A parallel$-$plate capacitor $($area $\backslash ($ A $\backslash )$ and separation $\backslash ($ d $\backslash ))$ initially contains air for which $\backslash (\backslash$varepsilon$\_\{\backslash$mathrm$\{$r$\}\}=1\backslash ).$ Its plates are connected, via a closed switch $\backslash (\backslash$mathbf$\{$S$\}\backslash ),$ to a battery which maintains a potential difference $\backslash ($ V$\_\{0\}\backslash )$ across them. In all subsequent calculations, assume that fringing effects are negligible

$($a$)$ Suppose the space between the plates is partially filled with a slab of linear dielectric material $($area $\backslash ($ A $\backslash )$ and thickness $\backslash ($ h $\backslash ))$ as shown above. Calculate the following:

$-$ The electric field in the air gap,

$-$ The electric field in the dielectric,

$-$ The capacitance $\backslash ($ C $\backslash )($expressed in terms of the vacuum capacitance $\backslash ($ C$\_\{0\}\backslash )).$

$($b$)$ The dielectric slab is now removed, S is opened and the slab returned to its original position between the plates $($note the sequence in which these changes are made$).$ Repeat the calculations for $($a$)$ above.