Animated Solution for Physics - Properties of Solids and Liquids: If $S$ is stress and $Y$ is Young's modulus of material of a wire, the energy stored in the wire per unit volume is

Imagine you are holding a thick rubber band or a metallic wire. When you pull it from both ends, you are applying a force, and in doing so, you are doing work. But where does this work go? It doesn't just vanish! According to the law of conservation of energy, the work you do against the internal interatomic forces of the material gets stored inside it. We call this stored energy the Elastic Potential Energy.

When dealing with materials, we often care more about the energy stored in a specific chunk of the material rather than the whole thing. This is where the concept of Energy Density, or energy stored per unit volume, comes into play.

For a wire stretched by a force, the total elastic potential energy $U$ is given by the beautiful relation:

Why the $\frac{1}{2}$? Because when you start pulling the wire, the restoring force is initially zero and builds up linearly to its maximum value $F$. The average force doing the work is therefore $\frac{F}{2}$.

To find the energy stored per unit volume (let's call it $u$), we simply divide the total energy by the volume of the wire:

Now, look at the options provided in the question. Do you see 'Strain' anywhere? No! The options are purely in terms of Stress ($S$) and Young's modulus ($Y$). This is a classic move in physics problems—we need to eliminate the variable we don't want.

To do this, we call upon our trusty tool: Hooke's Law. Within the elastic limit, Hooke's law tells us that stress is directly proportional to strain, and the constant of proportionality is Young's modulus $Y$:

We can easily rearrange this to isolate Strain:

Since the problem tells us to use $S$ for Stress, we write:

We are almost there! Let's take this expression for strain and substitute it back into our energy density equation:

Multiply the terms together, and the magic happens:

And there we have it! The energy stored per unit volume is elegantly expressed as $\frac{S^2}{2Y}$. This matches perfectly with option (b).

Pro Tip: Always be ready to manipulate this formula. If a question asks for the energy density in terms of strain instead of stress, you would substitute $\text{Stress}=Y\times \text{Strain}$ to get $u=\frac{1}{2}Y(\text{Strain}{)}^2$. Mastering these quick substitutions will save you precious time in the exam!