# Young Modulus in Finn's Thermal Physics

In Finn's Thermal Physics (equation 2.4), the Young modulus $$Y$$ of a stretched wire with tension $$F$$ is given to be

$$Y = \frac{L}{A} \left( \frac{\partial F}{\partial L}\right)_T$$

However, usually the Young modulus is defined to be the ratio of stress and strain, specifically

$$Y = \frac{\sigma}{\varepsilon} = \frac{F L_0}{A \Delta L} = \frac{FL_0}{A(L-L_0)}\implies dF = \frac{AY}{L_0} dL$$

this would suggest that the relationship given in Finn's textbook should actually be

$$Y = \frac{L_0}{A} \left( \frac{\partial F}{\partial L}\right)_T$$So I wondered why they have used $$L$$ instead of $$L_0$$ in the definition. Is it a mistake? Thanks!

When assuming infinitesimal strains, one can use either $$Y=\frac{L}{A}\left(\frac{\partial F}{\partial L}\right)_T$$ or $$Y=\frac{L_0}{A}\left(\frac{\partial F}{\partial L}\right)_T$$, depending on what's convenient; the difference is negligible. The former is related to the definition of the true strain $$e$$ from $$de=dL/L$$ (so that $$e=\ln(1+\Delta L/L)$$) and the latter to the definition of the engineering strain $$\varepsilon=\Delta L/L$$. The difference between the true strain and engineering strain is, of course, also negligible in this context.