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← Electric Charges, Fields, and Gauss's Law
8.6

Gauss's Law

Gauss's law relates the total electric flux through a closed surface to the net charge enclosed within it.

15–25% of exam
All TopicsPrevious8.5Electric Flux
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Context

What this topic is and why it exists

Imagine wrapping a gift in invisible shrink wrap — no matter how loosely or tightly you wrap it, the present inside doesn't change.
That's the essence of Gauss's law.
You construct an imaginary closed surface, called a Gaussian surface, around a charge distribution, and the total electric flux pushing through that surface depends only on the charge trapped inside: ∮ E · dA = qencq_{enc}qenc​/ε₀.
Blow the surface up to twice its size, and the flux stays the same because every field line that exits still originates from the same enclosed charge.
The real power of Gauss's law is strategic laziness.
You choose your Gaussian surface so that the electric field is either perfectly perpendicular to the surface (making E · dA simple multiplication) or perfectly parallel (making that piece contribute zero).
Spheres around point charges, cylinders around long wires, flat pillboxes straddling charged planes — each shape is chosen to exploit symmetry and collapse a nasty integral into arithmetic.
One more thing: if charge is spread out with some density function ρ(r), you find the enclosed charge by integrating that density over the volume inside your surface — qencq_{enc}qenc​ = ∫ρ dV.
Gauss's law is so fundamental it earned the title of Maxwell's first equation, the opening line of the complete story of electromagnetism.
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