All Problems Constant Magnetic Field, Magnetics
Problem 3.284
A thin iron ring with mean diameter \(d=50\) cm supports a winding consisting of \(N=800\) turns carrying current \(I=3.0 \mathrm{~A}\). The ring has a cross-cut of width \(b=2.0 \mathrm{~mm} .\) Neglecting the scattering of the magnetic flux at the gap edges, and using the plot shown in Fig. \(3.76,\) find the permeability of iron under these conditions.
3.284. From the theorem on circulation of the vector \(\mathrm{H}\) we obtain \[ B \approx \frac{\mu_{0} N I}{b}-\frac{\mu_{0} \pi d}{b} H=1.51-0.987 H(\mathrm{kA} / \mathrm{m}) \] Besides, \(B\) and \(H\) are interrelated as shown in Fig. \(3.76 .\) The required values of \(H\) and \(B\) must simultaneously satisfy both relations. Solving this system of equations by means of plotting, we obtain \[ H \approx 0.26 \mathrm{kA} / \mathrm{m}, B \approx 1.25 \mathrm{~T}, \text { and } \mu=B / \mu_{0} H \approx 4 \cdot 10^{3} \]