A capacitor stores \(6.0\,\mu\mathrm{C}\) when the potential difference is \(12\,\mathrm{V}\). Find its capacitance.

A parallel-plate capacitor has plate area \(2.5\times10^{-2}\,\mathrm{m^2}\) and separation \(1.0\,\mathrm{mm}\). Estimate its capacitance in vacuum.

An isolated conducting sphere has radius \(0.15\,\mathrm{m}\). Estimate its capacitance, then find the charge needed to raise it to \(900\,\mathrm{V}\) relative to infinity.

A parallel-plate capacitor is scaled so every length in the geometry is multiplied by \(s\): the plate side lengths become \(s\) times larger and the separation also becomes \(s\) times larger. Derive how \(C\) changes. Compare this with the scaling of an isolated conducting sphere and explain the common pattern.

Convert \(0.020\,\mu\mathrm{F}\) into nanofarads and picofarads.

A \(2.2\,\mathrm{nF}\) capacitor is connected across \(9.0\,\mathrm{V}\). Find the charge magnitude on either plate.

A capacitor has charge magnitude \(45\,\mathrm{nC}\) when its capacitance is \(1.5\,\mathrm{nF}\). Find the potential difference.

A vacuum parallel-plate capacitor has \(A=0.040\,\mathrm{m^2}\), \(d=2.0\,\mathrm{mm}\), and is connected across \(150\,\mathrm{V}\). Find \(C\), \(Q\), and the approximate field magnitude between the plates.

A vacuum parallel-plate capacitor must have capacitance \(100\,\mathrm{pF}\) with plate separation \(0.80\,\mathrm{mm}\). What plate area is required?

A parallel-plate capacitor has its plate area doubled and its separation tripled. By what factor does its capacitance change?

Capacitor B has twice the plate area and half the plate separation of capacitor A. Both are vacuum parallel-plate capacitors. Find \(C_B/C_A\).

An isolated conducting sphere has radius \(5.0\,\mathrm{cm}\) and is held at \(2.0\,\mathrm{kV}\) relative to infinity. Find its capacitance, charge, and surface electric field magnitude.

A parallel-plate capacitor has capacitance \(220\,\mathrm{pF}\), plate separation \(1.2\,\mathrm{mm}\), and charge \(3.3\,\mathrm{nC}\). Find the plate area and the electric field magnitude between the plates.

Derive the vacuum parallel-plate result \(C=\epsilon_0A/d\) from \(\sigma=Q/A\), \(E=\sigma/\epsilon_0\), and \(\Delta V=Ed\).

Derive the capacitance of an isolated conducting sphere of radius \(R\), using \(V=(1/4\pi\epsilon_0)Q/R\). Explain why the result does not depend on \(Q\).

An isolated vacuum parallel-plate capacitor keeps the same charge while its separation is increased from \(1.0\,\mathrm{mm}\) to \(2.5\,\mathrm{mm}\). By what factors do \(C\), \(\Delta V\), and the plate field \(E\) change?

A vacuum parallel-plate capacitor remains connected to a battery while its plate separation is halved. By what factors do \(C\), \(Q\), and \(E\) change?

Two isolated conducting spheres are far apart. Sphere A has radius \(R\) and charge \(Q_0\); sphere B has radius \(2R\) and is initially uncharged. They are connected by a thin wire until they reach a common potential, then disconnected. Find the final charge on each sphere.

Two concentric conducting spherical shells of radii \(a\) and \(b\), with \(b>a\), form a capacitor. Derive its capacitance in vacuum. Check the limits \(b\gg a\) and \(b-a\ll a\).

A long coaxial cylindrical capacitor has inner radius \(a\), outer conductor inner radius \(b\), and length \(L\), with \(L\gg b\). Derive its capacitance in vacuum by finding the field from a line charge and integrating the potential difference.