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When charge is distributed continuously, the field is found by integrating contributions from infinitesimal elements: dE = k*dq/$r^2$. Three density types describe the distribution: linear density lambda $\frac{C}{m}$ for wires, surface density sigma (C/$m^2$) for sheets, and volume density rho (C/$m^3$) for solid objects.

Important standard results that must be memorized for JEE:

Uniformly charged ring (charge Q, radius R) on axis at distance x: E = $\frac{kQx}{x^2 + R^2}$^$\frac{3}{2}$. At center (x=0): E = 0. Maximum field at x = $\frac{R}{sqrt}$(2): $E_{max}$ = 2$\frac{kQ}{3*sqrt(3}$*$R^2$). For x >> R: E ~ kQ/$x^2$ (point charge behavior).

Infinite line charge (lambda): E = $\frac{lambda}{2*pi*epsilon_0*r}$ = 2k*lambda/r, radially outward. Falls as 1/r.

Infinite plane sheet (sigma): E = $\frac{sigma}{2*epsilon_0}$, uniform, perpendicular to surface. Independent of distance.

Two parallel infinite sheets with equal and opposite charges (+sigma, -sigma): E = $\frac{sigma}{epsilon_0}$ between them (uniform), E = 0 outside. This is the parallel plate capacitor configuration.

Uniformly charged solid sphere (Q, radius R): Outside (r >= R): E = $\frac{kQ}{r}$^2. Inside (r < R): E = $\frac{kQr}{R}$^3 = rho*$\frac{r}{3*epsilon_0}$. The field increases linearly inside and decreases as 1/$r^2$ outside, with maximum at the surface.

Uniformly charged disk on axis at distance x: E = $\frac{sigma}{2*epsilon_0}$[1 - x/sqrt($x^2$ + $R^2$)].

The strategy for integration problems: identify the symmetry axis, decompose dE into components, note which components cancel by symmetry, and integrate only the surviving component. Always verify limits: does the result reduce to known cases (point charge at large distance, infinite sheet for large radius)?