In a simplified model of an undoped semiconductor, the actual distribution of energy states may be replaced by one in which there are $N_v$ states in the valence band, all these states having the same energy $E_v$, and $N_c$ states in the conduction band, all these states having the same energy $E_c$. The number of electrons in the conduction band equals the number of holes in the valence band. (a) Show that this last condition implies that
$$
\frac{N_c}{\exp \left(\Delta E_c / k T\right)+1}=\frac{N_v}{\exp \left(\Delta E_v / k T\right)+1},
$$
in which
$$
\Delta E_c=E_c-E_{\mathrm{F}} \text { and } \Delta E_v=-\left(E_v-E_{\mathrm{F}}\right) .
$$
(b) If the Fermi level is in the gap between the two bands and its distance from each band is large relative to $k T$, then the exponentials dominate in the denominators. Under these conditions, show that
$$
E_{\mathrm{F}}=\frac{\left(E_c+E_{\mathrm{v}}\right)}{2}+\frac{k T \ln \left(N_{\mathrm{v}} / N_c\right)}{2}
$$
and that, if $N_v \approx N_c$, the Fermi level for the undoped semiconductor is close to the gap's center.