A hollow cylinder has a charge q coulomb within it. If $\mathrm{\varphi }$ is the electric flux in unit of volt-meter associated with the curved surface B, the flux linked with the plane surface A in unit of volt-meter will be

Gauss's law states that the net electric flux through any closed surface is equal to the net charge inside the surface divided by ε₀·

$\text{ i.e., } {\mathrm{\varphi }}_{\text{total }}=\frac{\mathrm{q}}{{\mathrm{\epsilon }}_0}$

Let electric flux linked with surfaces A, B and C are ${\mathrm{\varphi }}_{\mathrm{A}}, {\mathrm{\varphi }}_{\mathrm{B}} \mathrm{and} {\mathrm{\varphi }}_{\mathrm{C}}$, respectively. That is

$$\begin{gathered} {\mathrm{\varphi }}_{\text{total }}={\mathrm{\varphi }}_{\mathrm{A}}+{\mathrm{\varphi }}_{\mathrm{B}}+{\mathrm{\varphi }}_{\mathrm{C}} \\ \text{ Since }{\mathrm{\varphi }}_{\mathrm{C}}={\mathrm{\varphi }}_{\mathrm{A}} \\ \therefore 2{\mathrm{\varphi }}_{\mathrm{A}}+{\mathrm{\varphi }}_{\mathrm{B}}={\mathrm{\varphi }}_{\text{total }}=\frac{\mathrm{q}}{{\mathrm{\epsilon }}_0} \\ \text{ or } {\mathrm{\varphi }}_{\mathrm{A}}=\frac{1}{2}\left(\frac{\mathrm{q}}{{\mathrm{\epsilon }}_0}-{\mathrm{\varphi }}_{\mathrm{B}}\right) \\ \text{ But } {\mathrm{\varphi }}_{\mathrm{B}}=\mathrm{\varphi } \left(\mathrm{given}\right) \\ \text{ Hence, }{\mathrm{\varphi }}_{\mathrm{A}}=\frac{1}{2}\left(\frac{\mathrm{q}}{{\mathrm{\epsilon }}_0}-\mathrm{\varphi }\right) \end{gathered}$$