The Optical Lever

An optical lever is a convenient device to magnify a small displacement and thus to make possible an accurate measurement of the displacement. Experiment M-11, Young's modulus, uses an optical lever to magnify the extension of a wire produced by a series of different loads.

The plate P carries a mirror M. The mirror mount has two points resting in a fixed groove, F, and at the other end has a single point resting on the object whose displacement one is measuring. Raising the object through a distance $\Delta$L will tilt the mirror through an angle $\theta$ or $\Delta$L/d radians (approximately) but will turn the light beam through an angle 2$\theta$.

Figure 1: Schematic of the optical lever.

Hence

$$\theta$$   =  $${\frac{{\Delta L}}{{d}}}$$ $$\sim$$ $${\frac{{\frac{1}{2} ( y_1-y_0)}}{{D}}}$$

if $\theta$ is small so that $\theta$ $\sim$ tan$\theta$. Therefore

2$$\theta$$ = $${\frac{{2\Delta L}}{{d}}}$$ = $${\frac{{y_1-y_0}}{{D}}}$$,  and  $$\Delta$$l = (y₁ - y₀)$$\left[\vphantom{ \frac{d}{2D}}\right.$$$${\frac{{d}}{{2D}}}$$$$\left.\vphantom{ \frac{d}{2D}}\right]$$  .

Note that with the telescope nearly perpendicular to the scale at the beginning then y₀ is close to the telescope, and the difference between two elongations ( $\Delta$L₂  - $\Delta$L₁) is very accurately given by

$$\Delta$$L₂  - $$\Delta$$L₁ = $${\frac{{y_2~d}}{{2D}}}$$ - $${\frac{{y_1~d}}{{2D}}}$$ = $${\frac{{( y_2-y_1) d}}{{2D}}}$$

where y_i is the scale reading. This relation holds so long as 2$\theta$ is small enough that tan 2$\theta$  $\sim$ 2$\theta$.