Finding time constants of a circuit? So this is a homework question and I am having trouble figuring out what they are asking. The potential difference (voltage) across the capacitor at time t > 0 is given by V_C(t)=q(t)/C. The quantity RC has the dimensions of time and is often called the time constant for the circuit. How many time constants does it take for a capacitor to charge to 90% of the applied voltage, V0? Justify your answer' So change in V, or V_C , or delta V is 90%. In other words we have 0.9V=q(t)/C? I have found in a previous question that q(t)=V0C(1−e^(−t/CR)) so 0.9V=(V_0C(1-e^((-t)/(CR))))/(C)

dredyue

dredyue

Answered question

2022-08-11

Finding time constants of a circuit?
So this is a homework question and I am having trouble figuring out what they are asking.
'The potential difference (voltage) across the capacitor at time t > 0 is given by V C ( t ) = q ( t ) / C. The quantity RC has the dimensions of time and is often called the time constant for the circuit. How many time constants does it take for a capacitor to charge to 90% of the applied voltage, V0? Justify your answer'
So change in V, or V C , or δ V is 90%. In other words we have 0.9 V = q ( t ) / C?
I have found in a previous question that q ( t ) = V 0 C ( 1 e t C R ) so
0.9 V = V 0 C ( 1 e t C R ) C
However I am not sure where to go from here, if I am even on the right track at all.

Answer & Explanation

Adelyn Mercado

Adelyn Mercado

Beginner2022-08-12Added 13 answers

If you look at the capacitor voltage curve, you notice that somewhere between 2 and 3 time constants, we have 90 % charge. We should be able to figure this out generally when not given the resistor and capacitor value.
We have the unknown:
Time Constant = τ = R C
Using:
0.9 V = V 0 C ( 1 e t C R ) C = V 0 ( 1 e t C R ) = V 0 ( 1 e t τ )
We want to solve for t, so we have:
t = τ ln ( 0.9 V V 0 V 0 )
However, the voltage across the capacitor is .9 of the the voltage source V 0 , so we can rewrite this as:
t = τ ln ( 0.9 V V 0 V 0 ) = τ ln ( 0.9 V 0 V 0 V 0 ) = τ ln ( 0.1 ) = ( 2.30259 ) τ = 2.30259 τ
In other words, it will take 2.30259 time constants to charge to 90 %

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