(III) A coaxial cable consists of a solid inner conductor of radius 𝑅₁ , surrounded by a concentric cylindrical tube of inner radius 𝑅₂ and outer radius 𝑅₃ (Fig. 28–45). The conductors carry equal and opposite currents I₀ distributed uniformly across their cross sections. Determine the magnetic field at a distance 𝑅 from the axis for: (a) 𝑅 < 𝑅₁ ; (b) 𝑅₁ < 𝑅 < 𝑅₂ ; (c) 𝑅₂ < 𝑅 < 𝑅₃ ; (d) 𝑅 > 𝑅₃ . (e) Let I₀ = 1.50 A, 𝑅₁ = 1.00 cm , 𝑅₂ = 2.00 cm , and 𝑅₃ = 2.50 cm Graph B from 𝑅 = 0 to 𝑅 = 3.00 cm.


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Verified step by step guidance

1. Use Ampere's Law, which states that the line integral of the magnetic field around a closed loop is equal to the permeability of free space times the current enclosed by the loop. The formula is \( \oint \vec{B} \cdot d\vec{l} = \mu_0 I_{\text{enc}} \).
2. For part (a) where \( R < R_1 \), consider a circular Amperian loop of radius \( R \) inside the inner conductor. Since the current is uniformly distributed, calculate the current enclosed by this loop as a fraction of the total current based on the area ratio.
3. For part (b) where \( R_1 < R < R_2 \), the entire current \( I_0 \) from the inner conductor is enclosed within the Amperian loop. However, there is no contribution from the outer conductor in this region.
4. For part (c) where \( R_2 < R < R_3 \), the current enclosed by the Amperian loop includes the total current from the inner conductor and a fraction of the current from the outer conductor, depending on the area of the outer conductor enclosed by the loop.
5. For part (d) where \( R > R_3 \), the Amperian loop encloses both conductors completely. Since the currents in the conductors are equal and opposite, the net current enclosed by the loop is zero, resulting in a magnetic field of 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 in systems with symmetrical current distributions, such as coaxial cables.

Magnetic Field Inside a Conductor

The magnetic field inside a conductor carrying a uniform current is determined by the current density and the geometry of the conductor. For a solid cylindrical conductor, the magnetic field increases linearly with distance from the center until reaching the surface. Inside a coaxial cable, the magnetic field behaves differently depending on whether the point of interest is inside the inner conductor, between the conductors, or outside the outer conductor.

Superposition Principle

The Superposition Principle states that in a linear system, the total response (in this case, the magnetic field) caused by multiple sources (currents) is the sum of the responses that would be caused by each source individually. In the context of the coaxial cable, this principle allows us to calculate the net magnetic field at any point by considering the contributions from both the inner and outer conductors, taking into account their equal and opposite currents.