A solid conductor with radius a is supported by insulating disks on the axis of a conducting tube with inner radius b and outer radius c (Fig. E28.43). The central conductor and tube carry equal currents I in opposite directions. The currents are distributed uniformly over the cross sections of each conductor. Derive an expression for the magnitude of the magnetic field (b) at points outside the tube (r > c).

Verified step by step guidance

1. Identify the region of interest for the magnetic field calculation, which is outside the tube where r > c.
2. Apply Ampere's Law, \( \oint \vec{B} \cdot d\vec{l} = \mu_0 I_{\text{enc}} \), to a circular Amperian loop of radius r centered on the axis. Here, \( I_{\text{enc}} \) is the net current enclosed by the loop.
3. Recognize that the currents in the central conductor and the tube are equal and opposite, thus their contributions to the magnetic field outside the tube cancel each other out. Therefore, the net enclosed current \( I_{\text{enc}} \) is zero.
4. With \( I_{\text{enc}} = 0 \), Ampere's Law simplifies to \( \oint \vec{B} \cdot d\vec{l} = 0 \). This implies that the line integral of the magnetic field around the loop is zero.
5. Conclude that the magnetic field at any point outside the tube (r > c) is zero, since the integral of a non-zero field over a closed loop would not sum to zero.

Key Concepts

Ampère's Law

Ampère's Law relates the integrated magnetic field around a closed loop to the electric current passing through that loop. It is mathematically expressed as ∮B·dl = μ₀I_enc, where B is the magnetic field, dl is a differential length element of the loop, μ₀ is the permeability of free space, and I_enc is the enclosed current. This law is fundamental for analyzing magnetic fields generated by current-carrying conductors.

Magnetic Field Due to a Long Straight Conductor

The magnetic field generated by a long straight conductor carrying a current can be determined using the formula B = (μ₀I)/(2πr), where B is the magnetic field, I is the current, r is the distance from the conductor, and μ₀ is the permeability of free space. This concept is crucial for understanding how the magnetic field behaves at various distances from the conductor, especially in configurations involving multiple conductors.

Superposition Principle in Magnetism

The superposition principle states that the total magnetic field at a point due to multiple sources is the vector sum of the magnetic fields produced by each source independently. In the context of the given problem, this principle allows us to calculate the net magnetic field outside the conducting tube by considering the contributions from both the central conductor and the surrounding tube, which carry equal but opposite currents.