Transcript B - Personal.psu.edu

Sources of the Magnetic
bb
Field
Lesson 8
Biot-Savart Law
Force between parallel conductors
Ampere’s Law
Use of Ampere’s Law
E Field
Interaction between stationary charges
is mediated by an Electric Field
E Field
B Field
Interaction between currents is
mediated by a Magnetic Field
B Field
Electric Dipoles align along
an E Field
E
-
+
E Dipole
Magnetic Dipoles align along
a B Field
B
S
N
B Dipole
Biot-Savart
Law
Biot-Savart Law
For a point source charge dq the
E and B Fields due sources
electric field produced at a position r from the source is
1 dq r
1 dq
ˆ
dE=
r
=
2
3
4pe0 r
4pe0 r
For a length of wire ds with current I
(this is the same as charge per unit length dq with velocity v )
the magnetic field produced at a position r from
the source d s is
I
m0 I
m
0
dB =
ds  ˆr =
ds  r
2
3
4p r
4p r
B Field due to source
m 0 = 4p  10 - 7 Tm
A
= Permeability
B Field due to loop
Right Hand Rules
Right Hand Rule I
Need B and v then get FB
Right Hand Rule II
Need I get B
Magnetic Fields from conductors
of different shapes
k
B
Field
from
wire
Magnitude of Magnetic Field about
 thin conductor
B(r ) = B(r ) =
R q ds
r
m0I
2pr


 




m0I

B(r) =
d B = 


2 sin q ds  k
r
p

2



 - 




2
2 
r
= s + R 
sin q = sin ( p - q ) =
2
2 ; r
s +r





 m 0 I 


r
=
B (r ) 

3 ds k




2
2
p
2
2


+r )
-  (s









B Field from closed wire
segment
I
r
q
X
m0I
q
B=
4 pr
B Field from closed loop
I
B
B=
m0 I
2r
Picture
I
B
k
Two parallel conductors
Parallel wires I
j
i
B2
I1
a
I2
L
B1
Force on wire
F1 =
1
due to field of wire 2
I1 L  B 2
= I1L j B 2
B2
F1 = -
m
0
m 0 I2
= i
2p a
I1 I2 L
m 0 I1 I2 L
j i = +
k
2pa
2pa
Force on wire
2 due to field of wire 1
I2 L  B 1 = - I2 L j  B 1
F2 =
B1 = -
F2 =
m
0
I1 I2 L
2pa
m
0
I2
2 pa
j i = -
i
m0
I1 I2 L
2pa
k
k
Two parallel conductors
j
i
Parallel wires II
B2
F1
a
F2
B1
Parallel wire rules
Parallel Currents Attract
Anti Parallel Currents
Repel
a = 1m
I1 = I2 = I = 1A
Definition
of Amp
m
F=
0
= 2  10 N
-7
m
2p
This defines the Amp . as:
2  10 N m
measured between wires one meter apart
This in turn defines the Coulomb as :
The quantity of charge that flows through any
cross section of a conductor in one second when
a steady current of one amp is flowing .
The current flowing when a force of
-7
Recall Gauss ' s Law

Gauss's
Law
for
E
Q=
d A =
e
E
EA
Gaussian surface
as E = constant and E || d A everywhere on surface
For magnetism

B  dA = 0
as magnetic field has ALWAYS been observed to
be produced by magnetic dipoles
In electrostatics, potential difference is defined by
 for B
Gauss's
Law
where integral is evaluated along ANY path
b
V
ab
=
E  ds
a
starting at
a
and finishing at

b, thus
E  ds = 0
as E is a conservative force per unit charge
For magnetism, in general


B  ds  0
and
B  ds =
Amperian Loop
B

ds
Amperian Loop
: B =
constant
and
B
ds

Amperes
B  d s Law
= B ( RI )
I
B
R
Amperian Loop

ds
Amperian Loop
for steady current
 m 0 
= 
 I (2 p R ) = m 0 I
2 p R 
true for ANY closed path
Amperes Law
Amperes Law II

B  ds = m 0 ITotal
ITotal = sum of all currents threading
loop
Equivalence of Laws
Gauss’s Law is
Equivalent to Coulombs
Law
Biot-Savart Law is
Equivalent to Amperes
Law.
Toroids and Solenoids I
I
Toroid
Solenoid


Toroids
and Solenoids
II
B
B  ds =
B  ds = B
loop
2
path 1

w

path 1
B  d s = m 0 NI = BL
loop
1
3 l
ds = BL
N
B = m0
L
4
Amperian Loop
I=
m 0 nI
Picture
One can perform the same Amperian
calculation
the Toroid III
Toroids
and for
Solenoids
and get the same result
B = m 0 nI
where n is the number of turns per length
N
n=
but now
2pR , R = Radius of Toroid
instead of
N
n=
L , for Solenoid
Galvanometer
Galvanometer