Q$1.($a$)$ Transfer the two$-$source network left to terminals a and $b($Figure Q$1$a$)$ into a singlesource circuit based on Thevenin's theory. Estimate the power consumption of the resistor R for $R=1,3,5$ and $7,$ respectively.

$(10$ marks$)$

$($b$)$ A ring iron$-$cored coil has a mean circumference length of $500$ mm and a uniform crosssectional area of $100$ mm $,$ with $1000$ turns tightly and uniformly wound on the core. There is a $2$ mm long air$-$gap in the core to increase the coil the linearity. The magnetic fringing effect in the air$-$gap and the coil leakage may be ignored. The iron core has a B $-$ H characteristic, as shown in Figure Q$1$b:

$\backslash$table$[[$H $($kATurns$/$m$),1,2,4,8,16],[$B $($Tesla$),0\mathrm{.}5,1,2,4,6]]$

Figure Q$1$b

A two$-$ampere DC current is applied to the coil to produce a strong magnetic field in the air gap. Estimate the magnetic flux density in the air$-$gap and the coil inductance at this current level.

$(15$ marks$)$