Fig. $1-16$

$1\mathrm{.}7.$ Assuming an ideal core $({}_3),$ calculate the flux density in the airgap of magnetic circuit shown in Fig. $1-16($a$).$

The electrical analog, Fig. $1-16($b$),$ may be reduced to the simpler Fig. $1-16($c$).$ From the latter, and Table $1-1,$

${}_g=$ airgap reluctance $=\frac{5{10}^{-3}}{{}_0(2040{10}^{-6})}=\frac{50}{8{}_0}$

${}_s=$ sleeve reluctance $=\frac{2{10}^{-3}}{{}_0(2020{10}^{-6})}=\frac{20}{4{}_0}$

${}_g-=$ total reluctance $={}_g+\frac{1}{2}{}_s=\frac{70}{8{}_0}$

${}_g=$ airgap flux $=\frac{F}{{}_s}=\frac{(50)(10)}{\frac{70}{8}{}_0}=\frac{400{}_0}{7}$

$B_g=$ airgap flux density $=\frac{{}_g}{A_g}=\frac{400\frac{{}_0}{7}}{2040{10}^{-6}}$

Substituting ${}_0=4{10}^{-7},$ we get $B_g=90mT,$ or $900$ gauss.