Units
of
magne-c
field:
Tesla


1
T
=
10000
Gauss


Surface
of
earth:
0.5×10‐4
T
Bar
magnet:


1.5
T
Strong
superconduc-ng
magnet:
30
T


dB =

µ0I4π

dS × r̂r2

dS

r

Current


Biot‐Savart
Law


dB

µ0 = 4π ×10

−7T im / A

Infinite
straight
wire
example


Z


a


a
.


b
.
c
.


d
.


e
.


The
segment
of
wire
(total
length
=
6R)
is
formed
into
the
shape
shown
and
carries
a
current
I.
What
is
the
magnitude
of
the


resul-ng
magne-c
field
at
the
point
P?


a.



b.


c.



d.


e.


The
segment
of
wire
(total
length
=
6R)
is
formed
into
the
shape
shown
and
carries
a
current
I.
What
is
the
magnitude
of
the


resul-ng
magne-c
field
at
the
point
P?


Ques-on
1:
Equal
currents
of
magnitude
I
travel
into
the
page
in
wires
M
and
N.
Eight
direc-ons
are
indicated
by
leYers
A


through
H.



The
direc-on
of
the
magne-c

field
at
point
P
is


a. 
B.
b. 
C.
c.  D.
d.  E.
e. 
F.


Solenoid


Z


a


B =µ0I2

a2

a2 + z2( )32

Approximate
solenoid
as
bunch
of
separate
loops


Magne-c
field
at
the
center
of
the
solenoid


L/2


L
:
length
of
solenoid
n:
coil
density
(#/m)
a:
radius
of
the
coil


Looks
like
integra-on
is
required
again,
but
how?


a


Z


dz
 Current
through
each
coil:
I
Number
of
coils
in
dz:
ndz


dB = nµ0I2

a2dz

a2 + z2( )32

B =µ0nI2

a2dz

a2 + z2( )32

−L /2

L /2

∫

B =µ0nIa

2

2L / 2

a2 a2 + L4

2⎛⎝⎜

⎞⎠⎟

12

−−L / 2

a2 a2 + L4

2⎛⎝⎜

⎞⎠⎟

12

⎡

⎣

⎢⎢⎢⎢⎢⎢

⎤

⎦

⎥⎥⎥⎥⎥⎥

Z


dz
Current
through
each
coil:
I
Number
of
coils
in
dz:
ndz


B =µ0nIa

2

2L / 2

a2 a2 + L4

2⎛⎝⎜

⎞⎠⎟

12

−−L / 2

a2 a2 + L4

2⎛⎝⎜

⎞⎠⎟

12

⎡

⎣

⎢⎢⎢⎢⎢⎢

⎤

⎦

⎥⎥⎥⎥⎥⎥

B =µ0nIa

2

2L

a2 a2 + L4

2⎛⎝⎜

⎞⎠⎟

12

If
L>>a


B ~ µ0nIa2

2L

a2 L4

2⎛⎝⎜

⎞⎠⎟

12

=µ0nI2

LL / 2

= µ0nI

Z


dz
What
about
the
field
at
the
very
end?


B =µ0nI2

a2dz

a2 + z2( )32

0

L

∫

B =µ0nIa

2

2L

a2 a2 + L2( )12

−0

a2 a2( )12

⎡

⎣

⎢⎢⎢

⎤

⎦

⎥⎥⎥

B =µ0nIa

2

2L

a2 (a2 + L2 )1/2

If
L>>a


B ~ µ0nIa2

2L

a2 L2( )12

=µ0nI2

LL=µ0nI2

B
µ0nI

Ampere’s
Law


Current


Amperian
Loop


B ⋅dS = µ0∫ I

Integral
evaluated
around
any
closed
path
where
I
is
the
total
current
passing
through

any
surface
defined
by
the
path


Example
1:
Inside
and
outside
of
a
wire


Example
2:
Solenoid


d


Magne-c
Flux


ΦB =

B ⋅dA∫

From
last
Thursday:
Torque
on
current
loop
in
magne-c
field


µ = IA

µ
:
magne-c
moment


Magne-c
field
applies
torque
on
magne-c
moment


Example:
Torque
on
magne-c
moment


µ B

Clicker
ques-on
#3:
Which
way
will
the
magne-c
field
try
to
orient
the
moment?


a.  Clockwise
b.  Counter
clockwise
c.  Out
of
the
board
d.  Into
the
board