Download Ch. 28: Sources of Magnetic Fields

Ch. 28: Sources of Magnetic Fields
Electric Currents Create Magnetic Fields
A long, straight wire
A current loop
A solenoid
Slide 24-14
Biot-Savart Law
•  Current produces a magnetic field
•  The Biot-Savart law gives
the magnetic field arising
from an infinitesimal
current element:
~ ⇥ r̂
µ0 I dL
~
dB =
4⇡
r2
•  Permeability of free space:
µ0 = 4⇡ ⇥ 10
7
T m/A
•  Integrating along the wire:
~ =
B
Z
~ ⇥ r̂
µ0 I dL
4⇡
r2
~ =
B
Z
~ ⇥ r̂
µ0 I dl
4⇡ r2
At location P, what is the direction of the
infinitesimal contribution dB created by the current
in dl?
A)  Up the page
B)  Directly away from dl
(in the plane of the page)
C)  Into the page
D)  Out of the page
E)  Some other direction
~r
~ =
B
Z
~ ⇥ r̂
µ0 I dl
4⇡ r2
P
What is the magnitude of
~ ⇥ r̂
dl
?
r2
θ
~r
dl
Origin
a)  dl sin ✓
r2
c)  dl cos ✓
r2
b) dl sin ✓
r
dl cos ✓
d)
r3
e)
dl
r2
Magnetic field of long straight wire
~ =
B
sin(⇥) = sin(
Z
⇥) = p
µ0 I a sin ✓
Bz =
dx
2
4⇡
a r
sin ✓ = sin(⇡ ✓) = d/r
• as a goes to infinity:
µ0 I
Bz !
2⇡d
x
Z
~ ⇥ r̂
µ0 I dl
4⇡ r2
x2 + y 2
µ0 I
) Bz =
4⇡
Z
a
a
d dx
(d2 + x2 )3/2
µ0 I
2a
p
Bz =
4⇡ d d2 + a2
Clicker Question
CT 32.14b
What is the direction of the Force
acting on the Red Wire?
I
A)  Up
B)  Right
C)  Left
D)  Into the Page
E)  Out of the Page
©University of Colorado, Boulder (2008)
I
CT 32.7a
A long straight wire carries current I out of the
page. An electron travels towards the wire from
the right. Which way is the force on the electron?
I
-
e
v
A:
B:
C: ←
D: ↓
E: 0
©University of Colorado, Boulder (2008)
B field due to a current loop
•  Only component of dB in x
direction doesn’t cancel out
µ0 I dL
dBx =
cos
2
2
4⇥ x + a
B=
µ0 Ia 2
(
2
2 x +a
2
32
)
•  For large distances (x >> a),
this reduces to
B=
µ0 Ia 2
2x 3
~ =
B
Z
~ ⇥ r̂
µ0 I dL
4⇡
r2
CT 32.16
What is the direction of the B- Field at point P?
A:
B:
C: 0
D: →
E: Other
©University of Colorado, Boulder
(2008)
Magnetic dipoles
•  A small current loop constitutes a
magnetic dipole.
•  Magnitude of B decreases
with distance as 1/x3
•  Its dipole moment is µ = IA,
with A the loop area.
•  For an N-turn loop, µ = NIA.
•  The direction of the dipole
moment vector is
perpendicular to the loop area.
•  The fields of electric and
magnetic dipoles are similar
far from their sources, but
differ close to the sources.
Clicker Question
CT 32.14c
Two loops of wire have current going around
in the same direction.
The magnetic forces on the loops are:
i2
A: Attractive
B: Repulsive
C: Net force is zero.
i1
©University of Colorado, Boulder (2008)
Ampère’s law
•  From Zthe Biot-Savart law,
~
~ =
B
µ0 I dL ⇥ r̂
4⇡
r2
•  Ampere’s law can be derived
I
~ = µ0 Ienc
~ · dl
B
•  For steady currents
•  The integral is taken
around any closed loop,
and Ienc is the current
encircled by that loop.
•  Useful only for current
distributions with
symmetry
Field due to a long cylindrical conductor
outside conductor: Iencl=I
µ0 I
2 Br = µ0 I =) B =
2 r
inside conductor: Iencl = J⇡r 2 , J =
I
⇡R2
2
µ0 Ir
r
2
2 Br = µ0 J r = µ0 I 2 =) B =
2 R2
R
Solenoids
•  A solenoid is a long, tightly wound
coil of wire.
•  When a solenoid’s length is
much greater than its
diameter, the magnetic field
inside is nearly uniform
except near the ends, and the
field outside is very small.
.
Clicker Question
•  In the ideal limit of an infinitely long solenoid, the field
inside the solenoid is uniform everywhere, and the field
outside is zero.
•  Application of Ampère’s law shows that the field of an
infinite solenoid is B = µ0nI, where n is the number of
turns per unit length
I
~ = BL = µ0 nIL
~ · dl
B
B = µ0 nI
Magnetism in matter
•  Magnetism in matter arises from atomic current loops associated
with valence electrons having orbital angular momentum and spin.
Classical picture of
magnetic dipole
moment arising
from orbiting
electron
Magnetism in matter
•  In ferromagnetic materials
like iron, strong interactions
among individual magnetic
dipoles result in large-scale
magnetic properties,
including strong attraction
to magnets.
Inducing a Magnetic Moment in a Piece of Iron
Electromagnet
•  Use current to induce magnetic moment in iron
Magnetism in matter
•  Paramagnetic materials
exhibit much weaker
magnetism.
•  Diamagnetic materials
respond oppositely, and are
repelled by magnets.
B = Bvac (1 + )
Hysterisis
•  Consider again a charged particle moving near a long wire with
current I.
Q
v
I
•  Recall the Lorentz force is given by
~
F~ = q~v ⇥ B
•  What frame of reference of reference do we use to determine the
velocity?
•  Answer: Same frame in which the magnetic field was determined.
•  What if we switch frames to a frame where the velocity is
zero?