An infinitely long hollow conducting cylinder with inner...

An infinitely long hollow conducting cylinder with inner radius (R/2) and outer radius $\mathrm{R}$ carries a uniform current density along its length. The magnitude of the magnetic field $\left|\mathrm{B}^{-}\right|$ as a function of the radial distance $\mathrm{r}$ from the axis is best represented by

An infinitely long hollow conducting cylinder with inner radius (R/2) and outer radius $\mathrm{R}$ carries a uniform current density
along its length. The magnitude of the magnetic field $\left|\mathrm{B}^{-}\right|$ as a function of the radial distance $\mathrm{r}$ from the axis is best represented by

Key Concepts

Magnetic Fields in Hollow Conductors

In hollow conductors, the current flows only in the conductive material surrounding a hollow region, affecting the distribution of the magnetic field both inside the hollow and in the conductive layers. Analyzing these configurations requires careful consideration of different radial zones, as the magnetic field expressions will vary depending on whether the Amperian loop lies inside the hollow region, within the conducting material, or outside the conductor.

Ampere's Law

Ampere's Law is a fundamental principle in magnetostatics that relates the magnetic field along a closed loop to the total current enclosed by that loop. It is especially useful in problems with high symmetry, such as cylindrical geometries, where it allows for the straightforward calculation of the magnetic field by equating the line integral of the magnetic field around the loop with the enclosed current multiplied by the permeability of free space.

Cylindrical Symmetry

Cylindrical symmetry simplifies the analysis of magnetic fields by ensuring that field properties depend only on the radial distance from the axis and not on the angular or axial positions. This symmetry allows one to choose circular Amperian loops centered on the axis, making it easier to derive expressions for the magnetic field in different radial regions of a cylindrical current distribution.

Uniform Current Density

A uniform current density means that the current is distributed evenly throughout the conductor's cross-sectional area. This uniformity allows for the proportional scaling of the enclosed current with the area when applying Ampere's Law, making it simpler to determine the magnetic field strength as a function of radial distance within the conductor.

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