Write the chemistry of recharging the lead storage battery,highlighting all the materials that are involved during recharging.

(N/A) lead storage battery consists of an anode of lead $(Pb)$,a cathode of a grid of lead packed with lead dioxide $(PbO_2)$,and a $38\%$ $H_2SO_4$ solution as an electrolyte.
When the battery is in use (discharging),the reactions are:
Anode: $Pb_{\text{(s)}} + SO_4^{2-}{_{\text{(aq)}}} \rightarrow PbSO_{4\text{(s)}} + 2e^{-}$
Cathode: $PbO_{2\text{(s)}} + SO_4^{2-}{_{\text{(aq)}}} + 4H^+{_{\text{(aq)}}} + 2e^- \rightarrow PbSO_{4\text{(s)}} + 2H_2O_{\text{(l)}}$
Overall reaction: $Pb_{(s)} + PbO_2(s) + 2H_2SO_4(aq) \rightarrow 2PbSO_4(s) + 2H_2O_{(l)}$
On charging the battery,the reverse reaction takes place,where $PbSO_4$ deposited on the electrodes is converted back to $Pb$ and $PbO_2$,and $H_2SO_4$ is regenerated:
$2PbSO_4(s) + 2H_2O_{(l)} \rightarrow Pb_{(s)} + PbO_2(s) + 2H_2SO_4(aq)$