energy - Why are so many energies represented by $frac{1}{2} ab^2$?

Why are so many energies in our universe mathematically represented by the equation $\frac{1}{2}ab^2$.

For example:

• Kinetic energy $$\frac{1}{2}mv^2$$


• Energy stored in a capacitor $$\frac{1}{2}CV^2$$


• Energy stored in an inductor $$\frac{1}{2}LI^2$$

These are just some I can remember off the top of my head ,I remember that there are more like this. Is this just a coincidence or is there a reason behind this ?

I tried to google it but couldn't find any relevant results.

Answer

This is purely because of dimensional reasons. Energy has dimension $\rm ML^2T^{-2}$, and you can construct this dimension from different combinations of dimesionful quantities. Note that, the mass $m$ and velocity $v$ has dimensions $\rm M$ and $\rm LT^{-1}$ respectively which makes the unique combination $mv^2$ to have the dimension $\rm ML^2T^{-2}$. Similarly, the capacitance $\rm C$ and potential difference (voltage) $\rm V$ has dimensions $\rm M^{-1}L^{-2} T^4 I^2$ and $\rm ML^2I^{-1}T^{-3}$ respectively and so on. For completeness, you can construct a few more. For example, the energy stored in an electric and magnetic field in a volume $\rm V$, elastic potential energy in a spring are respectively given by $\int_{\rm V}\frac{1}{2}\epsilon_0\textbf{E}^2dV$ and $\int_{\rm V}\frac{1}{2\mu_0}\textbf{B}^2dV$, $\frac{1}{2}kx^2$ etc.

Now, it remains to explain the factor of $\frac{1}{2}$. However, you should note that you can construct many other examples of dimension energy which do not carry the factor of $1/2$. For example, the gravitational potential energy of a particle of mass $m$ on the surface of earth is given by $$U(R)=-\frac{GMm}{R^2}$$ and so on. Also, energies need not necessarily have a quadratic dependence on some observable. For example, the thermal energy per particle is given by $\sim k_BT$.

The factor of $1/2$ is simply a matter of definition. For example, you define the kinetic energy to be $$T=\frac{1}{2}mv^2\tag{1}$$ or, alternatively as,

the work done required to change the velocity of a particle from $0$ to $v$.

When the energy is quadratic in some observable $\mathscr{z}$, the origin of this $1/2$ factor, if it appears, always come from an integral of the type $$\int \textbf{F}\cdot d\textbf{r}\sim \int \mathscr{z} d\mathscr{z}\sim \frac{1}{2}\mathscr{z}^2.$$ Now, the quantity $\mathscr{z}$ will vary from one situation to another. In the example of kinetic energy, $\mathscr{z}=v$.