<s>
A	O
solenoid	B-Algorithm
(	O
)	O
is	O
a	O
type	O
of	O
electromagnet	O
formed	O
by	O
a	O
helical	B-Application
coil	O
of	O
wire	O
whose	O
length	O
is	O
substantially	O
greater	O
than	O
its	O
diameter	O
,	O
which	O
generates	O
a	O
controlled	O
magnetic	O
field	O
.	O
</s>
<s>
The	O
concept	O
of	O
solenoid	B-Algorithm
was	O
introduced	O
in	O
1820	O
by	O
André-Marie	O
Ampère	O
who	O
coined	O
the	O
term	O
solenoid	B-Algorithm
in	O
1823	O
.	O
</s>
<s>
The	O
helical	B-Application
coil	O
of	O
a	O
solenoid	B-Algorithm
does	O
not	O
necessarily	O
need	O
to	O
revolve	O
around	O
a	O
straight-line	O
axis	O
;	O
for	O
example	O
,	O
William	O
Sturgeon	O
's	O
electromagnet	O
of	O
1824	O
consisted	O
of	O
a	O
solenoid	B-Algorithm
bent	O
into	O
a	O
horseshoe	O
shape	O
(	O
similarly	O
to	O
an	O
arc	O
spring	O
)	O
.	O
</s>
<s>
Solenoids	B-Algorithm
provide	O
magnetic	O
focusing	O
of	O
electrons	O
in	O
vacuums	O
,	O
notably	O
in	O
television	O
camera	O
tubes	O
such	O
as	O
vidicons	O
and	O
image	O
orthicons	O
.	O
</s>
<s>
Electrons	O
take	O
helical	B-Application
paths	O
within	O
the	O
magnetic	O
field	O
.	O
</s>
<s>
These	O
solenoids	B-Algorithm
,	O
focus	O
coils	O
,	O
surround	O
nearly	O
the	O
whole	O
length	O
of	O
the	O
tube	O
.	O
</s>
<s>
In	O
engineering	O
,	O
the	O
term	O
"	O
solenoid	B-Algorithm
"	O
refers	O
not	O
only	O
to	O
the	O
electromagnet	O
but	O
to	O
a	O
complete	O
apparatus	O
providing	O
an	O
actuator	B-Algorithm
that	O
converts	O
electrical	O
energy	O
to	O
mechanical	O
energy	O
.	O
</s>
<s>
An	O
infinite	O
solenoid	B-Algorithm
has	O
infinite	O
length	O
but	O
finite	O
diameter	O
.	O
</s>
<s>
"	O
Continuous	O
"	O
means	O
that	O
the	O
solenoid	B-Algorithm
is	O
not	O
formed	O
by	O
discrete	O
finite-width	O
coils	O
but	O
by	O
many	O
infinitely	O
thin	O
coils	O
with	O
no	O
space	O
between	O
them	O
;	O
in	O
this	O
abstraction	O
,	O
the	O
solenoid	B-Algorithm
is	O
often	O
viewed	O
as	O
a	O
cylindrical	O
sheet	O
of	O
conductive	O
material	O
.	O
</s>
<s>
The	O
magnetic	O
field	O
inside	O
an	O
infinitely	O
long	O
solenoid	B-Algorithm
is	O
homogeneous	O
and	O
its	O
strength	O
neither	O
depends	O
on	O
the	O
distance	O
from	O
the	O
axis	O
nor	O
on	O
the	O
solenoid	B-Algorithm
's	O
cross-sectional	O
area	O
.	O
</s>
<s>
This	O
is	O
a	O
derivation	O
of	O
the	O
magnetic	O
flux	O
density	O
around	O
a	O
solenoid	B-Algorithm
that	O
is	O
long	O
enough	O
so	O
that	O
fringe	O
effects	O
can	O
be	O
ignored	O
.	O
</s>
<s>
In	O
Figure	O
1	O
,	O
we	O
immediately	O
know	O
that	O
the	O
flux	O
density	O
vector	O
points	O
in	O
the	O
positive	O
z	O
direction	O
inside	O
the	O
solenoid	B-Algorithm
,	O
and	O
in	O
the	O
negative	O
z	O
direction	O
outside	O
the	O
solenoid	B-Algorithm
.	O
</s>
<s>
Since	O
we	O
are	O
dealing	O
with	O
a	O
long	O
solenoid	B-Algorithm
,	O
all	O
of	O
the	O
components	O
of	O
the	O
magnetic	O
field	O
not	O
pointing	O
upwards	O
cancel	O
out	O
by	O
symmetry	O
.	O
</s>
<s>
Now	O
consider	O
the	O
imaginary	O
loop	O
c	O
that	O
is	O
located	O
inside	O
the	O
solenoid	B-Algorithm
.	O
</s>
<s>
By	O
Ampère	O
's	O
law	O
,	O
we	O
know	O
that	O
the	O
line	O
integral	O
of	O
B	O
(	O
the	O
magnetic	O
flux	O
density	O
vector	O
)	O
around	O
this	O
loop	O
is	O
zero	O
,	O
since	O
it	O
encloses	O
no	O
electrical	O
currents	O
(	O
it	O
can	O
be	O
also	O
assumed	O
that	O
the	O
circuital	O
electric	O
field	O
passing	O
through	O
the	O
loop	O
is	O
constant	O
under	O
such	O
conditions	O
:	O
a	O
constant	O
or	O
constantly	O
changing	O
current	O
through	O
the	O
solenoid	B-Algorithm
)	O
.	O
</s>
<s>
We	O
have	O
shown	O
above	O
that	O
the	O
field	O
is	O
pointing	O
upwards	O
inside	O
the	O
solenoid	B-Algorithm
,	O
so	O
the	O
horizontal	O
portions	O
of	O
loop	O
c	O
do	O
not	O
contribute	O
anything	O
to	O
the	O
integral	O
.	O
</s>
<s>
Since	O
we	O
can	O
arbitrarily	O
change	O
the	O
dimensions	O
of	O
the	O
loop	O
and	O
get	O
the	O
same	O
result	O
,	O
the	O
only	O
physical	O
explanation	O
is	O
that	O
the	O
integrands	O
are	O
actually	O
equal	O
,	O
that	O
is	O
,	O
the	O
magnetic	O
field	O
inside	O
the	O
solenoid	B-Algorithm
is	O
radially	O
uniform	O
.	O
</s>
<s>
A	O
similar	O
argument	O
can	O
be	O
applied	O
to	O
the	O
loop	O
a	O
to	O
conclude	O
that	O
the	O
field	O
outside	O
the	O
solenoid	B-Algorithm
is	O
radially	O
uniform	O
or	O
constant	O
.	O
</s>
<s>
This	O
last	O
result	O
,	O
which	O
holds	O
strictly	O
true	O
only	O
near	O
the	O
center	O
of	O
the	O
solenoid	B-Algorithm
where	O
the	O
field	B-Application
lines	I-Application
are	O
parallel	O
to	O
its	O
length	O
,	O
is	O
important	O
as	O
it	O
shows	O
that	O
the	O
flux	O
density	O
outside	O
is	O
practically	O
zero	O
since	O
the	O
radii	O
of	O
the	O
field	O
outside	O
the	O
solenoid	B-Algorithm
will	O
tend	O
to	O
infinity	O
.	O
</s>
<s>
An	O
intuitive	O
argument	O
can	O
also	O
be	O
used	O
to	O
show	O
that	O
the	O
flux	O
density	O
outside	O
the	O
solenoid	B-Algorithm
is	O
actually	O
zero	O
.	O
</s>
<s>
Magnetic	O
field	B-Application
lines	I-Application
only	O
exist	O
as	O
loops	O
,	O
they	O
cannot	O
diverge	O
from	O
or	O
converge	O
to	O
a	O
point	O
like	O
electric	O
field	B-Application
lines	I-Application
can	O
(	O
see	O
Gauss	O
's	O
law	O
for	O
magnetism	O
)	O
.	O
</s>
<s>
The	O
magnetic	O
field	B-Application
lines	I-Application
follow	O
the	O
longitudinal	O
path	O
of	O
the	O
solenoid	B-Algorithm
inside	O
,	O
so	O
they	O
must	O
go	O
in	O
the	O
opposite	O
direction	O
outside	O
of	O
the	O
solenoid	B-Algorithm
so	O
that	O
the	O
lines	O
can	O
form	O
loops	O
.	O
</s>
<s>
However	O
,	O
the	O
volume	O
outside	O
the	O
solenoid	B-Algorithm
is	O
much	O
greater	O
than	O
the	O
volume	O
inside	O
,	O
so	O
the	O
density	O
of	O
magnetic	O
field	B-Application
lines	I-Application
outside	O
is	O
greatly	O
reduced	O
.	O
</s>
<s>
In	O
order	O
for	O
the	O
total	O
number	O
of	O
field	B-Application
lines	I-Application
to	O
be	O
conserved	O
,	O
the	O
field	O
outside	O
must	O
go	O
to	O
zero	O
as	O
the	O
solenoid	B-Algorithm
gets	O
longer	O
.	O
</s>
<s>
Of	O
course	O
,	O
if	O
the	O
solenoid	B-Algorithm
is	O
constructed	O
as	O
a	O
wire	O
spiral	O
(	O
as	O
often	O
done	O
in	O
practice	O
)	O
,	O
then	O
it	O
emanates	O
an	O
outside	O
field	O
the	O
same	O
way	O
as	O
a	O
single	O
wire	O
,	O
due	O
to	O
the	O
current	O
flowing	O
overall	O
down	O
the	O
length	O
of	O
the	O
solenoid	B-Algorithm
.	O
</s>
<s>
where	O
is	O
the	O
magnetic	O
flux	O
density	O
,	O
is	O
the	O
length	O
of	O
the	O
solenoid	B-Algorithm
,	O
is	O
the	O
magnetic	O
constant	O
,	O
the	O
number	O
of	O
turns	O
,	O
and	O
the	O
current	O
.	O
</s>
<s>
This	O
equation	O
is	O
valid	O
for	O
a	O
solenoid	B-Algorithm
in	O
free	O
space	O
,	O
which	O
means	O
the	O
permeability	O
of	O
the	O
magnetic	O
path	O
is	O
the	O
same	O
as	O
permeability	O
of	O
free	O
space	O
,	O
μ0	O
.	O
</s>
<s>
If	O
the	O
solenoid	B-Algorithm
is	O
immersed	O
in	O
a	O
material	O
with	O
relative	O
permeability	O
μr	O
,	O
then	O
the	O
field	O
is	O
increased	O
by	O
that	O
amount	O
:	O
</s>
<s>
In	O
most	O
solenoids	B-Algorithm
,	O
the	O
solenoid	B-Algorithm
is	O
not	O
immersed	O
in	O
a	O
higher	O
permeability	O
material	O
,	O
but	O
rather	O
some	O
portion	O
of	O
the	O
space	O
around	O
the	O
solenoid	B-Algorithm
has	O
the	O
higher	O
permeability	O
material	O
and	O
some	O
is	O
just	O
air	O
(	O
which	O
behaves	O
much	O
like	O
free	O
space	O
)	O
.	O
</s>
<s>
The	O
inclusion	O
of	O
a	O
ferromagnetic	O
core	O
,	O
such	O
as	O
iron	O
,	O
increases	O
the	O
magnitude	O
of	O
the	O
magnetic	O
flux	O
density	O
in	O
the	O
solenoid	B-Algorithm
and	O
raises	O
the	O
effective	O
permeability	O
of	O
the	O
magnetic	O
path	O
.	O
</s>
<s>
A	O
finite	O
solenoid	B-Algorithm
is	O
a	O
solenoid	B-Algorithm
with	O
finite	O
length	O
.	O
</s>
<s>
Continuous	O
means	O
that	O
the	O
solenoid	B-Algorithm
is	O
not	O
formed	O
by	O
discrete	O
coils	O
but	O
by	O
a	O
sheet	O
of	O
conductive	O
material	O
.	O
</s>
<s>
We	O
assume	O
the	O
current	O
is	O
uniformly	O
distributed	O
on	O
the	O
surface	O
of	O
the	O
solenoid	B-Algorithm
,	O
with	O
a	O
surface	O
current	O
density	O
K	O
;	O
in	O
cylindrical	O
coordinates	O
:	O
</s>
<s>
Inside	O
the	O
solenoid	B-Algorithm
,	O
far	O
away	O
from	O
the	O
ends	O
(	O
)	O
,	O
this	O
tends	O
towards	O
the	O
constant	O
value	O
.	O
</s>
<s>
For	O
the	O
case	O
in	O
which	O
the	O
radius	O
is	O
much	O
larger	O
than	O
the	O
length	O
of	O
the	O
solenoid	B-Algorithm
(	O
)	O
,	O
the	O
magnetic	O
flux	O
density	O
through	O
the	O
centre	O
of	O
the	O
solenoid	B-Algorithm
(	O
in	O
the	O
z	O
direction	O
,	O
parallel	O
to	O
the	O
solenoid	B-Algorithm
's	O
length	O
,	O
where	O
the	O
coil	O
is	O
centered	O
at	O
z	O
=	O
0	O
)	O
can	O
be	O
estimated	O
as	O
the	O
flux	O
density	O
of	O
a	O
single	O
circular	O
conductor	O
loop	O
:	O
</s>
<s>
Within	O
the	O
category	O
of	O
finite	O
solenoids	B-Algorithm
,	O
there	O
are	O
those	O
that	O
are	O
sparsely	O
wound	O
with	O
a	O
single	O
pitch	O
,	O
sparsely	O
wound	O
with	O
varying	O
pitches	O
(	O
varied-pitch	O
solenoid	B-Algorithm
)	O
,	O
or	O
those	O
with	O
a	O
varying	O
radius	O
for	O
different	O
loops	O
(	O
non-cylindrical	O
solenoids	B-Algorithm
)	O
.	O
</s>
<s>
They	O
are	O
called	O
irregular	O
solenoids	B-Algorithm
.	O
</s>
<s>
They	O
have	O
found	O
applications	O
in	O
different	O
areas	O
,	O
such	O
as	O
sparsely	O
wound	O
solenoids	B-Algorithm
for	O
wireless	B-Application
power	I-Application
transfer	I-Application
,	O
varied-pitch	O
solenoids	B-Algorithm
for	O
magnetic	O
resonance	O
imaging	O
(	O
MRI	O
)	O
,	O
and	O
non-cylindrical	O
solenoids	B-Algorithm
for	O
other	O
medical	O
devices	O
.	O
</s>
<s>
The	O
calculation	O
of	O
the	O
intrinsic	O
inductance	O
and	O
capacitance	O
cannot	O
be	O
done	O
using	O
those	O
for	O
the	O
traditional	O
solenoids	B-Algorithm
,	O
i.e.	O
</s>
<s>
A	O
table	O
of	O
inductance	O
for	O
short	O
solenoids	B-Algorithm
of	O
various	O
diameter	O
to	O
length	O
ratios	O
has	O
been	O
calculated	O
by	O
Dellinger	O
,	O
Whittmore	O
,	O
and	O
Ould	O
.	O
</s>
<s>
Similar	O
analysis	O
applies	O
to	O
a	O
solenoid	B-Algorithm
with	O
a	O
magnetic	O
core	O
,	O
but	O
only	O
if	O
the	O
length	O
of	O
the	O
coil	O
is	O
much	O
greater	O
than	O
the	O
product	O
of	O
the	O
relative	O
permeability	O
of	O
the	O
magnetic	O
core	O
and	O
the	O
diameter	O
.	O
</s>
<s>
That	O
limits	O
the	O
simple	O
analysis	O
to	O
low-permeability	O
cores	O
,	O
or	O
extremely	O
long	O
thin	O
solenoids	B-Algorithm
.	O
</s>
