Time Constant, Maximum Charge, and Current in an RC Circuit

[vysero]

Homework Statement

[/B]

With a 12V battery, a 5.00µF capacitor and a 8x105Ω resistor determine the following:
a) The time constant of the circuit
b) The maximum charge on the capacitor
c) The maximum current in the circuit
d) The charge on the capacitor as a function of time, q(t)
e) The current in the circuit as a function of time, I(t)
f) The time until the charge on the capacitor is 75% of it’s maximum value

A long time later the capacitor starts fully charged. At this new t=0, starting with a fully charged capacitor, the switch is moved to position b. Determine the following:
g) The charge on the capacitor as a function of time, q(t)
h) The current in the circuit as a function of time, I(t)
i) The time for the capacitor to reach 15% of it’s maximum value

Homework Equations

τ = RC
q = CV
V = IR

Charging:
q = CV(1-e^(-t/RC))
i = (V/R)e^(-t/RC)
V = (q/C) = V(1-e^(-t/RC))

Discharging:
q = qₒe^(-t/RC)
i = -(qₒ/RC)e^(-t/RC)

The Attempt at a Solution

a) RC = (5x10^-6)(8x10^5) = 4 seconds

b) I am not sure what the question is asking. Obviously it wants the maximum charge but does it want that maximum when t = 0 or at some other point? I said the answer was zero but I am not sure.

c) Once again it depends on the time. Right after t = 0 all the current is on the resistor but before that at t = 0 there is no current in the circuit. So for t = 0 the answer is zero but just after zero say like .01 sec the current is all on the resistor: R = V/I = 12/(8x10^5) = 1.5x10^-5 A.

d) Same problem not sure what time I should be looking at. At t = 0 the answer is zero.
e) At t = 0 it is zero again.

f) Not sure how to do this part.

The next few questions I think will be better if left alone until I get the previous set. Any help would be appreciated, mostly I need to know if what I am assuming the questions are asking is correct. Also, any other help would be appreciated.

 

[NascentOxygen]

(b) maximum means at any time, just whenever the maximum occurs

(c) you need to consider time of 0⁺, as you have

 

[vysero]

(b) maximum means at any time, just whenever the maximum occurs

(c) you need to consider time of 0⁺, as you have

So for b should I be using q = CV so q = (5x10^-6)(12) = 6x10^-5 C? Can I assuming I was correct about d,e both being zero? Still not sure how to do f.

EDIT: An update on part f. I want to say I need to solve for t when .75 = e ^(-t/RC) is this correct? I came up with t = -RCln(.75) = 1.151 sec.

 

[NascentOxygen]

(d) asks for q(t), i.e., how capacitor charge varies as a function of time. It will involve an exponential.

Something similar applies to (e).

(f) You will revise your answer once you get (d) right.

 

[vysero]

(d) asks for q(t), i.e., how capacitor charge varies as a function of time. It will involve an exponential.

Something similar applies to (e).

(f) You will revise your answer once you get (d) right.

So for d it asks for q(t) isn't that going to be an equation and not like a number? Is it looking for: q = CV(1-e^(-t/RC))?

 

[NascentOxygen]

So for d it asks for q(t) isn't that going to be an equation and not like a number? Is it looking for: q = CV(1-e^(-t/RC))?

That looks right, and you know values for most of those pronumerals.

 

[vysero]

That looks right, and you know values for most of those pronumerals.

Maybe I am making a math error then because I tried saying t = -RCln(1-(1/CV)

That looks right, and you know values for most of those pronumerals.

Okay so solving for t from that equation you get: t = -RCln(1-(q/CV)). At this point do I assuming RC = 4, q = 5x10^-6, and for CV do I use CV(.75)? If I do that I come up with: t = .4711 seconds for f.

 

[vysero]

Not feeling to confident in my answers atm...

 

[BvU]

So far I've seen

$\tau =$ 4 sec,
60 $\mu C$,
15 $\mu A$,
$q = CV(1-e^{-t/\tau})$ (or q = CV(1-e^(-t/RC)) ),
i = (V/R)e^(-t/RC).​

All good. No reason for lack of confidence.

Maybe I am making a math error then because I tried saying t = -RCln(1-(1/CV)

Maybe yes. You have an expression for q(t) and one for q_max.
An easy way to check that it's not the correct expression:
you can only take a logarithm of a number.
So whatever you take the logarithm of, it can not have a dimension (like 1/Coulomb)

same thing for exponential, and things like sine etc. -- where you need an angle, but angles are dimensionless too.​

But you correct yourself a little further down :

t = -RCln(1-(q/CV)) is good.
And you want to know the time when q = 0.75 q_max,
so q/CV = ... ?

And there's a simple check: at t = $\tau$, q is at 1 - e^{-1} = 63% of q_max, so the correct answer must be more than 4 seconds...

 

[vysero]

Okay so what I come up with is 5.45 sec for f and 7.59 sec for i. Which I am fairly confident in :D thanks for your help guys. One more thing does voltage work the same way? Like I have an equation for voltage Vc = V(1-e^(-t/RC)). Does that mean that voltage and charge will both take the same amount of time?

 

[NascentOxygen]

For a capacitor, its charge is related to its voltage as: Q = CV
or, more formally, q(t) = C.v(t), where C is a constant here.

So, there's a direct proportionality between v(t) and q(t). Exactly the same waveshape, just with a different scaling factor.

 

[BvU]

Well done !