Calculating Resistance in an RC Circuit: A Time Constant Conundrum

[QuarkCharmer]

Homework Statement

A 5 micro farad capacitor is in series with a resistor, a switch, and a 12VDC ideal battery. The switch is closed at t=0s. The time constant of the circuit is 4.0s.

Determine the value of the resistance R?

Homework Equations

The Attempt at a Solution

From another example of a series circuit with a resistor and cap and switch, we found the equation:

$$q(t) = CE(1-e^{\frac{-t}{\tau}})$$

where tau is the time constant, and is substituted for RC in the equation. Now to me, this seems simple, tau = RC...

so $$R = \frac{\tau}{5.0*10^{-6}}$$
where tau = 4.0 seconds

which is like 800,000 $\Omega$, which cannot be right! How do I tackle this problem?

 

[gneill]

There's nothing wrong with 800 kΩ for a resistor value. Heck, it's less than a megohm!

 

[QuarkCharmer]

Really? I was expecting a much smaller number so I thought I was wrong. Thank you.

The other parts of the problem ask for the max charge on the cap. Which is simply the lim as t goes to infinity, but we were told a rule of thumb is that the cap is fully charged when about 5 time constants have passed.

Then the next part asks the charge remaining on one plate after one time constant. Wouldn't that simply be the function q(t) evaluated at one time constant? The charge on one plate of the cap would just be negative the charge of the other right?

 

[gneill]

Really? I was expecting a much smaller number so I thought I was wrong. Thank you.

You're welcome.

The other parts of the problem ask for the max charge on the cap. Which is simply the lim as t goes to infinity, but we were told a rule of thumb is that the cap is fully charged when about 5 time constants have passed.

Yup, that's a handy rule of thumb. After five time constants the circuit conditions should reach about 99.3% of their final values.

Then the next part asks the charge remaining on one plate after one time constant. Wouldn't that simply be the function q(t) evaluated at one time constant? The charge on one plate of the cap would just be negative the charge of the other right?

Yes and yes.

 

[QuarkCharmer]

Thanks!