State Gauss's law in integral form.

What charge appears in Gauss's law: total charge in the universe, nearby charge, or net enclosed charge?

A closed surface encloses no net charge. What is the net electric flux through it?

A closed surface encloses \(+2.0\,\mathrm{nC}\). Find the net electric flux. Use \(\epsilon_0=8.85\times10^{-12}\,\mathrm{C^2\,N^{-1}\,m^{-2}}\).

A closed surface encloses \(-5.0\,\mathrm{nC}\). Find the net electric flux and state its sign meaning.

A Gaussian surface encloses charges \(+3.0\,\mathrm{nC}\), \(-1.0\,\mathrm{nC}\), and \(-4.0\,\mathrm{nC}\). Find the net flux.

The net flux through a closed surface is \(6.0\times10^3\,\mathrm{N\,m^2\,C^{-1}}\). Find the net enclosed charge.

A \(+4.0\,\mathrm{nC}\) charge is outside a closed surface. What is its contribution to the net flux through that surface?

A \(+4.0\,\mathrm{nC}\) charge is inside a closed surface, and a \(-7.0\,\mathrm{nC}\) charge is outside. Find the net flux through the surface.

A point charge \(q\) is at the center of a spherical Gaussian surface of radius \(r\). Use Coulomb's field to derive the net flux through the sphere.

A point charge \(q\) is moved from the center of a spherical Gaussian surface to an off-center position, still inside the surface. What happens to the net flux? Explain.

A closed surface has zero net flux. Does this prove that \(\vec E=0\) everywhere on the surface? Explain.

A closed surface encloses \(+q\) and \(-q\). A strong external field also passes through it. What is the net flux?

A closed surface encloses charge \(Q\). The surface is stretched into a different shape without crossing any charges. How does the net flux change?

A closed surface encloses charge \(Q\). Can Gauss's law always be used to find the electric field magnitude at every point on the surface from \(Q\) alone? Explain.

A closed surface encloses three point charges. Two are known, \(+2q\) and \(-5q\). The measured net flux is \(-q/\epsilon_0\). Find the third enclosed charge.

A point charge outside a closed surface creates a nonzero electric field on the surface. Give a flux argument for why its net flux contribution is zero.

Two closed surfaces \(S_1\) and \(S_2\) enclose the same net charge but have different shapes. Explain what must be the same and what may be different about their electric flux integrals.

A Gaussian surface is chosen so that \(\vec E\) is not constant and not everywhere normal to the surface. Explain why Gauss's law still holds but may not directly solve for \(E\).

Starting with the field of a point charge at the center of a sphere, derive \(\Phi_E=q/\epsilon_0\). Then argue why this result supports the enclosed-charge form of Gauss's law rather than a law depending on surface radius.