# After how many time constants is capacitor energy 1/4?

• David Day
In summary, the conversation discusses the question of how many time constants it takes for the stored energy of a discharged capacitor to reach one-fourth of its initial value. The first equation, U = Q2/2C, is used to correctly solve the problem. However, the initial attempt to use equation (2), U = Qmaxε/2, was not successful due to using the same ε on both sides of the equation, which is not valid when the capacitor is discharging. The error is corrected by using Q/C instead of ε on the RHS of the equation, leading to the same form as equation (1).

## Homework Statement

[/B]
After how many time constants
is the stored energy of a discharged capacitor one-fourth its initial value?

## Homework Equations

[/B]
(1) U = Q2/2C

(2) U = Qmaxε/2

(3) q(t) = Qie-t/RC

## The Attempt at a Solution

The solution can be correctly attained using the first equation:

(1/4) Q2/2C = (Qie-t/RC)2/2C

(1/4) = e-2t/RC

t = 0.693RC

However, I first attempted to use equation (2) but could not get the correct answer. I can't figure out what mistake or incorrect assumption I'm making. My attempt follows:

(1/4) Qmaxε/2 = Qmaxe-t/RCε/2

(1/4) = e-t/RC

As can be seen here, this answer is off from the first by an exponent of two. Q2/2C and Qmaxε/2 are mathematically equivalent, so I'm not sure where I'm making the error. Is it not correct to assume Qmax = Qi?

I'd really appreciate some help in clearing this up!

EDIT: I may see my problem. The ε in equation (2) is proportional to Q and will thus be changing, i.e, it's not equal to the ε on the other side of the equation except when t = 0, so I need to change it to Q/C, which will be the same form as equation (1). I'm not sure if this is the problem or not...

Last edited:
David Day said:
After how many time constants
is the stored energy of a discharged capacitor
Do you mean a "discharging" capacitor?
David Day said:
(1/4) Qmaxε/2 = Qmaxe-t/RCε/2
If the capacitor is discharging, can you use the same ε on the RHS of this equation as in the LHS?

Edit: I see you found the error in your equation by yourself while I was typing my response. Well done!

Last edited:
David Day

## 1. What is a time constant in relation to capacitors?

A time constant is a measure of how quickly a capacitor can charge or discharge. It is calculated by multiplying the resistance (in ohms) of the circuit by the capacitance (in farads) of the capacitor.

## 2. How is the energy stored in a capacitor related to its time constant?

The energy stored in a capacitor is directly proportional to its capacitance and the square of its voltage. This means that the energy stored in a capacitor will increase as the time constant increases.

## 3. Why is it important to know how many time constants it takes for a capacitor's energy to reach 1/4?

Knowing the number of time constants it takes for a capacitor's energy to reach 1/4 can help in understanding the behavior and performance of the capacitor in a circuit. It can also be used to calculate the amount of time needed for a capacitor to fully charge or discharge.

## 4. How can the time constant be calculated?

The time constant can be calculated by dividing the capacitance (in farads) by the resistance (in ohms) of the circuit. This will give you the time constant in seconds.

## 5. What factors can affect the time constant of a capacitor?

The time constant of a capacitor can be affected by the capacitance and resistance in the circuit, as well as the voltage applied to the capacitor. Additionally, the material and construction of the capacitor can also impact its time constant.