A hollow cylinder has a charge q coulomb within it. If φ is the...

Step 1: Total flux through cylinder using Gauss Law By Gauss Law, total flux through the cylinder is given by: (\phi_{\text{total }}=\frac{\mathrm{q}}{\varepsilon_0} \ldots (1)) Step 2: Flux through plane surface $A$ As surfaces (\mathrm{A} \& \mathrm{C}) is symmetrical so (\phi_{\mathrm{A}}=\phi_{\mathrm{C}}) (\phi_{\text{total }}=\phi_{\mathrm{A}}+\phi_{\mathrm{B}}+\phi_{\mathrm{C}}=\frac{\mathrm{q}}{\varepsilon_{\mathrm{o}}} \quad [\text{Using Equation (1)] }) Given Flux through curved surface (\phi_B=\phi) So, $$\begin{aligned} & 2 \phi_A+\phi=\frac{q}{\varepsilon_0} \\ & \Rightarrow \phi_A=\frac{1}{2}\left(\frac{q}{\varepsilon_0}-\phi\right) \end{aligned}$$)$