Download Lecture 15 Magnetostatic Field – Forces and the Biot

Lecture 15
Magnetostatic Field – Forces
and the Biot-Savart Law
Sections: 7.1, 8.1, 8.2
Homework: See homework file
LECTURE 15
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Magnetic Forces Review – Ampère’s Force Law
Ampère’s force law (aka motor equation)
Fm  IL(a I  B), N
magnetic flux density vector
(the magnetic field force vector)
L
I
B
Fm
video on wire with current in magnet
http://www.fyzikalni-experimenty.cz/en/electromagnetism/threadwith-current-in-a-magnetic-field/
• force does not depend on the position if I = const. and B = const.
(expected for a large flat field source)
You would like to increase the magnetic force Fm on the wire
N times while still using the same magnet. How would you
propose to achieve that?
LECTURE 15
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Magnetic Forces Review – Lorentz’ Force Law
Lorentz’ force law follows directly from Ampère’s force law
• for a very short current element IdL (B is locally constant)
dFm  I (dL  B), N where
IdL  J
ds dL  J dsdL
  JdV   v vdV , A×m
dV
Ia L
 dFm  (  v dv )( v  B), N

dQ
• integrate over charge volume
 Fm  Qv  B , N (holds for point charge Q)
LECTURE 15
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Magnetic Forces Review – Homework
Solve the problems below:
#1. Magnetic field B = 0.5az T is due to a magnet with flat poles.
What is the force exerted on a piece of wire with current I = 2 A of
length L = 10 cm and orientation given by the unit vector
1a y  1a z
Ans.: Fm  0.0707a x , N
aL 
2
#2. An electron (e = −1.602x10−19 C) is moving along a Cu wire
due to electric field E = 1 kV/m. Its mobility is μe = 0.0032
m2/(Vs). The wire is immersed in magnetic field B = 1.2 T, which
is orthogonal to the wire. Find the strength of the magnetic force
exerted on the electron.
Ans.: Fm ≈ 6.15x10−19 N
LECTURE 15
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Magnetic Forces – the 2nd of Ampère’s Force Laws
Ampère’s force law for two wires with current
F | Fm12 || Fm 21 | km
I1I 2 L

, N
• in vacuum (in SI)
0
km 
, 0  4 107 H/m
2
I1
F12

F12
F21
I1
I2
I2
F21
F 0 I1I 2
• force per unit length F   
, N/m
L 2 
 l1  l 2
, N/m
• compare with the 2-D Coulomb force law F  
2 0 
1
• note the typical 1/ρ behavior for a 2-D problem
LECTURE 15
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Magnetic Force – 2, Homework
#3. What is the magnitude of the current flowing in two parallel
wires, which are 10 cm apart (center to center), if the force per
unit length between them is Fꞌ = 10−3 N/m? The currents in both
wires have the same magnitude.
Ans.: 22.36 A
video on the magnetic forces between two wires with current
http://www.fyzikalni-experimenty.cz/en/electromagnetism/interactions-between-twoconductors-amperes-force-law/
https://www.youtube.com/watch?v=kapi6ZDvoRs
(some serious currents running here)
LECTURE 15
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Magnetic Field of Straight Wire with Current
 putting together the two Ampère laws
 I1I 2 L
Fm12  I1L  B2 and Fm12 
2 
 I2
| B 2 |
, T
2 
I
• right-hand rule of current direction and
magnetic field B (Oersted, Ørsted, 1820)
video on Oersted’s experiment
http://www.fyzikalni-experimenty.cz/en/electromagnetism/oerstedexperiment-longitudinal/#
• using cylindrical coordinates
 I
B
a
2 
l
• compare with the E field of a line charge E 
 a
2 
1
LECTURE 15
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Biot-Savart Law for the Magnetic Flux Density B
• Biot and Savart set out to prove that the magnetic field of a very
short current element IdL will produce the typical 1/R2 behavior
• by measuring the torque on a magnetic needle they found (1825)
 IdL
dB 
 2  sin  , T
4 R

Km
K m 0  10
B
I
 R
7
current element
 IdL  a R
, T
dB 

2
4
R
 IdL  R
dB 

4
R3
dL
analogy with dQ in electrostatics
• indeed this expression defaults to Ampère’s field in the case of a
straight wire
LECTURE 15
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Biot-Savart Law: Application to Infinite Straight Wire
 I ( dL  R )

dB 

B
3
4
4
R

I (a z  R )
 R3 dz
z

R   a   ( z  z )a z
|R | R   2  ( z  z ) 2

P(  ,  , z )
a z  R  a z    a   ( z  z )a z    a
Assuming z = 0

 B  a
4
R

I 
 (  2  z2 )3 / 2 dz

I 
z
 B  a
2  2  2  z 2

0
I
 a
2
LECTURE 15
z  Idz a z
0
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Biot-Savart Law for the Magnetic Intensity H
the magnetic force Fm is proportional to B in magnitude, i.e., B is
the magnetic “force vector” (as E is for the electric field)
B depends not only on the magnetic sources (current elements) but
also on the magnetic properties of medium (similar to E)
1 dQ
 IdL  a R
dE 
 2 a R , V/m
dB 

, T
2
4 R
4
R
the magnetic field vector H (field intensity) depends on the
sources only (similar to D in electricity)
H  B/
1 IdL  a R
, A/m
dH 

2
4
R
for a straight wire with current
dD 
H  a
LECTURE 15
1 dQ
 2 a R , C/m 2
4 R
I
2
, A/m
l
D  a
2
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Biot-Savart Law for Line and Volume Currents
• for a line current I along a contour
H( P ) 
1
4
I (Q )dL(Q )  a R (Q )
, A/m

2
RQP
C
R QP  r  r
C
Q
IdL
P
R QP
r
r
0
• in general, current distribution is given by current density J, A/m2
IdL  Jdv 
( Jdv )  a R
R2
1
J (Q )  a R (Q )
dv, A/m
 H( P ) 

2
4 v
RQP
dH 
LECTURE 15
J , A/m 2
S
I  J S , A
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Biot-Savart Law for Surface Currents
• surface current density K, A/m, defined for thin layers with current
I   J  ds  
S
w x
0
w
J z dx dy

0 

 I   K z dy, A
Kz
x
x
dy
y
K  K z a z , A/m
0
x
K

P
Jdx  J x, A/m
R QP
x 0
Jdv  (J x)ds  Kds
( Kds )  a R
1
dH 
H
2
R
4
H
K
K  aR
ds
 RQP
2
s
ds
dw
dL
r
r
0
dH is always orthogonal to the current element (IdL, Jdv, or Kds)
and it obeys the right-hand rule
LECTURE 15
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Surface Current Density – Examples
#1. Current I = 5 A flows along a thin strip of width 1 cm. What
is the surface current density if the distribution is uniform?
#2. A coil has 5000 tightly wound turns and it carries 1 A current.
What is the equivalent surface current density of the coil current if
the coil’s length is 15 cm?
L  15 cm
Total current of all turns:
It 
I 1 A
K
LECTURE 15
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You have learned:
that the magnetic field interacts with moving charges (currents)
that moving charges (currents) generate magnetic field
how to calculate the magnetic force exerted on one and two wires
through Ampère’s force laws
how to calculate the magnetic force exerted on a moving charge
(Lorentz’ force)
that the magnetic field of a current element obeys the inversesquare law and is orthogonal to the current direction (Biot-Savart)
how to use Biot-Savart’s law and the superposition principle to
compute the magnetic field of current distributions (line contours,
surfaces and volumes carrying current)
LECTURE 15
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