We present an alternative formulation of the magnetostatic boundary value problem that is useful for calculating the magnetic field around a magnetic material placed in the vicinity of steady currents. The formulation differs from the standard approach in that a single-valued scalar potential plus a vector field that depends on the given currents but not on the magnetic material are used to obtain the total magnetic field instead of a magnetic vector potential. We illustrate the method with examples.

1.
Julius
,
Electromagnetic Theory
, 1st ed. (
McGraw-Hill
,
New York and London
,
1941
), pp.
254
262
.
2.
J. D.
Jackson
,
Classical Electrodynamics
, 2nd ed. (
John Wiley & Sons
,
New York
,
1974
), pp.
191
194
.
3.
A very interesting alternative method for calculating the magnetic field around a current loop using rotation matrices is given in
Matthew I.
Grivich
and
David P.
Jackson
, “
The magnetic field of current-carrying polygons: An application of vector field rotations
,”
Am. J. Phys.
68
,
469
474
(
2000
).
4.
Oleg D.
Jefimenko
, “
New method for calculating electric and magnetic fields and forces
,”
Am. J. Phys.
51
,
545
551
(
1983
).
5.
O. C.
Zienkiewicz
,
John
Lyness
, and
D. R. J.
Owen
, “
Three-dimensional magnetic field determination using a scalar potential—A finite element solution
,”
IEEE Trans. Magn.
13
(
5
),
1649
1656
(
1977
).
6.
Oleg D.
Jefimenko
, “
Direct calculation of electric and magnetic forces from potentials
,”
Am. J. Phys.
58
,
625
631
(
1990
).
7.

This separation of the field into two parts follows from an application of the Poincare lemma: We know that for any given scalar field ψ, $∇→×(∇→ψ)=0$ always. The Poincare lemma then assures us that if a vector field $F→$ has the property that $∇→×F→=0$ in a given region of space, there exists a scalar function ϕ defined in that region such that $F→=∇→ϕ$.

8.
This type of analogy is not new and has been explored in the literature. An interesting mapping between the calculation of electrostatic and magnetic fields in two dimensions has been discussed by
Ying-yan
Zhou
, “
The analogy between the calculation of a two-dimensional electrostatic field and that of a two-dimensional magnetostatic field
,”
Am. J. Phys.
64
,
69
72
(
1996
).
9.
An interesting analogy between the calculation of the magnetic field of a solenoid and the electric field of a cylindrical capacitor filled with a dielectric ϵ is given by
L.
Lerner
, “
Magnetic field of a finite solenoid with a linear permeable core
,”
Am. J. Phys.
79
,
1030
1035
(
2011
).
10.
See, for example,
George
Arfken
,
Mathematical Methods for Physicists
, 2nd ed. (