Microscopes: simple and compound

The simple microscope (magnifier)

A simple microscope is a convex lens of short focal length held close to the eye. The object is placed between the lens and its focus, so the lens forms a virtual, erect, magnified image. For the clearest view, the image is at the near point, D = 25 cm from the eye. Then v = −D, and the thin lens formula gives m = v/u = 1 − v/f = 1 + D/f. If the object is at the focus, the image is at infinity and the eye is relaxed. The object then subtends an angle θ = h/f at the eye. At the near point, without the lens, it would subtend θ₀ = h/D. So m = θ/θ₀ = D/f. A shorter focal length gives a larger magnification.

Derivation: magnifying power of a compound microscope

The objective has a very short focal length fₒ. The eyepiece also has a short focal length fₑ, usually a little longer than fₒ.

1. 1
   The tiny object AB is placed just beyond the focus Fₒ of the objective. The objective forms a real, inverted, magnified image A′B′.
2. 2
   A ray from the top of AB, parallel to the axis, passes through the second focus of the objective. It makes an angle β with the axis there.
3. 3
   Use the two right triangles on either side of that focus: tan β = h/fₒ = h′/L. Here h′ is the height of A′B′ and L is the tube length.
4. 4
   So the objective magnifies by mₒ = h′/h = L/fₒ.
5. 5
   A′B′ is the object for the eyepiece. It lies at, or just inside, the eyepiece's focus. The eyepiece acts as a simple microscope: mₑ = 1 + D/fₑ for the final image at the near point, or mₑ = D/fₑ for the final image at infinity.
6. 6★
   Total magnifying power m = mₒ × mₑ. For the final image at infinity, m = (L/fₒ)(D/fₑ).
7. 7
   For the final image at the near point, m = (L/fₒ)(1 + D/fₑ). Short focal lengths fₒ and fₑ give a large m.

★ key step

How to draw: compound microscope (final image at the near point)

1. 1
   Draw a long horizontal axis. Draw a small convex lens on the left (objective) and a larger convex lens on the right (eyepiece). Draw an eye just right of the eyepiece.
2. 2
   Mark Fₒ on both sides of the objective, close to it. Mark Fₑ on both sides of the eyepiece.
3. 3
   Draw the object AB as a small upright arrow just beyond Fₒ, to the left of the objective.
4. 4
   From the top of AB, draw a ray parallel to the axis to the objective, and refract it through Fₒ on the right. Draw a second ray through the centre of the objective, undeviated.
5. 5
   Where these rays meet below the axis, draw the real, inverted, enlarged image A′B′. Place it just inside Fₑ, between Fₑ and the eyepiece.
6. 6
   From the tip of A′B′, draw a ray parallel to the axis to the eyepiece, and refract it through Fₑ on the right. Draw another ray through the centre of the eyepiece. These two rays spread apart.
7. 7
   Extend both rays backwards as dotted lines. They meet far to the left, below the axis, at the tip of the final image A″B″. It is virtual, inverted and much enlarged, at distance D from the eye.
8. 8
   Label objective, eyepiece, AB, A′B′, A″B″, Fₒ, Fₑ, L and D. Draw real rays as solid lines with arrows, and virtual rays as dotted lines.
9. 9
   For normal adjustment, draw A′B′ exactly at Fₑ instead. The rays then leave the eyepiece parallel, and the final image is at infinity.

Worked example: microscopes

Objective versus eyepiece

In a telescope the objective is the lens with the long focal length. Do not mix up the two instruments.