What must symmetry let you do before Gauss's law can directly solve for \(E\)?

What Gaussian surface matches spherical symmetry?

What Gaussian surface matches an infinitely long uniform line charge?

What Gaussian surface matches an infinite uniformly charged sheet?

Use Gauss's law to find the electric field magnitude \(0.20\,\mathrm{m}\) from a \(+5.0\,\mathrm{nC}\) point charge.

A long line has \(\lambda=3.0\,\mathrm{nC\,m^{-1}}\). Find \(E\) at \(r=0.050\,\mathrm{m}\).

An infinite sheet has \(\sigma=4.0\,\mathrm{nC\,m^{-2}}\). Find the electric field magnitude on either side.

A thin spherical shell has total charge \(+Q\). Find \(E\) for \(r<R\), where \(R\) is the shell radius.

A thin spherical shell has total charge \(+Q\). Find \(E\) for \(r>R\).

A uniformly charged solid sphere has total charge \(Q\) and radius \(R\). Derive \(q_{\mathrm{enc}}\) for a Gaussian sphere of radius \(r<R\).

Using the result for \(q_{\mathrm{enc}}\) inside a uniformly charged solid sphere, derive \(E(r)\) for \(r<R\).

A uniformly charged solid sphere has \(Q=8.0\,\mathrm{nC}\), \(R=0.20\,\mathrm{m}\). Find \(E\) at \(r=0.10\,\mathrm{m}\).

A very long insulating cylinder has uniform volume charge and radius \(R\). Explain why a coaxial Gaussian cylinder is useful and which parts of it contribute flux.

An infinite sheet has negative surface charge density \(-\sigma\). State the field magnitude and direction on both sides of the sheet.

Derive the field of a long uniform line charge using a cylindrical Gaussian surface, including why the end caps contribute no flux.

Derive the field of an infinite uniformly charged sheet using a pillbox Gaussian surface, including the factor of 2.

A charge distribution is spherically symmetric but not uniform. Explain how Gauss's law determines \(E(r)\) if \(q_{\mathrm{enc}}(r)\) is known.

A nonconducting spherical shell has inner radius \(a\), outer radius \(b\), and uniform volume charge density. Derive the form of \(E(r)\) in the three regions \(r<a\), \(a<r<b\), and \(r>b\) in terms of \(q_{\mathrm{enc}}(r)\).

A long cylinder and an infinite sheet both have positive charge. Compare how their electric field magnitudes depend on distance and connect the difference to Gaussian-surface area growth.

For spherical, cylindrical, and planar symmetry, derive the distance dependence of \(E\) by comparing how the Gaussian surface area grows while enclosed charge is fixed or grows with the surface size.