Charge Density and Setting Up the Field Integral

Coulomb's law works only for point charges, and a charged rod or disk is not a point. There's no single distance r you can plug in, since every bit of the rod sits a different distance from your field point. The fix is to chop the object into pieces so small that each one looks like a point charge dq. Each piece makes a tiny field dE = k dq / r², pointing from that piece toward or away from the field point. Add up all those dE contributions, and in the limit of infinitely many infinitely small pieces, that sum becomes an integral.

To write dq, you need a charge density that matches the object's shape. A thin rod or ring uses linear density lambda = Q/L, so dq = lambda dx. A flat sheet or shell uses surface density sigma = Q/A, so dq = sigma dA. A solid block or sphere uses volume density rho = Q/V, so dq = rho dV. If the charge is spread evenly, the density is constant and pulls straight out of the integral; if it varies with position, the density itself is a function you integrate along with dq.

The trap is direction. Every dE is a vector, and pieces on opposite sides of symmetric shapes often point partly in opposite directions. Components perpendicular to the axis of symmetry cancel in pairs, leaving only the components along that axis to survive the integral. This setup, dq and dE written in terms of one variable, is exactly what you need before Gauss's law can later hand you a shortcut for the right symmetry.