Magnetic field inside wire having parabolic current density
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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(theta). 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.