Question

For very large resistances it is easy to construct R-C circuits that have time constants of several seconds or minutes. How might this fact be used to measure very large resistances, those that are too large to measure by more conventional means?

Short answer

The large resistance would be measured by knowing the initial and discharging current for a known capacitance in a specific time constant.

Step-by-step solution

Equation for electric current in R-C circuit

In the R-C circuit, a resistor and capacitor are connected to an energy source in the electrical circuit.

The current flowing in the circuit can be written as:

$l=l_0{e}^{-t/RC}$

Here R is the resistance of the resister, and C is the capacitance of the capacitor. RC is the time constant for the capacitor.

Determining the fact to measure very large resistances

The equation for electric current in an R-C circuit is:

$$\begin{gathered} l=l_0{e}^{-t/RC} \\ \frac{l}{l_0}=e^{-t/RC} \\ \frac{-t}{RC}=\ln \frac{l}{l_0} \\ R=\frac{-t}{\ln \left(\frac{l}{l_0}\right)\times C} \end{gathered}$$

In the above equation, if the initial current $l_0$is known and the discharge current can be calculated and C is known. So, during the discharging process, the time of discharge can be determined. Now, all the variables are known in the equation, so then R can be measured.

Thus, the large resistance would be measured by knowing the initial and discharging current for a known capacitance in a specific time constant.