Exponential decay occurs throughout physics and engineering.
Examples include radioactive decay and the discharge of a capacitor.
Exponential decay is often also associated with growing functions, such as the growing voltage of a charging of a capacitor.
To understand exponential decay, consider this red arrow, which represents the distance to our destination.
Suppose we travel half the distance in the first hour.
Then, we travel half of the remaining distance in the second hour.
We travel half of the remaining distance after that in the third hour.
And we keep continuing this pattern for all eternity.
We keep getting closer to our destination, but we never reach it.
The length of the red arrow, as a function of time, is an example of what we call "Exponential Decay."
Exponential Decay is represented mathematically as follows.
The variable "tau" is the time constant.
The larger the value of tau, the slower the decay.
The smaller the value of tau, the faster the decay.
Let us return to our original example and our original time constant.
Everything is the same as before, except now the red arrow represents the distance from the start of our journey.
The red arrow keeps growing for all of eternity, yet it never grows beyond a certain point.
This is one of the reasons we should be warry of people who advocate strategies for what they call "continuous improvement", as often this is what they end up delivering.
This type of growth occurs throughout physics and engineering.
For example, if we have a voltage source in series with a resistor, this type of growth might represent the increasing voltage of a capacitor or the increasing current through an inductor.
The length of the red arrow plus the length of the black arrow is equal to the length of the blue arrow.
The blue arrow represents the total distance from the beginning to the destination.
The black arrow represents the remaining distance to the destination.
And the red arrow represents the distance we have travelled so far.
The lengths of each of these arrows, as a function of time, look like the following.
The red curve is equal to the blue curve minus the black curve.
We get this equation.
Sometimes, the physical quantities we are interested in do not have their beginning or their ending at zero.
We are interested in the sum of the lengths of the two red arrows and we are interested in the sum of the lengths of the two black arrows.
The waveforms have the same shape as in our previous example, except that they are now both offset by a constant.
The lengths of the orange and green arrows are constant.
Their difference is the length of the blue arrow, which is also constant.
The new equations, in terms of the length of the blue arrow, are as follows.
To get an intuitive feel for these equations, let us consider what happens when the time equals zero, and what happens when the time approaches infinity.
First, let us consider what happens when time equals zero.
When time is equal to zero, "e" raised to the power of zero is equal to one.
Hence, at time zero, the red curve is at the bottom line and the black curve is at the top line, as expected.
Let’s now consider what happens as time approaches infinity.
As time approaches infinity, "e" raised to the power of negative infinity is equal to zero.
Hence, at time infinity, the red curve is at the top line and the black curve is at the bottom line, as expected.
As discussed previously, we never actually reach our final value, at least mathematically.
Mathematically, if we take a number, and keep cutting it in half, we will eventually get to a point where we have a fraction less than one.
But in many physical systems, we can’t have only a fraction of a particle.
For example, a decaying radioactive substance must have an integer number of isotopes.
And a capacitor must have an integer number of electrons.
Whereas mathematical equations can take any value, the physical systems they describe can not.
This is where our mathematical models break down.
The mathematical models also break down if we consider the opposite of exponential decay, which is exponential growth.
This is discussed in the video titled, "Exponential Growth and its philosophical implications."