When charging a capacitor, as discussed in conjunction with Figure 21.38, how long does it take for the voltage on the capacitor to reach emf? Is this a problem?

The relation for charging of a capacitor,

\({\rm{V = emf}}\left( {{\rm{1 - }}{{\rm{e}}^{\frac{{{\rm{ - t}}}}{{{\rm{RC}}}}}}} \right)\)

Here,\({\rm{V}}\)is the voltage across the terminals of the capacitor due to charging, emf is the voltage across the terminals of the battery used in the circuit,\({\rm{t}}\)is time calculated from the start of the charging process,\({\rm{R}}\)is the resistance in the circuit, and\({\rm{C}}\)is the value for the capacitance of the capacitor.

The capacitor will take an infinite amount of time to achieve the same voltage as the source since the voltage grows asymptotically without ever reaching the \({\rm{E}}\)line.

This isn't a problem because the source soon approaches full emf; it takes only\({\rm{5RC}}\) to reach \({\rm{99}}{\rm{.33\% }}\)of full energy.