Optics: Simple and compound microscope

We want to form the image of an object seen through a converging lens. If the object is placed at left infinity, its image will be located exactly at the right focal point. If the object is placed at the left of the focal point of this converging lens, the related image is real and inverted.

The more we approach to this point from the left, the more the related image is large, and more far from the lens. If it is placed exactly at the focal point, the constructing rays of the image are parallel and the image is at the infinity.

Between the focal point and the center of the lens, the image is virtual, upright, and enlarged coming from the left. There is absolutely no image between the center of a converging lens and its right focal point.

1. The simple magnifier

The more an object is near the observer, the more the related angle formed by that object is large. For the human eye, the related length is about 25 cm. We write tg θ_o = h/25 cm. The length "h" is the height of the object.
tg θ = h/p

The angular magnification is defined as

m = θ/θ_o = (h/p)/(h/25 cm) = 25cm/p

The angular magnification is maximum when the angle θ is maximum, that is when the image formed by the converging lens is at q = - 25 cm.
Now, We want an expression of the angular magnification "m".
1/p + 1/q = 1/f, so
1/p = 1/f - 1/q = (q - f)/qf, then:
p = qf/(q - f)


The angles θ and θ_o are small, so θ_o = tg θ_o = h/25 cm, and θ = tg θ = h/p
Therefore:
m_max = (h/p)/(h/25cm) = 25cm/P = 25cm(q - f)/qf = 25cm(-25 cm - f)/(-25 cm)f with q = - 25 cm, we have:
m_max = 25 cm(25 cm - f)/25 cm f = 25 cm /f + 1

Exactly, when the object is near the focal point, (p = f) , the image should be formed at infinity to minimize eyestrain. we have:

m = (h/f)/(h/25 cm)= 25cm/f

This is the magnification of the simple magnifier.

2. Compound microscope

A compound microscope gives greater magnification than a single lens. It contains two converging lenses: The objective lens that has a short focal length f_o (< 1 cm) , and the eyepiece that has a focal length, f_e of a few cm.

The linear magnification by the objective is m_o = - p/q
The angular magnification by the eyepiece of the microscope is m_e = 25 cm / ƒ_e The total magnification m_t of the microscope is the product of the magnification by the objective = m_o = - q/p, and the magnification by the eyepiece = m_e = 25cm/f_e

m_t = - (q/p)(25cm/f_e)

Exactly, when the object is near the focal point, (p = f_o), and the first image is located at focal point of the eyepiece, the second image virtual, inverted and magnified is formed at infinity.
If we set:

T = distance between the two focal points f_e and f_e. It is called the tube length T. It is often set to 160 mm, then:
m_o = - q/p = - T/f_o

Therefore:

m_t = - (T/f₀)(25cm/f_e) = - 25cm T/f₀f_e)

This is the often used magnification of the compound microscope.