Final answer:
Current density in a long cylindrical wire is along the axis of the wire, and calculating both the current and the magnetic fields involves integrating current density across the cross-section and using Ampère's Law for magnetic field evaluations.
Step-by-step explanation:
When examining the characteristics of a long solid cylindrical wire carrying an electric current, it's important to comprehend how current density and magnetic fields relate within such conductors. Current density (J) is a vector quantity that refers to the amount of current flowing per unit area of cross-section in a specific direction. In a long cylindrical wire, the direction of current density is along the axis of the wire.
For various distributions of current density within the wire, such as J = cr or J = Cr², the resulting magnetic field can be determined using Ampère's Law, which relates the magnetic field to the current enclosed within a loop. Magnetic field strength varies based on whether the point of observation is inside or outside the wire's radius (referenced as R in these questions).
To calculate the current through the wire, we integrate the current density over the cross-sectional area of the wire. This involves the use of calculus to find the total current I. In another case, we might consider superposition of fields to determine the magnetic field in the presence of a cylindrical cavity.
Concerning magnetic fields, we can analyze these fields at various distances from the center of the wire, be it at a point inside, at the surface, or outside the wire. Ampère's Law is pivotal in this context and allows us to plot magnetic field strength as a function of the radial distance from the center of the wire.