91Ó°ÊÓ

Current density is a way to measure how much electric current is flowing per unit area through a given cross-section. In this exercise, we are dealing with a long, solid cylindrical wire where the current density is directed along the central axis. The expression provided is \( J = \frac{J_0 r}{a} \), where \( J_0 \) has a value of 420 A/m². This indicates that as you move away from the center outwards, the current density increases linearly with the distance \( r \). This sort of linear relationship implies that at the center (\( r = 0 \)), there is no current density, while at the surface of the wire (\( r = a \)), the current density reaches its maximum value.

Ampère's Law is a fundamental principle used to relate the magnetic field and the enclosed current in a loop. The law is expressed as \( \oint \vec{B} \cdot d\vec{l} = \mu_0 I_{encl} \), where \( \mu_0 \) is the permeability of free space. To utilize Ampère's Law for finding the magnetic field inside the cylindrical wire, a circular loop of radius \( r \) is imagined. The magnetic field \( B \) is tangent to this loop, making it uniform along the loop's path, so we simplify the integral to \( B \cdot (2 \pi r) = \mu_0 I_{encl} \). This law is a powerful tool because it connects the measurable magnetic field to the less tangible enclosed current, allowing us to calculate information about the current distribution within the wire.

The magnetic field inside a cylindrical wire is influenced by the enclosed current, \( I_{encl} \), which can be calculated through integration of the current density over the area enclosed by the loop. The enclosed current for any radius \( r \) is given by integrating the current density: \( I_{encl} = \int_0^r J \cdot 2\pi r' \, dr' = \frac{2\pi J_0}{a} \cdot \frac{r^3}{3} \bigg|_0^r = \frac{2 \pi J_0 r^3}{3a} \). Knowing \( I_{encl} \), the magnetic field \( B \) is found by simplifying the expression from Ampère's Law: \( B = \frac{\mu_0 I_{encl}}{2\pi r} \). Calculating \( B \) at specific points such as the center (\( r = 0 \)), halfway (\( r = a/2 \)), and at the surface (\( r = a \)) provides insight into how the magnetic field varies within the wire.

A cylindrical wire, in this context, is a solid conductor with a circular cross-section. Its main feature is that it has a uniform cross-sectional area along its length, helping simplify the analysis of current and magnetic fields. The radius \( a = 4.5 \) mm defines the maximum extent from the central axis to the wire's edge, influencing both the current density and resulting magnetic field. For a better understanding of the radial dependence of the magnetic field, consider how the wire's geometry affects the distribution of current flow and subsequently the field. The simplicity in calculating the magnetic field using symmetry of a cylindrical wire is a key learning point, as symmetries often simplify electromagnetic problems significantly in physics.