It is common to use the Biot–Savart law as a tool to explicitly calculate the magnetic field due to currents flowing in simply shaped wires such as circular loops and straight lines. In this work, by using the Biot–Savart law and its inherent geometric properties, we derive a very simple integral expression that allows a straightforward computation of the magnetic field due to arbitrarily shaped planar current-carrying wires, at a point that lies in the same plane as the current filament. Such an expression is conveniently written in terms of the wire’s shape $r=r(θ).$ We illustrate the usefulness of our result by calculating the magnetic field at specific points in the wire’s plane due to currents flowing in conic curves, spirals, and harmonically deformed circular circuits. Relevant asymptotic behavior is calculated in various limits of interest.

1.
R. A. Serway, Physics for Scientists & Engineers (Saunders, Philadelphia, 1990), 3rd ed., pp. 836–839.
2.
D. Halliday, R. Resnick, and J. Walker, Fundamentals of Physics (Wiley, New York, 1997), 5th ed., pp. 729–732.
3.
D. J. Griffiths, Introduction to Electrodynamics (Prentice–Hall, Englewood Cliffs, NJ, 1989), 2nd ed., pp. 207–211.
4.
J. D. Jackson, Classical Electrodynamics (Wiley, New York, 1975), 2nd ed., pp. 169–173.
5.
See, for example, M. A. Munem and D. J. Foulis, Calculus with Analytic Geometry (Worth, New York, 1978).
6.
W. R. Smythe, Static and Dynamic Electricity (McGraw–Hill, New York, 1969), 3rd ed., p. 320, problem 9C.
7.
J. H. Jeans, Mathematical Theory of Electricity and Magnetism (Cambridge U.P., Cambridge, 1948), 5th ed., p. 447, problem 6.
8.
See Ref. 1, problem 83, p. 872.
9.
M. R. Spiegel, Mathematical Handbook of Formulas and Tables (McGraw–Hill, New York, 1968), pp. 37–39.
10.
I. S. Gradshteyn and I. M. Ryzhik, Table of Integrals, Series, and Products (Academic, New York, 1994), pp. 907–913.
11.
See Ref. 10, formula 2.595.2, p. 212.
12.
J. A.
Miranda
and
M.
Widom
, “
Stability analysis of polarized domains
,”
Phys. Rev. E
55
(
3
),
3758
3761
(
1997
).
13.
J. A.
Miranda
, “
Closed form results for shape transitions in lipid monolayer domains
,”
J. Phys. Chem. B
103
(
8
),
1303
1307
(
1999
).
14.
See Ref. 10, formula 2.553.3, p. 180.
15.
J. D. Lawrence, A Catalog of Special Plane Curves (Dover, New York, 1972), pp. 184–190.
This content is only available via PDF.