Q.Obtain the expression for the electric field due to an infinitely long charged wire.

Step 1. Symmetry. By the symmetry of an infinite wire, the field can only point radially outward (or inward) from the wire and can only depend on the perpendicular distance $r$, never on position along the wire.

Step 2. Gaussian surface. Choose a cylinder of radius $r$ and length $l$, coaxial with the wire.

Step 3. Flux. The two flat end caps contribute zero flux (E runs parallel to them); the curved side contributes $\Phi_E=E\times(2\pi r l)$, since E is constant in magnitude and perpendicular to that surface everywhere.

Step 4. Enclosed charge. $Q_{\text{enclosed}}=\lambda l$. …

Choose a coaxial cylindrical Gaussian surface, argue only the curved side contr …

• Including the flat end caps in the flux calculation -- they contribute zero, since E …