The diagram shows a uniformly charged hemisphere of radius R. It has volume charge density $\mathrm{\rho }$. If the electric field at a point 2R distance above its centre is E, then what is the electric field at the point which is 2R below its centre?

Let us complete the sphere. Electric field at A due to lower part = electric field at B due to upper part = E (given)
Electric field at B due to lower part = electric field due to full sphere - electric field due to upper part

$=\frac{\mathrm{kQ}}{(2\mathrm{R}{)}^2}-\mathrm{E}=\frac{1}{4{\mathrm{\pi \epsilon }}_0}\frac{\mathrm{\rho }(4/3){\mathrm{\pi R}}^3}{4{\mathrm{R}}^2}-\mathrm{E}=\frac{\mathrm{\rho R}}{12{\mathrm{\epsilon }}_0}-\mathrm{E}$