How Long Does It Take for a Capacitor to Discharge in an RC Circuit?

[Mebmt]

1. Homework Statement

The capacitor in an RC circuit with a time constant of 25 ms is charged to 10 V. The capacitor begins to discharge at t = 0 s

2. Homework Equations
τ=RC
ΔVC=(ΔVC)e^-t/τ


3. The Attempt at a Solution
part a) At what time will the charge on the capacitor be reduced to half its initial value?

10 ln 5 = (10)e^-τ/.025s
ln 5 = -τ/.025s
ln 5(.025s)= τ
τ=0.04 s

part b) At what time will the energy stored in the capacitor be reduced to half its initial value?

Not sure, but any hints as to my next step would be appreciated...

 

[gneill]

1. Homework Statement

The capacitor in an RC circuit with a time constant of 25 ms is charged to 10 V. The capacitor begins to discharge at t = 0 s

2. Homework Equations
τ=RC
ΔVC=(ΔVC)e^-t/τ


3. The Attempt at a Solution
part a) At what time will the charge on the capacitor be reduced to half its initial value?

10 ln 5 = (10)e^-τ/.025s
ln 5 = -τ/.025s
ln 5(.025s)= τ
τ=0.04 s

Where did the 10 ln 5 in the first line come from?
The result for your time value looks a bit high.

part b) At what time will the energy stored in the capacitor be reduced to half its initial value?

Not sure, but any hints as to my next step would be appreciated...

What's an expression for the energy stored on a capacitor?

 

[ehild]

2. Homework Equations
τ=RC
ΔVC=(ΔVC)e^-t/τ

Is the same ΔVC on both sides? That would mean e^-t/τ=1 all time.

part a) At what time will the charge on the capacitor be reduced to half its initial value?

10 ln 5 = (10)e^-τ/.025s

Where does that ln5 come?

ehild

 

[Mebmt]

Where did the 10 ln 5 in the first line come from?

I used 10 V * ln 5 to signify that ln 5 was half of the initial value.

Instead I should have used 5
ΔV final=(ΔV initial)e^-t/τ

5 = 10e^-τ/.025s
0.5 = e^-τ/.025s
ln 0.5 = -τ/.025
ln 0.5(.025) = -τ
τ= 0.013 s

Where did the 10 ln 5 in the first line come from?
The result for your time value looks a bit high.


What's an expression for the energy stored on a capacitor?

Uc=0.5 C (ΔVc)^2 or Q = C ΔVc

Easy to solve if I have C or Q

 

[gneill]

Where did the 10 ln 5 in the first line come from?

I used 10 V * ln 5 to signify that ln 5 was half of the initial value.

Instead I should have used 5
ΔV final=(ΔV initial)e^-t/τ

Why not write v(t) = v_i e^-t/τ ?

5 = 10e^-τ/.025s
0.5 = e^-τ/.025s
ln 0.5 = -τ/.025
ln 0.5(.025) = -τ
τ= 0.013 s

Check the calculation on that last step.

Uc=0.5 C (ΔVc)^2 or Q = C ΔVc

Easy to solve if I have C or Q

C is just a proportionality constant; it won't affect the relative change in magnitude of the function.

Write the expression for the energy stored on a capacitor versus voltage. Plug in your expression for the voltage versus time.

 

[Mebmt]

Why not write v(t) = v_i e^-t/τ ?

Check the calculation on that last step.

Yes. Should have been 0.017 s or 17 ms

C is just a proportionality constant; it won't affect the relative change in magnitude of the function.

Write the expression for the energy stored on a capacitor versus voltage. Plug in your expression for the voltage versus time.

Uc= .5 Q (Δv)^2