Solution: Compound Microscope (Image at Least Distance of Distinct Vision)

Step 1 (Ray Diagram and Governing Definitions): [0.5 Mark]

The ray diagram must show:

• Objective lens ($f_o$) forming a real, inverted, magnified image ($A'B'$) of the object ($AB$).
• Eyepiece ($f_e$) acting as a simple magnifier to form a virtual, inverted, and magnified final image ($A''B''$) at the least distance of distinct vision ($D$).
• Correct labeling of angles $\alpha$ (angle subtended by object at eye) and $\beta$ (angle subtended by final image at eye).

(Note: In a standard board exam, the diagram must be drawn with a sharp pencil, showing rays passing through the optical centers and focal points.)

Step 2 (Step-by-Step Mathematical Working): [1.0 Mark]

The angular magnification $m$ is defined as the ratio of the angle subtended by the image at the eye ($\beta$) to the angle subtended by the object at the unaided eye ($\alpha$):

$$m = \frac{\beta}{\alpha} \approx \frac{\tan \beta}{\tan \alpha}$$

From the geometry of the diagram:

$$\tan \beta \approx \frac{A'B'}{B'E} = \frac{h'}{u_e}$$ $$\tan \alpha \approx \frac{h}{D}$$

Substituting these into the magnification formula:

$$m = \left( \frac{h'}{u_e} \right) \times \left( \frac{D}{h} \right) = \left( \frac{h'}{h} \right) \times \left( \frac{D}{u_e} \right)$$

Since the linear magnification of the objective is $m_o = \frac{h'}{h} = \frac{L}{f_o}$ (where $L$ is the tube length), and for the eyepiece using the lens formula $\frac{1}{v} - \frac{1}{u} = \frac{1}{f}$ with $v = -D$:

$$\frac{1}{-D} - \frac{1}{-u_e} = \frac{1}{f_e} \implies \frac{D}{u_e} = 1 + \frac{D}{f_e}$$

Step 3 (Final Concluded Answer): [0.5 Mark]

Combining the expressions, the total angular magnification $m$ for the image at the least distance of distinct vision is:

$$m = m_o \times m_e = \left( \frac{L}{f_o} \right) \left( 1 + \frac{D}{f_e} \right)$$

💡 EL10 Concept Insight:

Think of a compound microscope as a two-stage rocket: the first lens makes a small object look bigger, and the second lens acts like a magnifying glass to zoom in even further on that already-enlarged image so your eye can see the tiny details clearly.