# What is the 1/e Lifetime of a Graph?

• Richard Ros
In summary, the conversation discusses finding the value of τe (1/e life-time) from a given graph, where the units of voltage are "Volts" and the units of time are "seconds". The equation used is V(t) = Vi*e^-αt and the data is graphed using a semi-log scale for V and an arithmetic scale for t. The slope of the graph is used to find the value of α, which is then used to calculate τe as 1/α.
Richard Ros

## Homework Statement

Given that... The units of voltage are “Volts” and the units of time are “seconds”
Show τe (1/e life-time) on your graph and find a value for τe from your graph. ( The definition of τe is similar to the definition of τ1/2 (half-life))

V(t) t(s)
24 0.1
20 0.2
14 0.3
12 0.4
10 0.5
7.5 0.6
5.5 0.7
4.8 0.8
3.5 0.9
2.6 1.0

V(t) = Vi*e^-αt

## The Attempt at a Solution

I got the half life as .3 seconds. But I have no idea what is the life-time of a graph and how to even get it. I've tried googling it but to no avail. If anyone can help tell me the value and how you got it, I would greatly appreciate it. Thanks!

Richard Ros said:

## Homework Statement

Given that... The units of voltage are “Volts” and the units of time are “seconds”
Show τe (1/e life-time) on your graph and find a value for τe from your graph. ( The definition of τe is similar to the definition of τ1/2 (half-life))

V(t) t(s)
24 0.1
20 0.2
14 0.3
12 0.4
10 0.5
7.5 0.6
5.5 0.7
4.8 0.8
3.5 0.9
2.6 1.0

V(t) = Vi*e^-αt

## The Attempt at a Solution

I got the half life as .3 seconds. But I have no idea what is the life-time of a graph and how to even get it. I've tried googling it but to no avail. If anyone can help tell me the value and how you got it, I would greatly appreciate it. Thanks!
The 1/e lifetime is 1/α. Take the natural log of both sides of the equation. Graph the data using a semi-log scale for V and an arithmetic scale for t. The slope should be -α. See if you can figure out why this procedure works.

## What is the 1/e Life-Time of a Graph?

The 1/e life-time of a graph, also known as the characteristic time, is a measure of the time it takes for a system to reach a steady state or equilibrium. It is often used in the study of complex systems and can provide insight into the dynamics and stability of a system.

## How is the 1/e Life-Time of a Graph calculated?

The 1/e life-time of a graph is calculated by plotting the logarithm of the system's response over time and finding the time at which the response reaches 1/e of its maximum value. This value is then used to determine the characteristic time of the system.

## What factors can influence the 1/e Life-Time of a Graph?

The 1/e life-time of a graph can be influenced by various factors such as the complexity of the system, the presence of feedback loops, and the strength of interactions between components of the system. External factors such as environmental conditions and external inputs can also play a role in the characteristic time of a system.

## How can the 1/e Life-Time of a Graph be used in research?

The 1/e life-time of a graph is a useful tool in studying the behavior and dynamics of complex systems. It can provide insights into the stability and resilience of a system, as well as help in understanding how different factors can influence the behavior of the system.

## Are there any limitations to using the 1/e Life-Time of a Graph?

While the 1/e life-time of a graph can provide valuable information, it is important to note that it is not a universal measure and may not accurately reflect the behavior of all systems. Additionally, the calculation of the characteristic time can be affected by the choice of logarithmic scale and the specific methods used in the analysis.

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