qinm0= permeability constant
exactly m/AT104 7
-
8/13/2019 Skc Biot-Savart Law
5/28
Magnetic Field of a long straight wire Consider a thin, straight wire
carrying a constant current I alongthe y axis. To determine the totalmagnetic field B at the point P at adistance R from the wire:
Let ds = dx, then dssin q becomesdxsin q.
The contribution to the totalmagnetic field at point P from eachelement of the conductor ds is:
2
o
r
sindx
4
IdB
-
8/13/2019 Skc Biot-Savart Law
6/28
Express r in terms of R and x.
Express sin qin terms of R and r.
21
22xR
R
r
Rsin
2
122222
xRrxRr
2
o
2
o
r
sindx
4
I
r
sindx
4
IdB
23
22
o
23
22
o
212222
o
xR
dx
4
RI
xR
dxR
4
IB
xR
R
xR
dx
4
IB
-
8/13/2019 Skc Biot-Savart Law
7/28
From the table of integrals:
2
12222
322
xRR
x
xR
dx
R4
I211
R4
I
R4
IB
R4
IB
RRR4IB
xR
x
R4
RI
xRR
x
4
RIB
ooo
21
221
2
o
21
2221
22
o
21
222
o
21
222
o
R2
IB
o
-
8/13/2019 Skc Biot-Savart Law
8/28
Find B for a conductor of length l
From the table of integrals:
l
l
l
l
l
l
l
l
l
l
l
l
R 23
22
o
23
22
o
21
2222
o
2
o
2
o
xR
dx
4
RI
x
dxR
4
IB
xR
R
xR
dx
4
IB
r
sindx
4
I
r
sindx
4
IdB
2
12222
322
xRR
x
xR
dx
-
8/13/2019 Skc Biot-Savart Law
9/28
-
8/13/2019 Skc Biot-Savart Law
10/28
Just add up all of the contributionsds
to thecurrent, but now distance r=Ris constant,and .
Notice that . So the integralbecomes
For a complete loop, f= 2, so
Bat Center of a Circular Arc of Wire
fRdds
f f
m
0 02
0
4ds
R
idBB
R
iRd
R
iB
fmf
m f
44
0
02
0
R
iB
fm
4
0
sdr
R
iB
2
0m
B at center of a full circle
-
8/13/2019 Skc Biot-Savart Law
11/28
Find Magnetic Field on the Axis of aCircular Current Loop
Consider a circular loop of wire of radius R inthe yz plane and carrying a steady current I:
Note: Each element of length ds is
to the r from ds to point P.ds90sinds
& the direction of dB
from ds is at an angleqwith the x axis.
-
8/13/2019 Skc Biot-Savart Law
12/28
The direction of the net magnetic fieldis along the x axis and directed away
from the circular loop.
cosxR
ds
4
I
cos
xR
ds
4
I
cos
22
o
22
o
dB
dB
dBdBx
21
2222
o
xR
R
xR
ds
4
I
dB
ds
23
22
o
xR4
RIB
-
8/13/2019 Skc Biot-Savart Law
13/28
The sum of the elements of length dsaround the closed current loop is the
circumference; s = 2RThe net magnetic field B at point P is :
23
22
2
o
2
3
22
2
o
2322
o
xR2
RIB
xR4
RI2B
R2xR4
RIB
-
8/13/2019 Skc Biot-Savart Law
14/28
For large distances along the x axis fromthe current loop, where x is very large in
comparison to R:
3
2
o
6
2
o
3
2
2
o
23
2
2
o
23
22
2
o
x2
RIB
x2
RI
x2
RIB
x2
RI
xR2
RIB
-
8/13/2019 Skc Biot-Savart Law
15/28
B field for Circular Loop
R
B
x
x
0
R
iB
2
0m
23
22
2
o
xR2
RIB
3
2
o
x2
RIB
-
8/13/2019 Skc Biot-Savart Law
16/28
Remember how to Calculate Electric Field
Either:
Coulomb's Law:
Gauss' Law
What are the analogous equations for the
Magnetic Field?
-
8/13/2019 Skc Biot-Savart Law
17/28
Calculation of Magnetic Field
"High symmetry"
I
Two ways to calculate the Magnetic Field:
Biot-Savart Law:
Ampere's Law
-
8/13/2019 Skc Biot-Savart Law
18/28
Draw an amperian loop around a systemof currents (like the two wires at right). Theloop can be any shape, but it must be closed.
Add up the component of along the loop,for each element of length dsaround this
closed loop. The value of this integral is proportional to
the current enclosed:
Amperes Law
B
encisdB 0m
i1 i
2
Amperes Law
-
8/13/2019 Skc Biot-Savart Law
19/28
Magnetic Field Outside a LongStraight Wire with Current
We already used the Biot-Savart Law to showthat, for this case, .
Lets show it again, using Amperes Law:
First, we are free to draw an Amperian loop of
any shape, but since we know that themagnetic field goes in circles around a wire,lets choose a circular loop (of radius r).
ThenBand dsare parallel, and Bis constanton the loop, so
And solving for B gives our earlier expression.
r
iB
m
2
0
encisdB 0m
Amperes Law
enc
irBsdB0
2
m
r
iB
m
2
0
-
8/13/2019 Skc Biot-Savart Law
20/28
Magnetic Field Inside a LongStraight Wire with Current
Calculate B inside the wire. Draw a circular Amperian loop around the
axis, of radius r < R.
The enclosed current is less than the totalcurrent, because some is outside the
Amperian loop. The amount enclosed is
2
2
R
riA
A
Ji en
total
enc
inside a straight wire
2
2
002
R
riirBsdB
enc mm
rR
iB
2
0
2
mrR
~1/r~r
B
-
8/13/2019 Skc Biot-Savart Law
21/28
Two cylindrical conductors each carrycurrent Iinto the screen as shown.The conductor on the left is solid andhas radius R=3a. The conductor on
the right has a hole in the middle &carries current only between R=a &R=3a.
(a) BL(6a)< BR(6a) (b) BL(6a)= BR(6a) (c) BL(6a)> BR(6a)
(a) BL(2a)< BR(2a) (b) BL(2a)= BR(2a) (c) BL(2a)> BR(2a)
What is the relation between the magnetic field at R =
2a for the two cases (L=left, R=right)?
1B
1A
3a
a
3a
I I
2a
What is the relation between the magnetic field at R= 6a for the two cases (L=left, R=right)?
-
8/13/2019 Skc Biot-Savart Law
22/28
What is the relationbetween the
magnetic field at R =6a for the two cases(L=left, R=right)?
(a) BL(6a)< BR(6a) (b) BL(6a)= BR(6a) (c) BL(6a)> BR(6a)
1A
Amperes Law can be used to find the field in both cases. The Amperian loop in each case is a circle of radius R=6a in the plane
of the screen.
3a
a
3a
I I
2a
The field in each case has cylindrical symmetry, being everywheretangent to the circle. Therefore the field at R=6a depends only on the total currentenclosed!!
In each case, a total currentI
is enclosed.
-
8/13/2019 Skc Biot-Savart Law
23/28
(a) BL(2a)< BR(2a) (b) BL(2a)= BR(2a) (c) BL(2a)> BR(2a)
What is the relation between
the magnetic field at R = 2afor the two cases (L=left,R=right)?
1B
3a
a
3a
I I
2a
For the LEFT conductor:
Once again, the field depends only on how much current is enclosed.
For the RIGHT conductor:
I9
4I)3(
)2(I2
2L
a
a
I8
3I)3(
)2(I22
22R
aa
aa
-
8/13/2019 Skc Biot-Savart Law
24/28
Solenoids We saw earlier that a complete
loop of wire has a magnetic field atits center:
We can make the field stronger bysimply adding more loops. A many
turn coil of wire with current iscalled a solenoid.
R
iB
2
0m
The field near the wires is still circular,but farther away the fields blend into anearly constant field down the axis.
The actual field looks more like this:
-
8/13/2019 Skc Biot-Savart Law
25/28
Solenoids
We can use Amperes Law to calculate Binside the solenoid.
Characterize the windings in terms ofnumber of turns per unit length, n. Eachturn carries current i, so total current over
length his inh.
Compare with electric field in a capacitor. Like a capacitor, the field is uniform inside (except near the ends), but the
direction of the field is different.
Approximate that the field is constant inside and zero outside (just likecapacitor).
inhiBhsdBenc 00
mm
only section that has non-zerocontribution
inB0
m ideal solenoid
-
8/13/2019 Skc Biot-Savart Law
26/28
Toroids
Notice that the field of the solenoid sticks outboth ends, and spreads apart (weakens) at theends.
We can wrap our coil around like a doughnut, sothat it has no ends. This is called a toroid.
Now the field has no ends, but wraps uniformlyaround in a circle.
What is B inside? We draw an Amperian loopparallel to the field, with radius r. If the coil hasa total ofNturns, then the Amperian loopencloses currentNi.
iNirBsdBenc 00
2 mm
r
iNB
2
0
m inside toroid
-
8/13/2019 Skc Biot-Savart Law
27/28
Recall that a wire carrying a current in amagnetic field feels a force.
When there are two parallel wires carryingcurrent, the magnetic field from one causes aforce on the other.
When the currents are parallel, the two wires arepulled together.
When the currents are anti-parallel, the two wiresare forced apart.
Force Between Two Parallel Currents
FF
To calculate the force on bdue to a,abba
BLiF
d
ia
2
0
mR
iB
m2
0
d
LiiF
ba
ba
m
2
0 Force between two parallel currents
BLiFB
-
8/13/2019 Skc Biot-Savart Law
28/28
3. Which of the four situations below has thegreatest force to the right on the centralconductor?
A. I.
B. II.
C. III.
D. IV.E. Cannot
determine.
Forces on Parallel Currents
I.
II.
III.
IV.
Fgreatest?