What is the time constant of a simple RC circuit?

A capacitor charges from an ideal DC source of emf \(\mathcal E\). What is the final capacitor voltage after a very long time?

An initially uncharged capacitor charges through a resistor \(R\) from an ideal source \(\mathcal E\). What is the initial current?

What fraction of its final voltage has a charging capacitor reached after one time constant?

How many time constants are needed for a charging capacitor to reach \(90\%\) of its final voltage?

A discharging capacitor has been connected to a resistor for two time constants. What fraction of its initial voltage remains?

Find the time constant for \(R=20\,\mathrm{k}\Omega\) and \(C=50\,\mu\mathrm{F}\).

A \(10\,\mathrm{V}\) source charges a \(100\,\mu\mathrm{F}\) capacitor through \(2.0\,\mathrm{k}\Omega\). Find the initial current, the time constant, and \(V_C\) at \(t=0.20\,\mathrm{s}\).

A capacitor initially at \(12\,\mathrm{V}\) discharges through \(3.0\,\mathrm{k}\Omega\). The capacitance is \(200\,\mu\mathrm{F}\). Find the capacitor voltage after \(1.2\,\mathrm{s}\).

Derive the charging differential equation for a series \(RC\) circuit connected to a source \(\mathcal E\).

Derive the discharging equation \(q(t)=q_0e^{-t/RC}\) from the loop equation.

In charging a capacitor \(C\) from an ideal source \(\mathcal E\) through a resistor, how much energy is ultimately stored in the capacitor, how much energy is supplied by the source, and where does the difference go?

During charging, how long does it take for the current to fall to \(25\%\) of its initial value?

In a DC circuit after a very long time, how does an ideal capacitor branch behave? Explain using current.

Immediately after an uncharged capacitor is connected in series with a resistor and ideal source, what is the capacitor voltage and why is the initial current finite?

A capacitor initially at \(5.0\,\mathrm{V}\) is connected through a resistor to a \(12\,\mathrm{V}\) source. Write \(V_C(t)\) in terms of \(\tau=RC\).

A capacitor is connected from a divider node to ground. The divider has \(4.0\,\mathrm{k}\Omega\) from a \(12\,\mathrm{V}\) source to the node and \(6.0\,\mathrm{k}\Omega\) from the node to ground. The capacitor is \(10\,\mu\mathrm{F}\). Find the final capacitor voltage and the time constant.

During charging from zero, at what time does the capacitor voltage equal the resistor voltage? Express the answer in terms of \(\tau\).

Use \(I=C\,dV_C/dt\) to explain why capacitor voltage cannot change discontinuously in a circuit with finite current.

Design a series resistance so a \(100\,\mu\mathrm{F}\) capacitor reaches \(95\%\) of its final charging voltage in \(2.0\,\mathrm{s}\).