Question $2.$ Inductance

$[20$ marks$]$

An inductor was constructed with a square uniform cross$-$sectional area of $A_c=4c{m}^2.$ The core material has a relative permeability of ${}_r=2000.$ The inductor was constructed with a single air gap, $g,$ of length $l_g=0\mathrm{.}5mm,$ as shown below in Figure $2.$

Using the inductor shown in Figure $2$ calculate:

$($a$)$ The reluctance of the core and reluctance of the air gap. $($Assume no fringing effects$).$

$[5$ marks$]$

$($b$)$ The total inductance when the left leg is wound with $N=50$ turns as shown in Figure $2.($Assume no fringing effects$).$

$[5$ marks$]$

$($c$)$ The total inductance, when the fringing flux factor is given by:

$F=\frac{(a+l_g)(b+l_g)}{ab}$

where a and $b$ are the width and length of the cross$-$section of the core.

$[5$ marks$]$

$($d$)$ The current in the inductor, I, when the flux density within the core is $500$ mT $.($Include the effect of fringing as determined in part $($c$)).$

$[5$ marks$]$

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