# Optical instrument microscope – problems and solutions

1. The focal length of the objective lens is 2 cm and the focal length of the ocular lens is 5 cm. The distance between the objective lens and the ocular lens is 30 cm. Determine the overall magnification and the object distance from the objective lens when the eye is relaxed.

__Known :__

The focal length of objective lens (f_{ob}) = 2 cm

The focal length of ocular lens (f_{oc}) = 5 cm

Distance between the lenses (l) = 30 cm

Near point (N) = 25 cm

__Solution :__

__(a) the overall magnification__

The formula of the overall magnification :

M = m_{ob }M_{oc}

*M = the overall magnification, m*_{ob}* = the magnification of the objective lens, M*_{oc}* = the ocular angular magnification*

**The magnification of the objective lens when the eye is relaxed ****(m**_{ob}**) :**

The image distance from the objective lens (d_{ob}’) :

d_{ob}’ = l – f_{oc }= 30 cm – 5 cm = 25 cm

The object distance from the objective lens (d_{ob}) :

*The objective lens is a converging lens so the focal length has the plus sign.*

*The image distance has plus sign because the image is real or the rays pass through the image.*

The magnification of the objective lens :

**The ocular angular magnification when the eye is relaxed (M**_{ok}**) :**

M_{oc} = N / f_{oc }= 25 cm / 5 cm = 5X

**The overall magnification :**

M = m_{ob }M_{oc} = (12.5)(5) = 62.5X

__(b) The object distance from objective lens (d___{ob}__)__

The object distance from the objective lens is 2 cm.

2. A microscope consists of a 5X objective and a 20X ocular. The distance between the lenses is 15 cm. (a) Determine the overall magnification if the eye is relaxed (b) determine the focal length of the ocular lens (c) the focal length of the objective lens

__Known :__

The magnification of the objective lens (m_{ob}) = 5X

The magnification of ocular lens (M_{oc}) = 20X

Near point (N) = 25 cm

The distance between the lenses (l) = 15 cm

__Solution :__

__(a) The overall magnification (M)__

M = m_{ob }M_{oc}

M = (5)(20) = 100X

__(b) The focal length of the ocular lens (f___{oc}__)__

The formula of angular magnification of ocular lens (M_{oc}) when the eye is relaxed :

M_{oc} = N / f_{oc}

The focal length of the ocular lens :

f_{oc} = N / M_{oc} = 25 cm / 20 = 1.25 cm

__(c) the focal length of the objective lens (f___{ob}__)__

The formula of the magnification of the objective lens when the eye is relaxed :

__The distance of the real image from the objective lens (d___{ob}__’) :__

d_{ob}’ = l – f_{oc }= 15 cm – 1.25 cm = 13.75 cm

__The object distance from the objective lens (d___{ob}__) :__

__The focal length of the objective lens (f___{ob}__) :__

The focal length of the objective lens = 2.29 cm

3. The focal length of the objective lens is 0.9 cm and the focal length of the ocular lens is 2.5 cm. The microscope is used by a normal eye without accommodation and the overall magnification is 90 times. Determine the distance between the object and the objective lens. N = 25 cm.

__Known :__

The focal length of the objective lens (f_{ob}) = 0.9 cm

The focal length of the ocular lens (f_{ok}) = 2,5 cm

The overall magnification (M) = 90 times

The near point of a normal eye (N) = 25 cm

*The eye’s accommodation is minimum.*

__Wanted:__ The distance between the object and the objective lens (s_{ob})

__Solution :__

The equation of the total angular magnification when the accommodation is minimum :

Calculate the object distance about the objective lens (s_{ob}) using the equation of relation between the focal length of the objective lens (f_{ob}), the distance between the object and the objective lens (s_{ob}) and the distance between the image and the objective lens (s_{ob}‘) :

4. The focal length of the objective lens is 1.8 cm and the focal length of the ocular lens is 6 cm. The microscope is used by a normal eye without accommodation, the distance between the objective lens and the ocular lens is 24 cm. Determine the object distance from the objective lens.

__Known :__

The focal length of the objective lens (f_{ob}) = 1.8 cm

The focal length of the ocular lens (f_{ok}) = 6 cm

The distance between the objective lens and the ocular lens = the length of the microscope (l) = 24 cm

*The accommodation is a minimum.*

__Wanted:__ Distance between the object and the objective lens (s_{ob})

__Solution :__

When the accommodation is minimum, the distance between the final image and the ocular lens is infinity, as shown in the figure below.

The distance between the objective lens and the ocular lens (l) = the focal length of the ocular lens (f_{ok}) + the distance between the image and the objective lens (s_{ob}’).

s_{ob}’ = l – f_{ok} = 24 cm – 6 cm = 18 cm

Calculate the distance between the object and the objective lens (s_{ob}) using the equation of relation between the focal length of the objective lens (f_{ob}), *the distance between the object and the objective lens *(s_{ob}) and the distance between the image and the objective lens (s_{ob}‘) :

The distance between the object and the objective lens is 2 cm.

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