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8/12/2019 4 Magnetic Field Due to a Current-Carrying Wire, Biot-Savart Law APC
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Magnetic Field
dueto
a Current-Carrying Wire
Biot-Savart
Law
AP Physics C
Mrs. Coyle
Hans Christian Oersted, 1820
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Magnetic fields are caused by currents.
Hans Christian Oersted in 1820s showed
that a current carrying wire deflects acompass.No Current in the Wire
Current in the Wire
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Right Hand Curl Rule
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Magnetic Fields of Long Current-Carrying
Wires
B = mo I
2p rI = current through the wire (Amps)
r = distance from the wire (m)
mo = permeability of free space
= 4px 10-7T m / A
B = magnetic field strength (Tesla)
I
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Magnetic Field of a Current
Carrying Wire
http://www.walter-
fendt.de/ph14e/mfwire.htm
http://www.walter-fendt.de/ph14e/mfwire.htmhttp://www.walter-fendt.de/ph14e/mfwire.htmhttp://www.walter-fendt.de/ph14e/mfwire.htmhttp://www.walter-fendt.de/ph14e/mfwire.htmhttp://www.walter-fendt.de/ph14e/mfwire.htm8/12/2019 4 Magnetic Field Due to a Current-Carrying Wire, Biot-Savart Law APC
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What if the current-carrying wire is not
straight? Use the Biot-Savart Law:
2
0
4 r
rdsB
Id
p
m
Note:
dBis perpendicular to ds and r
Assume a small segment ofwire dscausing a field dB:
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Biot-Savart Law allows us to calculate
the Magnetic Field Vector
To find the total field, sum up the contributionsfrom all the current elements Ids
The integral is over the entire current distribution
24
Io d
r s rB
2
0
4 i
iiI
r
rdsB
p
m
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Note on Biot-Savart Law
The law is also valid for a current consisting
of charges flowing through space
dsrepresents the length of a small segmentof space in which the charges flow.
Example: electron beam in a TV set
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Comparison of Magnetic to Electric Field
Magnetic Field
B proportional to r2
Vector
Perpendicular to FB, ds, r
Magnetic field lines haveno beginning and no end;
they form continuous
circles
Biot-Savart Law
Amperes Law (where
there is symmetry
Electric Field
E proportional to r2
Vector
Same direction as FE
Electric field lines beginon positive charges and
end on negative charges
Coulombs Law
Gausss Law (where
there is symmetry)
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Derivation of B for a Long, Straight
Current-Carrying Wire
Integrating over all the
current elements gives
2
1
1 2
4
4
Isin
Icos cos
o
o
B d
a
a
sin d dx
s r k
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If the conductor is an infinitely long,
straight wire, q1= 0 and q2= p
The field becomes:
2IoB
a a
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B for a Curved Wire Segment
Find the field at point Odue to the wire segment
AACC:
B=0 due to AA and CC
Due to the circular arc:
qs/R,will be in radians
4
Io
B R
24
Io
d
r
s rB
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B at the Center of a Circular Loop of Wire
Consider the previous
result, with q= 2p
I I
I
24 4
2
o o
o
B
R R
B
R
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Note The overall shape of the magnetic field of
the circular loop is similar to the magneticfield of a bar magnet.
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B along the axis of a Circular Current Loop
Find Bat pointP
2
32 2 2
2
Io
x
RB
x R
24
Io
d
r
s rB
If x=0, B same as at center of a loop
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If x is at a very large distance away
from the loop.
x>>R:
2 2
3 3
2 2 2 22
I Io o
x
R RB
xx R
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Magnetic Force Between Two Parallel
Conductors
The field B2due to
the current in wire 2
exerts a force on
wire 1 ofF1= I1B2
1 2
12
I Io
F
a
I 2
22
o
B
a
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Magnetic Field at Center of a SolenoidB = mo NI
L
N: Number of
turnsL: Length
n=N/L
------------------------L----------------
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Direction of Force Between Two
Parallel Conductors
If the currents are in the:
samedirection the wires attracteach other.
oppositedirections the wires repeleach other.
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Magnetic Force Between Two
Parallel Conductors, FB
Force per unit length: 1 2
2
I IB o
F
a
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Definition of the Coulomb
The SI unit of charge, the coulomb, is
defined in terms of the ampere
When a conductor carries a steady current
of 1 A, the quantity of charge that flows
through a cross section of the conductor in
1 s is 1 C
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Biot-Savart Law: Field produced by current
carrying wires
Distance afrom long straight wire
Centre of a wire loop radius R
Centre of a tight Wire Coil with Nturns
Force between two wires
a
II
l
F
p
m
2
210
a
IB
p
m
2
0
R
IB
2
0m
R
NI
B 2
0m
2
0
4 r
rdsB
Id
p
m