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Length of wire forming solenoid

Short answer

Turn density n = B/(μ₀I) = 1768 m⁻¹, total turns N = nℓ ≈ 4600. Wire length = N π d ≈ 188 m.

A solenoid 2.6 m long and 1.3 cm in diameter carries a current of 9A. The magnetic field inside the solenoid is 20mT. Find the length of the wire forming the solenoid.

Inside a long solenoid, B = μ₀nI, where n = N/ℓ is the number of turns per metre. Each turn is a circle of circumference πd.

Given: ℓ = 2.6 m, d = 1.3 cm = 0.013 m, I = 9 A, B = 20 mT = 0.020 T.

n = B/(μ₀ I) = 0.020 / (4π×10⁻⁷ × 9)
  = 0.020 / 1.131×10⁻⁵
  = 1768 turns/m

N = n ℓ = 1768 × 2.6 = 4598 turns

Length of wire = N × π d = 4598 × π × 0.013
               = 4598 × 0.04084
               = 187.8 m

Answer: length of wire ≈ 188 m (about 4600 turns).

✓ Answer checked by Claude Opus 5.5 (high effort)

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