Find the energy stored in an \(8.0\,\mu\mathrm{F}\) capacitor charged to \(50\,\mathrm{V}\).

A capacitor has \(C=5.0\,\mu\mathrm{F}\) and charge \(Q=30\,\mu\mathrm{C}\). Find its voltage and stored energy.

A uniform electric field of magnitude \(1.5\times10^5\,\mathrm{V\,m^{-1}}\) fills a volume \(2.0\times10^{-5}\,\mathrm{m^3}\) in vacuum. Estimate the field energy in that volume.

An isolated parallel-plate capacitor has plate area fixed and charge fixed. Its plate separation is doubled. Determine how \(C\), \(V\), \(E\), and \(U\) change, and explain why the energy change does not contradict energy conservation.

A capacitor has \(Q=20\,\mu\mathrm{C}\) and \(V=30\,\mathrm{V}\). Find its stored energy.

A capacitor with \(C=4.0\,\mu\mathrm{F}\) stores \(2.0\times10^{-3}\,\mathrm{J}\). Find its voltage and charge.

If the voltage across a fixed capacitor is doubled, by what factor does the stored energy change?

An isolated capacitor keeps the same charge while its capacitance is tripled. By what factor does its stored energy change?

A vacuum electric field has magnitude \(4.0\times10^5\,\mathrm{V\,m^{-1}}\). Find the electric field energy density.

A vacuum parallel-plate capacitor has \(A=0.020\,\mathrm{m^2}\), \(d=1.0\,\mathrm{mm}\), and \(V=200\,\mathrm{V}\). Find its stored energy using both \(U=\frac12CV^2\) and \(u_E=\frac12\epsilon_0E^2\) times volume.

Derive the three equivalent capacitor energy forms \(U=Q^2/(2C)\), \(U=\frac12CV^2\), and \(U=\frac12QV\).

A \(6.0\,\mu\mathrm{F}\) capacitor is charged to \(40\,\mathrm{V}\) and then discharged through a lamp. How much energy is available to the lamp?

A vacuum parallel-plate capacitor has area \(0.010\,\mathrm{m^2}\) and breakdown field \(3.0\times10^6\,\mathrm{V\,m^{-1}}\). If the plate separation is \(0.50\,\mathrm{mm}\), estimate the maximum stored energy before breakdown.

A \(3.0\,\mu\mathrm{F}\) capacitor charged to \(18\,\mathrm{V}\) is connected in parallel with an uncharged \(6.0\,\mu\mathrm{F}\) capacitor. Find the final stored energy and the energy dissipated.

A battery remains connected to a parallel-plate capacitor while the plate separation is doubled. Determine how \(C\), \(Q\), \(E\), and capacitor stored energy change. Explain where the energy difference goes.

An isolated parallel-plate capacitor with fixed \(Q\) has plate separation increased by \(\Delta d\). Derive the work required in terms of \(Q\), \(\epsilon_0\), \(A\), and \(\Delta d\), neglecting edge effects.

A capacitor is charged by a battery from zero to final voltage \(V\). Show that the battery supplies \(CV^2\) of energy while the capacitor stores only \(\frac12CV^2\). State where the other half goes in a real circuit.

Use energy to derive the attractive force between parallel capacitor plates held at fixed voltage \(V\). Treat \(C=\epsilon_0A/d\) and explain why the relevant force magnitude is \(F=\frac12\epsilon_0AV^2/d^2\).

Use energy to derive the attractive force between isolated parallel capacitor plates carrying fixed charge \(Q\). Show that the force magnitude is \(F=Q^2/(2\epsilon_0A)\).

Two capacitors \(C_1\) and \(C_2\) initially at voltages \(V_1\) and \(V_2\) are connected positive-to-positive. Derive the energy dissipated and prove it is nonnegative.