$1-13.$ A core with three legs is shown in Figure P$1-10.$ Its depth is $5$ cm $,$ and there are $400$ turns on the center leg. The remaining dimensions are shown in the figure. The core is composed of a steel having the magnetization curve shown in Figure $1-10$c$.$ Answer the following questions about this core:

$($a$)$ What current is required to produce a flux density of $0\mathrm{.}5$ T in the central leg of the core?

$($b$)$ What current is required to produce a flux density of $1\mathrm{.}0$ T in the central leg of the core? Is it twice the current in part $($a$)?$

$($c$)$ What are the reluctances of the central and right legs of the core under the conditions in palt $($a$)?$

$($d$)$ What are the reluctances of the central and right legs of the core under the conditions in part $($b$)?$

$($e$)$ What conclusion can you make about reluctances in real magnetic cores?