Electric Fields of Continuous Charge Distributions: Lines, Rings, Disks, and Planes

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What is a continuous charge distribution?

A model that treats charge as spread continuously along a line, across a surface, or through a volume.

What do linear, surface, and volume charge densities measure?

Charge per length ($\lambda$), area ($\sigma$), and volume ($\rho$), respectively.

What is the principle of superposition for electric fields?

The net field is the vector sum of the fields contributed by the charge elements.

What does $\hat r$ indicate in a charge element's field contribution?

A unit vector pointing from the source element toward the observation point.

How does the field of an infinite uniform line depend on perpendicular distance $z$?

Its magnitude is proportional to $1/z$.

Quiz questions

1. At a point above the midpoint of a uniformly charged straight segment centered on the $x$-axis, what happens to the horizontal field components from elements at $+x$ and $-x$?Answer: They cancel because the elements are symmetrically placed.
2. If the perpendicular distance from an infinite uniformly charged line is doubled, how does the field magnitude change?Answer: It is reduced to half its original value.
3. A uniformly charged ring has total charge $q$ and radius $R$. Which expression gives the axial field at distance $z$ from its center?Answer: $\frac{1}{4\pi\varepsilon_0}\frac{qz}{(z^2+R^2)^{3/2}}\,\hat z$
4. How can a uniformly charged circular disk be divided to calculate its axial electric field?Answer: Into concentric rings with area elements $dA=2\pi r'\,dr'$.
5. Two parallel infinite planes carry equal and opposite surface charge densities. What is the net field outside the planes?Answer: Zero, because the two fields have equal magnitudes and opposite directions.