Potential of a Continuous Charge Distribution

Electric field from a continuous object is a pain because every little piece dq contributes a vector, pointing a different direction, and you have to break each one into components before adding. Potential skips all that. Every dq contributes a scalar, k dq / r, and scalars just add. No angles, no components, no worrying about symmetry cancelling one direction but not another. That is the whole reason 9.2 exists: same physical setup, dramatically easier integral.

The method is always the same three steps. Write dq in terms of a density (lambda dx for a rod, sigma dA for a disk, or just total charge Q if every piece is equidistant, as on a ring). Write r, the distance from that piece to your point, in terms of your integration variable. Then integrate k dq / r over the whole object. For a rod of length L with linear density lambda, charge a distance x away along its line, you get V = k lambda ln of a ratio of distances, found by integrating k lambda dx / x from one end to the other. For a ring, every point is the same distance √(x²+R²) from an axial point, so the integral is trivial: V = kQ/√(x²+R²), no integration really needed since r never changes.

A disk is a stack of rings, so you integrate the ring result over radius from 0 to R, which is how V(0) = sigma R/(2 epsilon₀) appears. This same stacking idea reappears once you take the gradient of V to recover E, which is coming next: differentiating these potential expressions is almost always easier than the vector field integrals they replace.