Alex Slocum of MIT has some great design tool spreadsheets.
Energy in a magnetic field, Ramo et. al.:
d
U
H
=
∫
V
H
⋅
d
B
d
V
{\displaystyle dU_{H}=\int _{V}^{}H\cdot dB\,dV}
For linear materials, where
B
=
μ
H
{\displaystyle B=\mu H}
U
H
=
1
2
∫
V
B
⋅
H
d
V
{\displaystyle U_{H}={\frac {1}{2}}\int _{V}^{}B\cdot H\,dV}
U
H
=
B
2
V
2
μ
{\displaystyle U_{H}={\frac {B^{2}V}{2\mu }}}
U
H
=
B
2
A
l
2
μ
{\displaystyle U_{H}={\frac {B^{2}Al}{2\mu }}}
F
=
d
U
H
d
l
{\displaystyle F={\frac {dU_{H}}{dl}}}
F
=
B
2
A
2
μ
{\displaystyle F={\frac {B^{2}A}{2\mu }}}
Force per unit area (pressure) is
P
=
B
2
2
μ
{\displaystyle P={\frac {B^{2}}{2\mu }}}
In the case of free space (air),
μ
o
=
4
π
⋅
10
−
7
H
m
{\displaystyle \mu _{o}=4\pi \cdot 10^{-7}{\frac {H}{m}}}
P
=
57.7
l
b
i
n
2
{\displaystyle P=57.7\,{\frac {lb}{in^{2}}}}
P
=
230.8
l
b
i
n
2
{\displaystyle P=230.8\,{\frac {lb}{in^{2}}}}
In a closed magnetic circuit:
B
=
μ
N
I
l
{\displaystyle B={\frac {\mu NI}{l}}}
F
=
μ
N
2
I
2
A
2
l
2
{\displaystyle F={\frac {\mu N^{2}I^{2}A}{2l^{2}}}}
To build a strong electromagnet, a short geometry, with large area is preferred. Note that most Ferromagnetic materials saturate around 1-2 Tesla. This occurs at a field intensity of:
H
=
20
A
m
p
e
r
e
⋅
t
u
r
n
s
i
n
c
h
{\displaystyle H=20\,{\frac {Ampere\cdot turns}{inch}}}
For this reason, there is no point in building an electromagnet with a higher field intensity.
