Microscope Magnification Calculator

How to calculate magnification

Magnification is the size of the image divided by the actual size of the specimen, with both lengths in the same unit: M = I ÷ A, or magnification = image size ÷ actual size. Rearranged, the actual size is A = I ÷ M and the image size is I = M × A, so any two of the three give the third. The AQA GCSE and A-level biology specifications write it the same way. A microscope’s total magnification is a separate number, eyepiece × objective, which the second panel works out.

Magnification has no unit, because it is one length divided by another, and it is written with a multiplication sign, as ×500. It only comes out right when the two lengths share a unit. Images are measured with a ruler in millimetres and cells are quoted in micrometres, and 1 mm = 1000 µm, so a size in millimetres is multiplied by 1000 before it is divided by one in micrometres. Electron micrographs take the same step one prefix further down, 1 µm = 1000 nm, and the SI prefixes table has the rest.

Using the calculator

The first panel solves M = I ÷ A for whichever term you mark, with a unit menu on each size, and prints the working with the conversion written out. Under it, “1 mm on the image” is the length of specimen one millimetre of the image covers, and “Scale bar” gives a round length no longer than the specimen and how long to draw it at that magnification.

The second panel takes the lens settings: the total magnification, the diameter and area of the field of view, and a cell size estimated from how many cells span the field. The two panels are kept apart on purpose, because the magnification of a drawing or a photograph is not the magnification of the lenses it was made through.

Worked example: the magnification of a drawing

A cell is drawn 45 mm wide, and its real width is 90 µm. What is the magnification of the drawing?

• Put the image size in micrometres: 45 mm × 1000 = 45,000 µm
• M = I ÷ A = 45,000 µm ÷ 90 µm
• M = ×500

Going the other way, an electron micrograph at ×12,000 shows a bacterium 36 mm long. Its actual length is 36 mm ÷ 12,000 = 0.003 mm, which is 3 µm. Above ×1, an actual size larger than the image means the division was done the wrong way up.

Total magnification of a microscope

A compound light microscope magnifies in two stages. The objective forms an enlarged image inside the tube and the eyepiece enlarges that image again, so the total is the product: total magnification = eyepiece × objective. A ×10 eyepiece on a ×40 objective gives ×400, and on a ×100 oil immersion objective ×1000. Some microscopes add a tube factor or a magnification changer, which multiplies in the same way: Evident’s microscopy primer works through a ×5 objective with ×15 eyepieces and a 1.25× tube factor, for ×93.75. Each stage is a converging lens forming an image, which the thin lens calculator solves and the ray diagram simulator draws.

That figure describes the lenses, not anything drawn through them. The 90 µm cell above, seen at ×400 and drawn 45 mm wide, makes a drawing at ×500, because a drawing’s magnification is set by how large it is drawn. Label a drawing with its own magnification or with a scale bar.

Field of view and cell size

The field of view is the diameter of the circle of specimen you can see. It is set by the eyepiece’s field number, the diameter in millimetres of the fixed diaphragm inside it, which is engraved after its magnification: a 10×/18 eyepiece has a field number of 18. Dividing by the objective magnification, and by any tube factor, gives the field at the specimen: field of view = FN ÷ (objective × factor). The eyepiece magnification is not in it.

With a 10×/18 eyepiece the field is 18 ÷ 4 = 4.5 mm across with a ×4 objective, 1.8 mm with ×10, 0.45 mm with ×40 and 0.18 mm with ×100. To estimate the size of a cell, count how many span the diameter and divide: five cells across the 0.45 mm field, which is 450 µm, are about 90 µm each. That is an estimate rather than a measurement, an average over the cells counted.

If the eyepiece shows no field number, measure the field at low power with a clear ruler and scale it. With the same eyepiece the field’s diameter is inversely proportional to the total magnification, so 4.5 mm measured at ×40 becomes 4.5 × 40 ÷ 400 = 0.45 mm at ×400. Choose A field I measured above and the calculator does that step.

Common mistakes

• Dividing sizes in different units. 45 mm ÷ 90 µm as bare numbers gives 0.5, a thousand times too small. Convert first: 1 mm = 1000 µm and 1 µm = 1000 nm.
• Using the lens setting as a drawing’s magnification. A cell seen at ×400 can be drawn at ×500 or at ×50, depending on how large it is drawn. The drawing’s magnification is its size over the cell’s.
• Multiplying by the magnification to get the actual size. The actual size is the image size divided by M, so above ×1 it always comes out smaller than the image.
• Adding the lens magnifications. A ×10 eyepiece and a ×40 objective make ×400, not ×50.
• Measuring the high power field with a ruler. At ×400 the field is about 0.45 mm across, narrower than one millimetre division, so measure at low power and scale it.

What this calculator leaves out

Resolution. Magnification says how large the image is, not how much detail it holds. An ordinary light microscope cannot resolve detail finer than about 0.2 µm, which is Abbe’s limit, wavelength ÷ (2 × numerical aperture), for green light of 550 nm through an oil immersion objective of numerical aperture 1.4. Past about 1000 times the numerical aperture, roughly ×1400 for that objective, the image only gets bigger. The calculator warns when the lens settings pass that, and when an actual size falls below 0.2 µm.

Graticule calibration. At A-level, cells are measured with an eyepiece graticule calibrated against a stage micrometer: if 40 graticule divisions line up with 100 µm of the stage micrometer, each division is 100 ÷ 40 = 2.5 µm with that objective. The calculator does not do that step, but a size worked out that way goes straight into the actual size field.

Photographs and screens. A magnification printed beside an image is only right at the size it was printed. Once a micrograph is zoomed on a screen or resized, measure against its scale bar instead, which changes size with it.

Converting units first? Use the length conversion table.