A pair of straight parallel thin wires, such as a lamp cord, each of radius r, are a distance 𝓁 apart and carry current to a circuit some distance away. Ignoring the field within each wire, show that the inductance per unit length is (μ₀/π) ln[(𝓁 - r) /r].

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Start by recalling the formula for the inductance per unit length of a pair of parallel wires. The inductance arises from the magnetic field generated by the current in one wire and the flux linkage with the other wire. The magnetic field outside a long straight wire carrying current I is given by Ampere's law: B = (μ₀I) / (2πr), where r is the radial distance from the wire.

Consider the magnetic flux linkage between the two wires. The magnetic field from one wire at a distance x from its center contributes to the flux linkage with the other wire. The flux per unit length is calculated by integrating the magnetic field over the region between the wires, from r (the surface of the wire) to (𝓁 - r) (the closest point to the other wire).

The flux linkage per unit length, λ, is given by: λ = ∫[r to (𝓁 - r)] (B dx), where B = (μ₀I) / (2πx). Substitute B into the integral: λ = ∫[r to (𝓁 - r)] [(μ₀I) / (2πx)] dx.

Solve the integral: λ = (μ₀I) / (2π) ∫[r to (𝓁 - r)] (1/x) dx. The integral of 1/x is ln(x), so λ = (μ₀I) / (2π) [ln(𝓁 - r) - ln(r)]. Using the logarithmic property ln(a) - ln(b) = ln(a/b), this simplifies to λ = (μ₀I) / (2π) ln[(𝓁 - r) / r].

Finally, the inductance per unit length, L, is defined as the flux linkage per unit current: L = λ / I. Substituting λ, we get L = [(μ₀I) / (2π) ln((𝓁 - r) / r)] / I. Simplify by canceling I, resulting in L = (μ₀ / π) ln[(𝓁 - r) / r].

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주요 개념

질문에 올바르게 답하기 위해 반드시 이해해야 하는 핵심 개념들은 다음과 같습니다.

Inductance

Inductance is a property of an electrical circuit that quantifies the ability of a conductor to induce an electromotive force (EMF) in itself or in another conductor due to a change in current. It is measured in henries (H) and is crucial for understanding how circuits respond to alternating current (AC) and the behavior of inductors in various applications.

Magnetic Field Due to Current

When an electric current flows through a conductor, it generates a magnetic field around it. For parallel wires, the magnetic field produced by one wire affects the other, leading to interactions that are essential for calculating inductance. The strength and direction of this magnetic field depend on the current and the distance from the wire.

Logarithmic Relationship in Inductance

The formula for inductance per unit length in parallel wires involves a logarithmic relationship, which arises from the geometry of the wires and their separation. Specifically, the inductance is proportional to the natural logarithm of the ratio of the distance between the wires minus the radius of the wires to the radius itself, reflecting how the magnetic field lines interact in space.