Solution
1. Analysis of the Optical System
The problem involves a composite lens with two distinct regions, each having a different focal length. The incident light is a parallel beam (cylindrical beam).
- Inner Region (Lens 1):
Diameter = 1.0 cm $\Rightarrow$ Radius $r_1 = 0.5$ cm.
Focal Length $f_1 = 10$ cm.
Rays passing through this region converge at $F_1$ (10 cm from lens) and then diverge beyond it. - Outer Region (Lens 2):
Diameter = 5.0 cm $\Rightarrow$ Radius $r_2 = 2.5$ cm.
Focal Length $f_2 = 20$ cm.
Rays passing through this region converge at $F_2$ (20 cm from lens).
2. Condition for Smallest Spot Diameter
The smallest light spot will occur in the region between the two foci ($f_1 < x < f_2$).
- The beam from the inner lens is diverging after passing $F_1$.
- The beam from the outer lens is still converging towards $F_2$.
The diameter of the spot is minimized at the specific distance $x$ where the outer boundary of the diverging inner beam intersects the inner boundary of the converging outer beam. Mathematically, we equate the radii of the two beams at position $x$.
3. Mathematical Formulation
Let $y_1$ be the radius of the inner (diverging) beam and $y_2$ be the radius of the outer (converging) beam at distance $x$.
For the Inner Beam (Geometry of Similar Triangles):
The ray starts at height $r_1$ at the lens ($x=0$) and crosses the axis at $x=10$.
For $x > 10$, the height $y_1$ is given by the slope $\tan \theta_1 = \frac{r_1}{f_1}$.
$$ \tan \theta_1 = \frac{0.5}{10} = \frac{1}{20} $$
$$ y_1 = (x – 10) \tan \theta_1 = \frac{x – 10}{20} $$
For the Outer Beam:
The ray starts at height $r_2$ at the lens ($x=0$) and crosses the axis at $x=20$.
The height $y_2$ at distance $x$ is determined by the slope $\tan \theta_2 = \frac{r_2}{f_2}$.
$$ \tan \theta_2 = \frac{2.5}{20} = \frac{1}{8} $$
Since the beam is converging towards $x=20$, the radius is:
$$ y_2 = (20 – x) \tan \theta_2 = \frac{20 – x}{8} $$
4. Solving for position $x$
Equating the radii $y_1 = y_2$:
$$ \frac{x – 10}{20} = \frac{20 – x}{8} $$Multiply by 40:
$$ 2(x – 10) = 5(20 – x) $$ $$ 2x – 20 = 100 – 5x $$ $$ 7x = 120 $$ $$ x = \frac{120}{7} \text{ cm} $$