Final answer:
The current density and magnetic fields in cylindrical wires are determined using principles of electromagnetism, specifically Ampère's law and the Biot-Savart law, with the calculations depending on whether the current density is uniform or varies with radius.
Step-by-step explanation:
The question pertains to the calculation of current density and magnetic fields in cylindrical wires under various circumstances, including wires with uniform and non-uniform current densities and the presence of cavities or holes within the wires. These calculations are typically grounded in the use of Ampère's law and the Biot-Savart law in electromagnetic theory.
Regarding a cylindrical wire with a current density that varies as a function of radius r, the magnetic field both inside and outside the wire depends on this varying current density. When the wire has a uniform current density, the calculation simplifies, and the magnetic field within the wire varies linearly with r, while outside the wire, it is inversely proportional to r (following Ampère's law for a long straight conductor).
For non-uniform current density, specifically when given by J(r) = Cr², the resulting current I through the wire is found by integrating the current density over the wire's cross-sectional area. Similarly, the magnetic field is calculated by integrating Biot-Savart law contributions or using Ampère's law for the appropriate regions of the wire, taking into account the symmetry and the specific configuration given.