Question

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

Short answer

For different regions, the magnetic field \( |\vec{B}| \) will behave as follows: \(/|\(0\leq r\< R/2)\): \( |\vec{B}| = 0 \), \(/|\(R/2\leq r\< R)\): \( |\vec{B}| \) linearly increases with \( r \), \(/|\(r \geq R)\): \( |\vec{B}| = \frac{\mu_0 I_{\text{enc}}}{2\pi r} \) and decreases inversely as \( r \) increases.

Step-by-step solution

Understand the Ampère's Circuital Law

Ampère's Circuital Law states that the line integral of the magnetic field \( \vec{B} \) around any closed loop is equal to the permeability of free space \( \mu_0 \) times the current \( I \) enclosed by that loop. In mathematical terms, \( \oint_{\text{loop}} \vec{B} \cdot d\vec{l} = \mu_0 I_{\text{enc}} \). For symmetry in this problem, we choose an Amperian loop which would be a circle centered on the axis of the cylinder.

Apply Ampère's Law for Region Inside the Hollow Part

For the region inside the hollow part of the cylinder \( (0 < r < R/2) \), no current is enclosed by the Amperian loop. Therefore, \( I_{\text{enc}} = 0 \), leading to \( \oint \vec{B} \cdot d\vec{l} = \mu_0 \times 0 \), which implies \( \vec{B} = 0 \) inside the hollow region.

Apply Ampère's Law for Region within the Conducting Material

For \( R/2 < r < R \) within the conducting material, the current enclosed by the Amperian loop is less than the total current of the cylinder. The current density \( J \) is uniform, so the enclosed current \( I_{\text{enc}} \) can be found by \( I_{\text{enc}} = J \times \text{area} \) where the area is \( \pi (r^2 - (R/2)^2) \). By Ampère's Law, \( \oint \vec{B} \cdot d\vec{l} = \mu_0 I_{\text{enc}} \), and since the magnetic field is uniform along the loop, we can simplify to \( B(2\pi r) = \mu_0 J \pi (r^2 - (R/2)^2) \). Solving for \( B \) gives us the linearly increasing function with \( r \) in this region.

Apply Ampère's Law for Region outside the Cylinder

For \( r > R \) outside the cylinder, all the current of the cylinder is enclosed by the Amperian loop. Using Ampère's Law, \( B(2\pi r) = \mu_0 I_{\text{enc}} \), we know that all the current in the cylinder contributes to the magnetic field at distance \( r \), which results in the magnetic field decreasing inversely with distance from the center, \( B(r) = \frac{\mu_0 I_{\text{enc}}}{2\pi r} \).

Key concepts

Magnetic Field Inside Conductors

When we consider the magnetic field inside conductors, Ampère's Circuital Law becomes a valuable tool for analyzing magnetic field patterns. In simple terms, this law helps us understand the relationship between the current flowing through a conductor and the magnetic field it produces. The key aspect of this law is that it ties together the magnetic field along a closed loop to the current penetrating that loop.

If we take a hollow conducting cylinder as in our example and apply Ampère's Circuital Law, we discover that inside the hollow region, no current is present. According to the law, this absence of current results in a magnetic field strength of zero within that region. It's fascinating how the magnetic field remains non-existent where there is no current, which is an intuitive outcome given that the source of the magnetic field is the flow of electric charge. In other words, no charge flow, no magnetic field.

Current Density

Current density, represented by the symbol \(J\), is a measure of the electric current per unit area of cross-section. It's a vital concept because it helps us determine how much current is flowing through different areas of a conductor. Considering our cylinder with uniform current density, it means that the current flows consistently across all sections of the conductor's material.

When Ampère's Circuital Law is employed for the region within the conducting cylinder, the enclosed current \(I_{\text{enc}}\) is calculated based on this current density. It's proportional to the product of the current density and the cross-sectional area through which the current is flowing. So, by applying this notion to different radial sections within the cylinder, it becomes possible to understand the varying contributions of different segments to the overall magnetic field within the material. This guides us in anticipating that the strength of the magnetic field would increase as we move outward because the enclosed current increases with the area.

Uniform Current Distribution

A uniform current distribution can greatly simplify our calculations when using Ampère's Circuital Law, as it means that the current is spread out evenly throughout the conductor. In the case of our cylindrical conductor, this assumption allows us to conclude that the larger the radius of our Amperian loop (up to the outer radius of the cylinder), the greater the portion of total current enclosed. As a result, the magnetic field strength follows a linear trend with increasing radius within the conductor's material.

For the current distribution to be uniform, the charges must be moving with a consistent density over the entire cross-sectional area of the conductor. This is an essential facet of the problem because it allows us to assert that at any point within the conductor, the local magnetic field generated is directly linked to the current density and the area chosen for our Ampère's loop. The consequence of uniform current distribution is a predictable magnetic field pattern, which simplifies the process of analysis and enhances our understanding of electromagnetism in conductive materials.