Exercise 3.2.4: Two parallel charged sheets

Each sheet alone produces the field of Problem 3.2.3, of magnitude ${\rho }_S/(2{\epsilon }_0)$, directed away from the sheet when ${\rho }_S>0$ (towards it when ${\rho }_S<0$). By superposition, adding the $z$ components in each region:

For ${\rho }_{S2}=-{\rho }_{S1}$ (a capacitor) the field is ${\rho }_{S1}/{\epsilon }_0$ (along $+a_z$) between the sheets and zero outside, as in Problem 3.2.4. For ${\rho }_{S2}={\rho }_{S1}$ the field is zero between the sheets and ${\rho }_{S1}/{\epsilon }_0$ outside, pointing away from the pair (for ${\rho }_{S1}>0$). Figure 3.2.28 compares the two cases.

In every case the field jumps by ${\rho }_S/{\epsilon }_0$ across each sheet, as Gauss's law applied to a thin pillbox requires: