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Constant Magnetic Field, Magnetics

Problem 3.284

A thin iron ring with mean diameter d=50d=50 cm supports a winding consisting of N=800N=800 turns carrying current I=3.0 AI=3.0 \mathrm{~A}. The ring has a cross-cut of width b=2.0 mm.b=2.0 \mathrm{~mm} . Neglecting the scattering of the magnetic flux at the gap edges, and using the plot shown in Fig. 3.76,3.76, find the permeability of iron under these conditions.

Reveal Answer
3.284. From the theorem on circulation of the vector H\mathrm{H} we obtain B≈μ0NIb−μ0πdbH=1.51−0.987H(kA/m) B \approx \frac{\mu_{0} N I}{b}-\frac{\mu_{0} \pi d}{b} H=1.51-0.987 H(\mathrm{kA} / \mathrm{m}) Besides, BB and HH are interrelated as shown in Fig. 3.76.3.76 . The required values of HH and BB must simultaneously satisfy both relations. Solving this system of equations by means of plotting, we obtain H≈0.26kA/m,B≈1.25 T, and μ=B/μ0H≈4⋅103 H \approx 0.26 \mathrm{kA} / \mathrm{m}, B \approx 1.25 \mathrm{~T}, \text { and } \mu=B / \mu_{0} H \approx 4 \cdot 10^{3}
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