π₯ The Move
Gaussβs Law with symmetry β pick a surface that makes \(E\) constant over the surface (or zero on parts). Use for infinite planes, lines, and cylinders; spheres with spherical symmetry. Avoid when the symmetry doesnβt let \(E\) be constant on your surface.
π Canonical Problem
An infinitely long cylinder of radius \(R\) has uniform volume charge density \(\rho\). Find the electric field magnitude \(E(r)\) for (i) \(r<R\) and (ii) \(r\ge R\).
- Given: \(\rho,\ R,\ \varepsilon_0\)
- Find: \(E(r)\) (radial) as a function of \(r\)
- Assume: infinite cylinder; end effects neglected.
Sketch notes (no figure):
- Choose a coaxial Gaussian cylinder: radius \(r\), length \(L\).
- By symmetry, \(\vec E = E(r)\,\hat r\) is radial and constant on the curved surface; flux through endcaps is zero.
- Gauss: \(\displaystyle \Phi_E=\oint \vec E\cdot d\vec A = \dfrac{Q_{\text{enc}}}{\varepsilon_0}\).
π Setup (Preview)
- Surface + symmetry. Use curved area only: \(\Phi_E=E(r)\,(2\pi rL)\).
- Charge enclosed. Inside: \(Q_{\text{enc}}=\rho\,(\pi r^2 L)\). Outside: \(Q_{\text{enc}}=\rho\,(\pi R^2 L)\).
Full derivation, exam-speed form, and traps at Black Belt.