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Grade 11-12 (age 16-18)

high school physics Electric Field and Gauss's Law

Wrap a charge in a closed surface like a balloon, and the number of field lines passing out through it stays the same no matter how you change the balloon's size or shape. Written as an equation, Gauss's law says the electric flux through a closed surface equals the enclosed charge divided by ε₀. When the charge is spread with good symmetry, as in a uniformly charged ball, this law gives the electric field with very little work. In the picture, changing the charge Q changes the number of field lines and the flux in proportion to Q, and the two Gaussian surfaces of different sizes are always crossed by the same number of lines.

Intuition of Gauss's Law

🎈 Think of a Balloon
①Wrap a charge in a balloon, and a fixed number of field lines pass out through the balloon
②Blow the balloon up or squash it, and every line from the charge still crosses the balloon once, so the total flux stays the same
③So the electric flux through a closed surface depends only on the charge inside, and its value is Q/ε₀

Gaussian Surfaces and Electric Flux

4 μC
💡 Reading the Picture
①The larger the charge, the more field lines there are and the larger the flux through each surface
②For a positive charge the lines point outward; for a negative charge they come in from outside
③S₁ and S₂ are different sizes, but every line crosses each surface once, so the flux through the two is the same
④Lines from a charge outside a surface go in and come back out, so they add nothing to the total flux

The Equation of Gauss's Law

Gauss's Law
∮ E · dA = Qε₀
Electric flux through a closed surface = enclosed charge ÷ permittivity of free space
Applying It to Spherical Symmetry
E × 4πr² = Qε₀ → E = Q4πε₀r² = kQr²
On a sphere around a point charge, E is the same everywhere on the surface, so Coulomb's law follows directly from Gauss's law
🔑 Where Gauss's Law Pays Off
①For a charge distribution with good symmetry, Gauss's law gives the electric field easily
②Spherical symmetry: a point charge or a uniformly charged sphere
③Cylindrical symmetry: an infinitely long line of charge, with field E = λ/(2πε₀r)
④Inside a conductor E = 0, and any excess charge gathers on the surface

Worked Examples

Example 1
A closed Gaussian surface encloses charges of +2Q and −Q. What is the total electric flux through it?
1
The total flux depends only on the net charge inside the surface.
Φ = Qencε₀
2
Substitute the net enclosed charge Qenc = 2Q − Q = Q.
Φ = 2Q − Qε₀ = Qε₀
▸ Q/ε₀
Charges of opposite sign cancel when added. Some of the lines leaving +2Q end on −Q, so the lines that get out through the surface amount to Q.
Example 2
If the radius of a spherical Gaussian surface around a point charge Q is doubled, how does the total electric flux change?
1
Gauss's law looks only at the charge inside, whatever the shape or size of the surface.
Φ = Qε₀ (independent of r)
2
The enclosed charge Q is unchanged, so the flux is unchanged too.
Φ(2r) = Qε₀ = Φ(r)
▸ No change
As with S₁ and S₂ in the picture, a larger surface is crossed by the same number of lines. The field E on the surface, however, drops in inverse proportion to r², so keep flux and field apart.

Summary

Gauss's Law
∮ E · dA = Qenc / ε₀
ε₀ = 8.85 × 10⁻¹² C²/(N·m²)
CSAT-style problem, adapted
In vacuum the field at distance r from a point charge Q is E. What is the field at distance 2r?
①2E
②E2
③E4
④4E
⑤E
▸ ③ E4
1
Applying Gauss's law with spherical symmetry gives E = kQ/r², so the field is inversely proportional to the square of the distance.
E = kQr2
2
If r doubles, r² becomes 4 times larger, so E becomes 1/4 as large.
E(2r) = kQ(2r)2 = E4
🎯 Exam Points
①Gauss's law ∮E·dA = Q/ε₀ is about the total flux through a closed surface
②Choose a Gaussian surface that uses the symmetry so that E is constant on it
③With spherical symmetry, a spherical Gaussian surface gives E = kQ/r²
④The electric field inside a conductor is 0, and the charge sits only on the surface
⑤In electrostatics, Gauss's law and Coulomb's law carry the same content
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Electromagnetic Induction

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