At for help with problems like this one. A very long, hollow...

At for help with problems like this one. A very long, hollow cylinder is formed by rolling up a thin sheet of copper. Electric charges flow along the copper sheet parallel to the axis of the cylinder. The arrangement is, in effect, a hollow tube of current $I$. Use Ampère's law to show that the magnetic field (a) is $\mu_{0} I /(2 \pi r)$ outside the cylinder at a distance $r$ from the axis and $(\mathrm{b})$ is zero at any point within the hollow interior of the cylinder. (Hint: For closed paths, use circles perpendicular to and centered on the axis of the cylinder.)

Key Concepts

Ampère's Law

Ampère's Law is a fundamental principle in electromagnetism stating that the line integral of the magnetic field around any closed path is equal to the permeability constant times the total current enclosed by that path. This law is especially powerful in systems with high symmetry, as it allows for the direct calculation of magnetic fields by relating them to the current that produces them.

Cylindrical Symmetry

Many current-carrying systems, such as a long hollow cylinder, possess cylindrical symmetry where the magnetic field depends only on the radial distance from the axis. This symmetry simplifies the use of Ampère's Law, as choosing a circular path concentric with the cylinder ensures that the magnetic field is constant along the path, making the integration straightforward.

Amperian Loops

Amperian loops are closed paths used in the application of Ampère's Law. In systems with cylindrical symmetry, selecting circular loops perpendicular to the cylinder’s axis leverages the symmetry by keeping the magnitude of the magnetic field uniform along the loop. This approach simplifies the calculation of the integral in Ampère's Law, both outside the conductor (where the loop encloses current) and inside the hollow region (where the loop encloses no net current).

Field Cancellation in Hollow Regions

For hollow current-carrying conductors, applying Ampère's Law shows that the magnetic field inside a hollow cavity is zero because any Amperian loop drawn within the cavity encloses no current. This demonstrates that, despite the presence of currents in the conductor, the magnetic contributions cancel out in the hollow region, leading to a zero magnetic field.

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