1Identify the System Type

The problem describes a rectangular object producing a rectangular image. Generally, for 3D objects, longitudinal magnification varies with position, causing distortion. The only optical configuration where magnification is constant—mapping a rectangle to a rectangle without distortion—is a Telescopic (Afocal) System.

In this configuration, the distance $l$ between the lenses is the sum of their focal lengths.

$$ l = f_1 + f_2 = 20 \text{ cm} + 10 \text{ cm} = 30 \text{ cm} $$

2Transverse Magnification ($b_2$)

For a telescopic system, the magnitude of transverse magnification $|m|$ is defined by the ratio of the focal length of the second lens to the first:

$$ |m| = \frac{f_2}{f_1} = \frac{10}{20} = 0.5 $$

Calculating the height of the image $b_2$:

$$ b_2 = b_1 \times |m| = 2 \text{ mm} \times 0.5 = 1 \text{ mm} $$

3Longitudinal Magnification ($a_2$)

The longitudinal magnification $m_L$ relates to the transverse magnification $m$ via the square relationship:

$$ m_L = m^2 $$

$$ m_L = (0.5)^2 = 0.25 $$

Calculating the width of the image $a_2$:

$$ a_2 = a_1 \times m_L = 1 \text{ mm} \times 0.25 = 0.25 \text{ mm} $$