In this note we synthesize and extend expressions for the magnetic field at the center of very short and very long current-carrying coils. Elementary physics textbooks present the following equation for the magnetic field inside a very long current-carrying coil (solenoid):
$Bsol=μ0NLI,$
(1)
where I is the current, N the number of windings, and L the coil length. It is obtained directly from Ampère's law, ignores end effects (hence it assumes an infinite coil), and is valid over the entire cross-section of the coil. The field is in the axial direction and it follows the right-hand rule convention. For N superposed loops of radius R each carrying current I, the multi-loop magnetic field at the center is given by
$Bml=μ0N2RI.$
(2)
1.
D. B.
Brick
and
A. W.
Snyder
, “
External dc field of a long solenoid
,”
Am. J. Phys.
33
,
905
909
(
Nov. 1965
).
2.
B. B.
Dasgupta
, “
Magnetic field due to a solenoid
,”
Am. J. Phys.
52
,
258
(
March 1984
).
3.
M.
, “
On the magnetic field of a solenoid
,”
Am. J. Phys.
54
,
1065
(
Dec. 1986
).
4.
S. Y.
Mak
and
K.
Young
, “
Voltage measurements in a loop surrounding a solenoid: Some further ‘paradoxes’ due to stray capacitance
,”
Am. J. Phys.
56
,
254
(
March 1988
).
5.
J. D.
Templin
, “
Exact solution to the field equations in the case of an ideal, infinite solenoid
,”
Am. J. Phys.
63
,
916
920
(
Oct. 1995
).
6.
R. J.
Protheroe
and
D.
Koks
, “
The transient magnetic field outside an infinite solenoid
,”
Am. J. Phys.
64
,
1389
1393
(
Nov. 1996
).
7.
T.
Takeuchi
, “
Magnetic field of a double-layer solenoid
,”
Phys. Teach.
35
,
306
307
(
May 1997
).
8.
J.
Farley
and
R. H.
Price
, “
Field just outside a long solenoid
,”
Am. J. Phys.
69
,
751
754
(
July 2001
).
9.
C.-T.
Chia
and
Y.-F.
Fang
, “
The magnetic field along the axis of a long finite solenoid
,”
Phys. Teach.
40
,
288
289
(
May 2002
).
10.
O.
Espinosa
and
V.
Slusarenko
, “
The magnetic field of an infinite solenoid
,”
Am. J. Phys.
71
,
953
954
(
Sept. 2003
).
11.
D. J.
Griffiths
,
Introduction to Electrodynamics
(
Pearson
,
2013
).