Capacitor charging with unknown resistor

[cdotter]

Homework Statement

Capacitor voltage as a function of time is recorded for a simple direct current RC series charging capacitor. How could we determine the resistance of the resistor?

Homework Equations

$$q=Qe^\frac{-t}{RC}$$

The Attempt at a Solution

I understand the question I'm just having trouble figuring it out. :grumpy:

If I know the voltage at a certain time t, couldn't I say that q=Cv? Then

$$Cv=Qe^\frac{-t}{RC} \Rightarrow R=\frac{-t}{C\;ln(\frac{Cv}{Q})}$$

Where Q=CV and V=the voltage of the capacitor after it's finished charging?

Or is q=Cv incorrect?

 

[cdotter]

Anyone?

 

[chaoseverlasting]

That would work. Another way is to differentiate the equation wrt time to get current and to measure current at different points in time. From those values too you can find the resistance. I think this principle is used in measurement of resistances using the Anderson (?) Bridge. I might be thinking of another bridge though.

 

[cdotter]

I didn't think of that. Thank you.

 

[rwisz]

I would suggest using the equation for the charging of a capacitor, which is q=CV(1-e^(-t/RC)). This equation takes into account the initial voltage of the capacitor, as well as the voltage at any given time during the charging process. From this equation, we can see that the voltage of the capacitor is dependent on the capacitance, resistance, and time.

To determine the resistance of the unknown resistor, we can rearrange the equation to solve for R. This would give us R = -t/(C*ln(1-(q/CV))). We can then plug in the known values of capacitance, time, and voltage at a specific point in time to solve for the resistance. This process can be repeated for multiple points in time to get an average resistance value.

Additionally, we can also plot the voltage as a function of time and use the slope of the line to determine the resistance. The slope of the line would be equal to -1/RC, so we can rearrange the equation to solve for R. This method may be more accurate if the data is not perfectly linear.

In summary, by using the equation for capacitor charging and rearranging it to solve for resistance, or by plotting the data and using the slope of the line, we can determine the resistance of the unknown resistor in the circuit.