A long, hollow wire has inner radius R₁ and outer radius R₂. The wire carries current I uniformly distributed across the area of the wire. Use Ampère's law to find an expression for the magnetic field strength in the three regions 0 < r < R₁, R₁ < r < R₂, and R₂ < r.

Verified step by step guidance

1. Use Ampère's law, which states that the line integral of the magnetic field **B** around a closed loop is equal to μ₀ times the total current enclosed by the loop: ∮B·dl = μ₀I_enclosed. This will be applied to each of the three regions separately.
2. For the region 0 < r < R₁ (inside the hollow part of the wire): Since there is no current enclosed within this region (the current is distributed only in the conducting shell between R₁ and R₂), I_enclosed = 0. Substituting this into Ampère's law gives B = 0 in this region.
3. For the region R₁ < r < R₂ (inside the conducting shell): The current enclosed by a circular Amperian loop of radius r is proportional to the area of the loop within the conducting region. The current density J is given by J = I / (π(R₂² - R₁²)). The enclosed current is then I_enclosed = J × π(r² - R₁²). Substituting this into Ampère's law and solving for B gives B = (μ₀I(r² - R₁²)) / (2πr(R₂² - R₁²)).
4. For the region r > R₂ (outside the wire): The total current enclosed by an Amperian loop of radius r is the entire current I, since all the current is within the wire. Substituting I_enclosed = I into Ampère's law and solving for B gives B = μ₀I / (2πr).
5. Summarize the results: The magnetic field strength in the three regions is as follows: (1) B = 0 for 0 < r < R₁, (2) B = (μ₀I(r² - R₁²)) / (2πr(R₂² - R₁²)) for R₁ < r < R₂, and (3) B = μ₀I / (2πr) for r > R₂.

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. Mathematically, it is expressed as ∮B·dl = μ₀I_enc, where B is the magnetic field, dl is a differential length element along the loop, μ₀ is the permeability of free space, and I_enc is the enclosed current. This law is fundamental for analyzing magnetic fields in systems with symmetry, such as wires carrying current.

Magnetic Field Inside a Conductor

In a conductor carrying a uniform current, the magnetic field varies with distance from the center. For a hollow wire, the magnetic field is zero inside the inner radius (0 < r < R₁) since there is no enclosed current. Between the inner and outer radii (R₁ < r < R₂), the magnetic field can be calculated using Ampère's Law, considering the current enclosed by the Amperian loop.

Magnetic Field Outside a Conductor

For regions outside a current-carrying conductor (R₂ < r), the magnetic field can be determined by treating the entire current as if it were concentrated at the center. The magnetic field strength decreases with distance from the wire and is given by B = (μ₀I)/(2πr), where r is the distance from the center of the wire. This behavior is a consequence of the symmetry of the current distribution and the application of Ampère's Law.