# Discharging capacitor half life

What is meant by discharging capacitor half life (the description). I seem to be getting different description, I would just like for someone to confirm it here for me please.

uart
If discharged through a resistor the capacitor voltage reduces exponentially via the equation

$$v = V_0 \, e^{-\frac{t}{RC}}$$

Mathematically it's easy to represent an exponential of one base in other other base.

In this case the above exponential can be re-written as

$$v = V_0\, 2^{-\frac{t}{\log(2) \, RC}}$$

where "log" is the natural logarithm.

From the above equation you can see that the "half life" is $RC/\log_e(2)$

Last edited:
Thanks Uart,

I understand that, how would you describe half life (not mathematically or through equations).

If discharged through a resistor the capacitor voltage reduces exponentially via the equation

$$v = V_0 \, e^{-\frac{t}{RC}}$$

Mathematically it's easy to represent an exponential of one base in other other base.

In this case the above exponential can be re-written as

$$v = V_0\, 2^{-\frac{t}{\log(2) \, RC}}$$

where "log" is the natural logarithm.

From the above equation you can see that the "half life" is $RC/\log_e(2)$

uart
Thanks Uart,

I understand that, how would you describe half life (not mathematically or through equations).

Well obviously, it's the time that you have to wait until the voltage is half of it's original value. That's how I'd describe it.

Delta2
Homework Helper
Gold Member
Half life of a quantity is the time it needs so that the quantity is reduced to half of its original value.

In the example of Uart, half life of the voltage is the time it gets for the voltage to reduce to the half of its starting value , that is the time it needs to go from $$V_0$$ to $$\frac{V_0}{2}$$

Thanks.

sophiecentaur
Gold Member
2020 Award
It may be worthwhile pointing out that this 'half life' applies wherever you start. So, every period of one half life decreases the voltage by a factor of two. This 'exponential' function is the only one with this property afaik. It applies in many examples of decay and growth processes (even bloody compound interest!).

uart
you can see that the "half life" is $RC/\log_e(2)$
Just correcting a typo above. That should of course have been $RC \, \log_e(2)$