Figure 30-42 shows a cross section of a long cylindrical...

Figure $30-42$ shows a cross section of a long cylindrical conductor of radius $a$, carrying a uniformly distributed current $i$. Assume that $a=$ $2.0 \mathrm{~cm}$ and $i=100 \mathrm{~A}$, and plot the magnitude of the magnetic field $|\vec{B}(r)|=B(r)$ over the range $0<r<6.0 \mathrm{~cm} .$

Key Concepts

Plotting Magnetic Field as a Function of Radius

Plotting the magnetic field involves using the expressions derived from Ampère's Law for both the interior and exterior regions of the conductor. This graphically illustrates the linear increase of the magnetic field within the conductor and its inverse dependence outside, providing a visual understanding of the transition between these regimes.

Magnetic Field Distribution Inside and Outside the Conductor

For a cylindrical conductor with a uniform current distribution, the magnetic field inside (r < a) increases linearly with radial distance because the enclosed current scales with the area of the circle. Outside the conductor (r > a), the entire current is enclosed, and the magnetic field decreases inversely with the distance from the axis. This distinction is key for plotting the field across different regions.

Ampère's Law

Ampère's Law states that the line integral of the magnetic field around any closed path is proportional to the total current enclosed by that path. It is a fundamental tool used to derive magnetic field distributions in systems with high symmetry, such as cylindrical conductors.

Cylindrical Symmetry

Cylindrical symmetry implies that the physical properties, including the magnetic field, depend only on the radial distance from the axis of the cylinder and not on the angular or axial positions. This simplification is crucial in applying Ampère's Law and deducing the magnetic field distribution around a cylindrical conductor.

Uniform Current Distribution

A uniform current distribution means that the current is evenly spread across the conductor's cross-sectional area, resulting in a constant current density. This is important for determining how much current is enclosed by an Ampèrian loop at different radial distances within the conductor.

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