Question

Which is the correct formula to use for plotting a straight line graph to find the time constant of a capacitor circuit from the gradient?

Answer 1

Option 2.

Explanation:

Consider a capacitor of capacitance `C` with an initial charge `Q_0` allowed to discharge through a resistor of resistance `R` as shown in the figure below.

The charge across the plates of the capacitor decays exponentially with time according to the expression

`Q(t)=Q_0e^(-t/(RC))`........ (1)

If we plot a graph between charge `Q` vs time `t` it wll be an exponential decay graph. However, if we take natural logarithm of both side we get

`lnQ(t)=ln(Q_0e^(-t/(RC)))`
`=>lnQ(t)=lnQ_0-t/(RC)`

Rewring and comparing with slope- intercept form of equation for straight line we see that

`lnQ(t)=-t/(RC)+lnQ_0`
`y=mx+c`

Slope of the straight line `=-1/(RC)-=-1/tau`, where `tau` is the time constant of the `RC` circuit and `lnQ_0` is the intercept on the `y`-axis.