Figure 29-87 shows a cross section of a hollow cylindrical...

Figure $29-87$ shows a cross section of a hollow cylindrical conductor of radii $a$ and $b$, carrying a uniformly distributed current $i$. (a) Show that the magnetic field magnitude $B(r)$ for the radial distance $r$ in the range $b < r < a$ is given by $$B=\frac{\mu_{0} i}{2 \pi\left(a^{2}-b^{2}\right)} \frac{r^{2}-b^{2}}{r}$$ (b) Show that when $r=a,$ this equation gives the magnetic field magnitude $B$ at the surface of a long straight wire carrying current $i$ when $r=b,$ it gives zero magnetic field; and when $b=0,$ it gives the magnetic field inside a solid conductor of radius $a$ carrying current $i$. (c) Assume that $a=2.0 \mathrm{~cm}, b=1.8 \mathrm{~cm},$ and $i=100 \mathrm{~A},$ and then plot $B(r)$ for the range $0 < r < 6 \mathrm{~cm}$

Figure $29-87$ shows a cross section of a hollow cylindrical conductor of radii $a$ and $b$, carrying a uniformly distributed current $i$. (a) Show that the magnetic field magnitude $B(r)$ for the radial distance $r$ in the range $b < r < a$ is given by
$$B=\frac{\mu_{0} i}{2 \pi\left(a^{2}-b^{2}\right)} \frac{r^{2}-b^{2}}{r}$$
(b) Show that when $r=a,$ this equation gives the magnetic field magnitude $B$ at the surface of a long straight wire carrying current $i$ when $r=b,$ it gives zero magnetic field; and when $b=0,$ it gives the magnetic field inside a solid conductor of radius $a$ carrying current $i$. (c) Assume that $a=2.0 \mathrm{~cm}, b=1.8 \mathrm{~cm},$ and $i=100 \mathrm{~A},$ and then plot $B(r)$ for the range $0 < r < 6 \mathrm{~cm}$

Key Concepts

Boundary Conditions in Magnetic Fields

When dealing with electromagnetic fields, particularly at the interfaces between different materials or regions (such as the inner and outer surfaces of a hollow conductor), it is crucial to apply boundary conditions. These conditions ensure continuity (or prescribe the correct discontinuity) of the magnetic field and relate the field values to known physical situations, such as zero field in specific regions or matching with the well-known magnetic field expressions outside a long straight wire.

Magnetic Field Profiles in Conductors

The magnetic field produced by a current-carrying conductor varies with the radial distance from the conductor's center. In systems with hollow or solid cross sections, the magnetic field profile must be determined separately for different regions (e.g., inside the conductor, in the hollow region, and outside the conductor) using methods like Ampere's Law combined with the appropriate current distribution. Analyzing these profiles helps in understanding the transition of magnetic fields across different material boundaries.

Cylindrical Symmetry

Cylindrical symmetry implies that the physical properties of the system, including the magnetic field, do not change with rotation about the central axis. This symmetry simplifies the analysis of the magnetic field by ensuring that the field is tangential and constant along circular paths at a fixed radial distance from the center, enabling the straightforward use of Ampere’s Law.

Uniform Current Distribution

In many conductor problems, the current is distributed uniformly across the cross-sectional area. This uniformity ensures that the current density is constant, which simplifies the calculation of the portion of the current enclosed by an Amperian loop at a given radius. Understanding this concept is essential when deriving explicit expressions for the magnetic field in regions within or around the conductor.

Ampere's Law

Ampere’s Law relates the line integral of the magnetic field around a closed loop to the total current enclosed by that loop. It is a central tool in magnetostatics for determining magnetic fields in systems with high symmetry. In cases such as cylindrical conductors, the symmetry allows a simplified application of the law, reducing the problem to one involving integrals over circular paths, thereby yielding explicit expressions for magnetic field magnitudes as functions of radial distance.

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