Capacitance and resistance
The time constant of a capacitance C and a resistance R is equal to CR, and represents the time to change the voltage on the capacitance from zero to E at a constant charging current E / R (which produces a rate of change of voltage E / CR across the capacitance). Similarly, the time constant CR represents the time to change the charge on the capacitance from zero to CE at a constant charging current E / R (which produces a rate of change of voltage E / CR across the capacitance).
If a voltage E is applied to a series circuit comprising a discharged capacitance C and a resistance R, then after time t the current i, the voltage v_{R} across the resistance, the voltage v_{C} across the capacitance and the charge q_{C} on the capacitance are:
i = (E / R)e^{  t / CR} v_{R} = iR = Ee^{  t / CR} v_{C} = E  v_{R} = E(1  e^{  t / CR}) q_{C} = Cv_{C} = CE(1  e^{  t / CR})
If a capacitance C charged to voltage V is discharged through a resistance R, then after time t the current i, the voltage v_{R} across the resistance, the voltage v_{C} across the capacitance and the charge q_{C} on the capacitance are:
i = (V / R)e^{  t / CR} v_{R} = iR = Ve^{  t / CR} v_{C} = v_{R} = Ve^{  t / CR} q_{C} = Cv_{C} = CVe^{  t / CR}
Inductance and resistance
The time constant of an inductance L and a resistance R is equal to L / R, and represents the time to change the current in the inductance from zero to E / R at a constant rate of change of current E / L (which produces an induced voltage E across the inductance).
If a voltage E is applied to a series circuit comprising an inductance L and a resistance R, then after time t the current i, the voltage v_{R} across the resistance, the voltage v_{L} across the inductance and the magnetic linkage y_{L} in the inductance are:
i = (E / R)(1  e^{  tR / L}) v_{R} = iR = E(1  e^{  tR / L}) v_{L} = E  v_{R} = Ee^{  tR / L} y_{L} = Li = (LE / R)(1  e^{  tR / L})
If an inductance L carrying a current I is discharged through a resistance R, then after time t the current i, the voltage v_{R} across the resistance, the voltage v_{L} across the inductance and the magnetic linkage y_{L} in the inductance are:
i = Ie^{  tR / L} v_{R} = iR = IRe^{  tR / L} v_{L} = v_{R} = IRe^{  tR / L} y_{L} = Li = LIe^{  tR / L}
Rise Time and Fall Time
The rise time (or fall time) of a change is defined as the transition time between the 10% and 90% levels of the total change, so for an exponential rise (or fall) of time constant T, the rise time (or fall time) t_{1090} is:
t_{1090} = (ln0.9  ln0.1)T » 2.2T
The half time of a change is defined as the transition time between the initial and 50% levels of the total change, so for an exponential change of time constant T, the half time t_{50} is:
t_{50} = (ln1.0  ln0.5)T » 0.69T
Note that for an exponential change of time constant T:

over time interval T, a rise changes by a factor
1  e^{ 1} (» 0.63) of the remaining change.

over time interval T, a fall changes by a factor
e^{ 1} (» 0.37) of the remaining change.

after time interval 3T, less than 5% of the total change remains.

after time interval 5T, less than 1% of the total change remains.
