Math Problem Statement

Consider two wires both aligned with the 𝑧-axis at positions 𝑦 = ±𝑦0 with 𝑦0 = 6[cm]. The wires have radius 𝑟0 = 5[mm] and they are carrying equal and opposite current ±𝐼0 = ±1 [mA]. If we assume the current is uniformly distributed around the surface of the wire and that 𝑦0 ≫ 𝑟0, verify that the inductance per unit length of this structure is L^'=μ_r μ_0 (cosh^(-1) (〖y_0/r〗_0 ))/π

Solution

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Math Problem Analysis

Mathematical Concepts

Electromagnetic Theory
Inductance
Hyperbolic Functions
Magnetic Fields

Formulas

L' = (μr * μ0 * cosh^(-1)(y0 / r0)) / π
Magnetic field: B(r) = (μ0 * I0) / (2π * r)
cosh^(-1)(x) = ln(x + sqrt(x^2 - 1))

Theorems

Ampère's Law
Magnetic Energy in Parallel Wires
Mutual Inductance

Suitable Grade Level

Undergraduate Level - Physics/Engineering