Physics 202, Lecture 13
Today’s Topics

Magnetic Forces: Hall Effect (Ch. 27.8)

Sources of the Magnetic Field (Ch. 28)
 B field of infinite wire
Force between parallel wires
 Biot-Savart Law
Examples: ring, straight wire
 Ampere’s Law:
 infinite wire, solenoid, toroid
The Hall Effect (1)
Potential difference on current-carrying conductor in B field:
positive charges moving
counterclockwise: upward force,
upper plate at higher potential
negative charges moving
clockwise: upward force
Upper plate at lower potential
Equilibrium between electrostatic & magnetic forces:
V
Fup = qvd B
Fdown = qEind = q H
VH = vd Bw = "Hall Voltage"
w
1
The Hall Effect (2)
VH = vd Bw =
I = nqvd A = nqvd wt
Hall coefficient:
Text: 27.47
RH !
IB
nqt
VH
1
=
IB nqt
Hall effect: determine sign,density of charge carriers
(first evidence that electrons are charge carriers in most metals)
Magnetic Fields of Current Distributions
Recall electrostatics: Coulomb’s Law)
!
!
! kqq!
F = qE
F = 2 r̂
r
Two ways to calculate E directly:
– Coulomb’s Law
!
dE =
(point charges)
"Brute force"
1 dq
r̂
4!" 0 r 2
– Gauss’ Law
! ! qin
E
! i dA = "
0
"High symmetry"
! 0 = (4" k)#1
ε0 = 8.85x10-12 C2/(N m2):
permittivity of free space
2
Magnetic Fields of Current Distributions
!
! r̂
!
F = km # Idl ! # I "dl " ! 2
r
! !
!
F = " Idl ! B
(2 circuits)
2 ways to calculate B:
– Biot-Savart Law
(“Brute force”)
!
! µ0 I dl " r̂
dB =
4! r 2
– Ampere’s Law
(“high symmetry”)
! !
"! Bidl = µ0 I enclosed
×I
–AMPERIAN LOOP
µ0 = 4! km
µ0 = 4πx10-7 T⋅m/A:
permeability of free space
Magnetic Field of Infinite Wire
Result (we’ll derive this both ways):
! µ0 I
B=
"ˆ
2! r
right-hand rule
closed circular loops centered on current
3
Quick Question 1: Magnetic Field and Current
Which figure represents the B field generated by the
current I?
xxxxxx
xxxxxx
xxxxxx
xxxxxx
xxxxxx
.......
.......
.......
.......
.......
......
......
......
......
......
xxxxxx
xxxxxx
xxxxxx
xxxxxx
xxxxxx
Quick Question 2: Magnetic Field and Current
Which figure represents the B field generated by the
current I?
4
Forces between Current-Carrying Wires
A current-carrying wire experiences a force in an external
B-field, but also produces a B-field of its own.
Magnetic Force:
Ib
!
F
d
!
F
Ia
Parallel currents: attract
!
F
d
Ib
!
F
Ia
Opposite currents: repel
Example: Two Parallel Currents
Force on length L of wire b due to field of wire a:
Ba =
! ! µ I I L
Fb = " I b dl ! Ba = 0 b a
2# d
µ0 I a
2! d
d
Ib
!
F
!
F
Ia
x
Force on length L of wire a due to field of wire b:
Bb =
µ0 I b
2! d
! ! µ I I L
Fa = " I a dl ! Bb = 0 a b
2# d
Text: 28.4, 28.58
5
Biot-Savart Law...
…add up the pieces
!
!
! µ0 I dl ! r̂ µ0 I dl ! r!
dB =
=
4! r 2
4! r 3
dl
θ
r
X
dB
I
B field “circulates” around the wire
Use right-hand rule: thumb along I,
fingers curl in direction of B.
B Field of Straight Wire, length L
(Text example 28-13) B field at point P is:
B=
µ0 I
(cos !1 # cos ! 2 )
4"a
When L=infinity
(text example 28-11):
µI
B= 0
2!a
θ1
θ2
(return to this later with Ampere’s Law)
Text: 28.43
6
B Field of Circular Current Loop on Axis
Use Biot-Savart Law (text example 28-12)
µ0 IR 2
Bx =
2( x 2 + R 2 ) 3 / 2
Bcenter =
µ0 I
2R
See also: center of arc (text example 28-13)
Text: 28.37
B of Circular Current Loop: Field Lines
B
7
Ampere’s Law
I
Ampere’s Law:
"!
! !
Bi dl = µ0 I encl
any
closed path
ds
 applies to any closed path, any static B field
 useful for practical purposes
only for situations with high symmetry
 Generalized form (valid for time-dependent fields):
Ampere-Maxwell (later in course)
Ampere’s Law: B-field of ∞ Straight Wire
Use symmetry and Ampere’s Law:
×I
! "
B
#! i dl = µ0 I encl
Choose loop to be circle of radius R centered on the
wire in a plane ⊥ to wire.
Why?
Magnitude of B is constant (function of R only)
Direction of B is parallel to the path.
! "
# B i dl = # Br d! = 2" rB = µ I
0 encl
= µ0 I
B=
µ0 I
2! r
Quicker and easier method for B field of infinite wire
(due to symmetry -- compare Gauss’ Law)
8
B Field Inside a Long Wire
Text example: 28-6
Total current I flows through wire of
radius a into the screen as shown.
What is the B field inside the wire?
By symmetry -- take the path
to be a circle of radius r:
! !
B
" idl = B2! r
•Current passing through circle:
! !
B
" idl = B2! r = µo I encl
xxxxx
xxxxxxxx
xxxxxxxxx
r
xxxxxxxx
xxxxx
a
r2
I encl = 2 I
a
µ Ir
⇒
Bin = 0 2
2! a
B Field of a Long Wire
Inside the wire: (r < a)
B=
µ0 I r
2! a 2
• Outside the wire:
(r>a)
B=
µ0 I
2! r
a
B
r
Text: 28.31
9
Ampere’s Law: Toroid
(Text example 28-10)
N = # of turns
Show:
µ0 NI
Bin =
2! r
Ampere’s Law: Toroid
Toroid: N turns with current I.
•
Bθ=0 outside toroid!
•
(Consider integrating B on circle
outside toroid: net current zero)
•
Bθ inside: consider circle of radius r,
centered at the center of the toroid.
! !
Bid
" s = B2! r = µo I enclosed
B=
•
•
•
•
•
xx x
x
x
x
x
x
r xx
x
x x xx
x
•
• B•
•
•
•
•
•
µ0 NI
2! r
10
B Field of a Solenoid
Inside a solenoid: source of uniform B field
• Solenoid: current I flows through a wire
wrapped n turns per unit length on a
cylinder of radius a and length L.
L
a
To calculate the B-field, use Biot-Savart
and add up the field from the different loops.
If a << L, the B field is to first order contained within the solenoid,
in the axial direction, and of constant magnitude.
In this limit, can calculate the field using Ampere's Law!
Ampere’s Law: Solenoid
The B field inside an ideal solenoid is:
B = µ0nI
n=N/L
segment 3
at ∞
ideal solenoid
11
Solenoid: Field Lines
Right-hand rule: direction of field lines
north pole
south pole
Like a bar magnet! (except it can be turned on and off)
Text: 28.68
12