| 8.4 Electric Fields of Charge Distributions |
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Finding the electric field from a continuous charge distribution requires breaking the distribution into infinitesimal elements dq, computing the field contribution dE from each element using Coulomb's law, and integrating over the entire distribution. The principle of superposition guarantees that the total field is the vector sum of all contributions. Before setting up any integral, symmetry analysis determines which field components cancel, reducing the vector integral to a single-component scalar integral.
Infinite Line Charge
- The components parallel to the line cancel by symmetry, leaving only the perpendicular (radial) component
- Integration over the entire line then yields:
E = λ/(2πε₀r)
- The field falls off as 1/r, in contrast to the 1/r² dependence of a point charge
Finite Line
- The integration limits are finite, so the result depends on the geometry — specifically the angles subtended at the field point
- In the limiting case of large distance, the result must reduce to the point-charge field E = Q/(4πε₀r²)
Ring of Charge
- All elements are equidistant from a point on the axis, so the distance factor is constant and comes out of the integral
- By symmetry, only the axial component survives, giving:
E = Qx/(4πε₀(x² + R²)^(3/2))
Semicircular Arc
- Uses angular integration with charge element dq = λR dθ
- Symmetry leaves only the component along the bisecting axis, yielding the field at the center:
E = λ/(2πε₀R)
Infinite Uniformly Charged Cylinder
The infinite uniformly charged cylinder uses Gauss's law with a coaxial cylindrical Gaussian surface.
- Inside the cylinder, the enclosed charge grows as r² while the Gaussian surface area grows as r, so the field increases linearly with distance from the axis:
E = ρr/(2ε₀)
- Outside the cylinder (where a is the cylinder radius), all of the charge is enclosed, and the growing Gaussian surface area causes the field to fall off as 1/r:
E = ρa²/(2ε₀r)
- At the surface (r = a), both expressions give E = ρa/(2ε₀), confirming physical consistency
Every derived expression should be checked against known limiting cases — at large distances, extended distributions must reduce to the point-charge result. This is a standard consistency check on AP Physics C FRQs.