Charge distribution

Also called Continuous charge distribution

A charge distribution is charge spread along a line, over a surface or through a volume rather than concentrated at a point. AP Physics C finds its electric field by integrating the contributions of its pieces; AP Physics 2 stays with small numbers of point charges.

Once charge is spread out, there is no single rr to put in Coulomb's law, so the field has to be assembled piece by piece.

The C: E&M treatment. Topic 8.4 is called Electric Fields of Charge Distributions, and EK 8.4.A.1 says expressions for the field of specified distributions can be found using integration and the principle of superposition. The sheet prints both the field and the potential in that form:

E⃗=14πε0∫dqr2r^V=14πε0∫dqr\vec{E} = \frac{1}{4\pi\varepsilon_0}\int \frac{dq}{r^2}\hat{r} \qquad V = \frac{1}{4\pi\varepsilon_0}\int \frac{dq}{r}

EK 8.4.A.2 adds that symmetry considerations simplify the analysis, which is the door into Gauss's law.

The boundary statement is a closed list. C: E&M expects calculus to be used for the field of an infinitely long uniformly charged wire or cylinder at a distance from its central axis, a thin ring of charge at a location along the axis of the ring, a semicircular arc or part of a semicircular arc at its centre, and a finite wire or line charge at a point collinear with it or along its perpendicular bisector. Anything else is out of scope for a calculus setup.

AP Physics 2 goes the other way. Its boundary statements cap you at four or fewer charged objects for field and potential calculations, allowing more only in situations of high symmetry, and keep fields inside insulators qualitative.

What you integrate is a charge density.

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