Gauss's Law and Electric Flux

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This accessible text edition preserves the document's narrative. See the interactive edition for typeset equations, diagrams and playback.

Gauss's Law

Gauss's law relates the total electric flux through a closed surface to the net enclosed charge: _S E dA=Q_ enc_0. For a point charge Q at the center of a spherical Gaussian surface, spherical symmetry gives E(4 r^2)=Q_0, E(r)=Q4_0r^2. Thus E r^-2, while the sphere area is A=4 r^2. Their product remains constant: _E=E(4 r^2)=Q_0.

Differential Form

For a volume charge density , Q_ enc=_V \,dV. Using the divergence theorem gives E=_0.

Physical Interpretation

A Gaussian surface is an imaginary closed surface used to calculate electric flux. For a positive point charge the field points radially outward; for a negative charge it points inward. A charge outside the Gaussian surface may affect the local electric field, but contributes zero net flux through the complete closed surface.

Symmetry

For a point charge, a spherical Gaussian surface is convenient. For an infinitely long line charge, a coaxial cylindrical surface is convenient. For an infinite plane charge, a pillbox surface is convenient.

ANITEX Interactive Visualization

The following block is interpreted by the NEXMASON Science Viewer.

What to Observe

During playback, the Gaussian sphere expands and contracts. The electric-field arrows should become shorter as the radius increases, while the displayed surface area increases. The total flux remains constant because the enclosed charge remains unchanged. Reversing the sign of Q reverses the field direction and the sign of the electric flux.