The current density in a cylindrical conductor of radius \(R\), varies as \(J(r)=J_{0} r / R\) (in the region from zero to \(R\) ). Express the magnitude of the magnetic field in the regions \(rR .\) Produce a sketch of the radial dependence, \(B(r)\)

To grasp how a magnetic field behaves around a conductor, we first need to delve into Ampere's Law. This fundamental law of magnetism is akin to a magnetic version of Gauss's Law for electricity. It connects the magnetism around a current-carrying conductor to the current itself. In essence, Ampere's Law states that the line integral of the magnetic field \( \vec{B} \) around a closed loop is directly proportional to the electric current \( I_{enclosed} \) passing through any surface bounded by the loop.

Ampere's Law is mathematically expressed as:\
\[ \oint \vec{B} \cdot \vec{dL} = \mu_{0} I_{enclosed} \]
In this expression, \( \vec{dL} \) is an infinitesimal vector element of the closed loop, \( \mu_{0} \) is the permeability of free space, and \( I_{enclosed} \) is the current enclosed by the loop. This intricate dance between current and magnetic field allows us to solve for the magnetic field given a specific distribution of current.

Current density, symbolized as \( J \), is a measure of how much current \( I \) flows through a unit area. It gives you a sense of how densely packed the current is within a conductor. The current density is particularly useful when the current is not uniformly distributed, as is the case in our cylindrical conductor scenario.

For our problem, the current density varies linearly with the radial distance \( r \) from the axis of the cylinder as:
\[ J(r) = J_{0} \frac{r}{R} \]
Here, \( J_{0} \) is the current density at the surface \( r = R \) of the cylinder. This linear relationship allows the intensity of the current to increase gradually from the center to the outer edge, and it significantly influences the resulting magnetic field inside the conductor.

To solve for the magnetic field inside our cylindrical conductor, we analyze the current enclosed by a hypothetical loop, which is necessary in applying Ampere's Law. The concept of 'current enclosed by a loop' refers to the total current passing through the area bounded by that loop. In our case, we're considering circular loops concentric with the cylinder.

For a loop of radius \( r \), using the given current density, we derive the enclosed current as:

• For \( r < R \) (inside the conductor), the enclosed current is the integral of the current density over the loop's area, producing \( I_{enclosed} = \pi J_0 r^{2} \).
• For \( r > R \) (outside the conductor), the enclosed current is the total current through the conductor's cross-sectional area, which is \( I_{enclosed} = \pi J_0 R^{2} \).

This distinction is crucial because it dictates the form of the magnetic field in different regions around the conductor.

A visual interpretation of the magnetic field strength as a function of distance from the conductor's center, or the magnetic field sketch, can provide intuitive understanding. To illustrate the behavior predicted by our findings, we sketch the magnetic field (B(r)) against the radial distance ( r ).

Inside the conductor \( (r < R) \), the magnetic field increases linearly with r due to the current density's linear relationship to r . Conversely, outside the conductor \( (r > R) \), it diminishes according to the inverse of the radial distance. The point where r = R marks the peak of the magnetic field, coinciding with the surface of the conductor. The sketch thus serves as a visual summary, revealing the increase in magnetic field up to the conductor's boundary and its decrease beyond that point.