Applying Gauss's Law to Symmetric Charge Distributions · seed 1 · A4, ink-friendly. The answer key prints on its own page for grown-ups.

Letting symmetry do the integral

Science · Electricity & Magnetism · ages 18-19
Name ______________________   Date ____________

  1. Spheres, infinite cylinders, and infinite planes are the three shapes that qualify.

    Circle one:   True   False

  2. The law is always true, but when is it useful for finding the field?

    • Always, for every charge shape
    • Only when symmetry cooperates
    • Never, it is purely decorative

  3. What is the first move in the problem solving strategy?

    • Integrate the charge density at once
    • Identify the spatial symmetry of the charge
    • Guess the final field direction

  4. Inside a uniformly charged sphere, which charge counts?

    • The full total charge always
    • Zero charge no matter what
    • Only the fraction enclosed by your surface

  5. What does a uniformly charged sphere look like from outside?

    • Exactly like a point charge at its center
    • Like no charge at all
    • Like a uniform field everywhere

  6. An infinite line of charge gives what distance behavior?

    • Uniform field at all distances
    • Field growing with distance squared
    • Field fading as one over distance

  7. An infinite sheet gives a uniform field on each side. Why is that plausible?

    • The flat surface area trick makes distance drop out
    • Sheets hold no charge at all
    • Uniform means the field must be zero

  8. A student applies the shortcut to a lumpy, uneven charge blob. What goes wrong?

    • Nothing, the shortcut fits every shape
    • No matching surface has constant aligned field, so E cannot be pulled out
    • The enclosed charge is always zero for blobs
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Answer key

For grown-ups. Fold this page away before handing over the rest.

Letting symmetry do the integral W1-mt_1LZboKPJ4D-s1

  1. True · Exactly these symmetries keep the field uniform over a matching surface.
  2. Only when symmetry cooperates · You need constant field along the surface normal to pull E out of the sum.
  3. Identify the spatial symmetry of the charge · Symmetry picks the surface, and everything else follows from that choice.
  4. Only the fraction enclosed by your surface · The right hand side of the law sees just the enclosed part.
  5. Exactly like a point charge at its center · The outside surface encloses the whole charge, giving the point charge field.
  6. Field fading as one over distance · The cylinder area grows with distance, so the field falls as one over distance.
  7. The flat surface area trick makes distance drop out · The pillbox flux grows with patch area the same way enclosed charge does.
  8. No matching surface has constant aligned field, so E cannot be pulled out · Without symmetry the flux sum stays an integral, and the law alone cannot isolate E.
Worksheet · LightMySky