Q.Derive the equation for effective focal length for lenses out of contact.

Step 1. For a thin lens, a ray parallel to the axis at height $h$ striking the lens is deviated through angle $\delta$, with $\tan\delta\approx\delta=h/f$ (from the triangle relating the height $h$, focal length $f$ and small deviation angle).

Step 2. Two lenses of focal length $f_1,f_2$, separated by distance $d$, are struck by a single parallel ray at height $h_1$ on the first lens; it is deviated by $\delta_1=h_1/f_1$, then travels to the second lens, arriving at height $h_2$, where it is deviated a second time by $\delta_2=h_2/f_2$. The two deviations add: $\delta=\delta_1+\delta_2=\dfrac{h_1}{f_1}+\dfrac{h_2}{f_2}$.

Step 3. From the geometry of the ray's straight-line travel across the separation $d$ between the two lenses, $h_2-h_1=-d\tan\delta_1\approx-d\,\delta_1=-\dfrac{dh_1}{f_1}$ (the ray converges towards the axis by this amount as it crosses the gap), so $h_2=h_1\left(1-\dfrac{d}{f_1}\right)$.

Step 4. Substituting $h_2$ into the Step 2 total-deviation expression: $\delta=\dfrac{h_1}{f_1}+\dfrac{h_1(1-d/f_1)}{f_2}=h_1\left(\dfrac{1}{f_1}+\dfrac{1}{f_2}-\dfrac{d}{f_1f_2}\right)$. …

Trace the deviation a parallel ray picks up at each of the two separated lenses, relate the ray height at the second lens back to its height at the fir …

• Forgetting this formula is valid ONLY for an object at infinity (parallel incident rays), and cannot be used directly for a finite object distan …