$4.$ We inject electrons into a p$-$type semiconductor $5$ microns long such that the concentration

varies linearly from $1020$ cm$-3$ to zero from left to right. If the mobility of the electrons is $500$ cm$2/$V$-$

s$,$ what is the current density if the electric fields are negligible?

$5.$ We shine $1017$ photons$/$cm$2/$s which are all absorbed near the surface x$=0$ of a p$-$type

semiconductor, raising the temperature of the sample to $500$K$.$ If the minority carrier lifetime is

$200$ ns in this material, electron mobility is $2000$ cm$2/$V$-$s$,$ hole mobility is $500$ cm$2/$V$-$s$,$ calculate

the electron diffusion current density $20$ microns from the surface.