1. Identify the System Type

The problem states that a rectangular object produces a rectangular image. For a 3D object (having longitudinal width $a_1$), the longitudinal magnification $m_L$ usually varies with position. The only optical configuration where the 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. Refer BYU 27

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

2. Transverse Magnification ($b_2$)

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

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

The height of the image $b_2$ is:

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

3. Longitudinal Magnification ($a_2$)

The longitudinal magnification $m_L$ is related to the transverse magnification $m$ by the relation:

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

The width of the image $a_2$ is:

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