(4A) Examples of numerical calculations in microscopy - magnifying power of a microscope

See also (6) More on magnification and measuring the size of a cell using a graticule and stage micrometer

Calculations involving scale drawings and magnification I've dealt with above.

Note: In these exemplar calculations I've used the symbol ≡ to indicate equivalence.

You need to be able to use the prefixes centi (10^-2), milli (10^-3), micro (10^-6) and nano (10^-9) and express answers in standard form when carrying out calculations involving magnification, real size and image size using the magnification formula (below).

The reason for this is that that the real size of the objects under investigation with a microscope are very small!

Magnification formulae

Magnification of image

e.g. solving magnification problems and the relative sizes of object e.g. a cell and its observed image in a microscope.

The formula 'triangle' for magnification is shown on the right (size = length).

When using the formula:

magnification of image = the image size ÷ object size

make sure the image size and object size are in the same length units!

Rearrangements: image size = magnification x object (specimen) size

and the real object size = image size ÷ magnification

Note: The quoted magnification of image in a textbook is quite valid and important.

However the same cannot be said for the image on a computer screen.

The screen resolution might be different from one screen to another, even for the same original image, so the magnification will appear to be different.

Two examples of calculations of image size magnification

I'm just using two previously used cell diagrams.

Ex 1.

A plant cell calculation of magnification of image

From the microscope scale the real width of the plant cell is 0.10 mm.

On a paper printout, the width of the plant cell is 15 mm.

image magnification = image size / actual real size of object = 15 / 0.10 = 150

 

Ex 2.

An onion cell calculation

From the microscope scale, two of the onion cells have on average a real total length of ~400 µm.

So the real cell length is ~200 µm

On a  paper printout, the length of two cells on average is ~5.0 cm,.

So the image cell length = ~2.5 cm

You must convert one of the numbers, so that both numbers have the same units.

2.5 cm ≡ 25 mm ≡ 25 x 10³  = 25 000 µm (1 mm = 1 x 10³ µm)

Image magnification = size of image / size of object = 25 000 / 200 = 125x

Ex 3. A micrograph

A micrograph of a red blood cell is 35 mm long.

If the red blood cell has a diameter of 7.0  µm, what is the image magnification?

You first need to do a unit conversion: 35 mm ≡ 35 x 10³ µm

Therefore image magnification = image size / real image size = 35 x 10³ / 7.0 = 5000

 

Total magnification of microscope - do NOT confuse with the above 'image magnification'

If you know (and you should!) know the magnifying power (x...) of both the eyepiece lens and the objective lens, it is quite easy to calculate the total magnifying power of the microscope.

The following simple formula for magnification is:

total magnification = eyepiece lens magnifying power x objective lens magnifying power

e.g. if the eyepiece lens power is x 5 and the objective lens power is x 50, the total magnification is 5 x 50 = x250

Simple rearrangement allows you to calculate the magnification necessary for a particular lens for a specified total magnification.

e.g. If an eyepiece lens has a magnification of 20x, what must the magnification of the objective lens (z) be to give a total magnification of 800x?

total magnification = 800 = 20 x z

z = magnification of the objective lens = 800 / 20 = 40x

And: If an objective lens has a magnification of 30x, what must the magnification of the eyepiece lens (y) be to give a total magnification of 1500x?

total magnification = 1500 = y x 30

y = magnification of eyepiece lens is 1500 / 30 = 50x

Further note on units

metres (m), centimetres (1 cm ≡ 1.0 x 10^-2 m), millimetres (1 mm ≡ 1.0 x 10^-3 m)

micrometres (1 µm ≡ 1.0 x 10^-6 m), nanometres (1 nm ≡ 1.0 x 10^-9 m)

See the questions below for other examples of unit conversions and presenting lengths in standard form.

 

Examples of microscope scale calculations

BUT, first a note on conversion factors for length

With the naked eye you can see (resolve) objects of width ~0.04 mm (0.04 x 10³ = 40 µm)..

Most animal and plant cells are ~0.01 to 0.1 mm in 'diameter' (10 to 100 µm), so some can be seen with the naked eye, but most can only be resolved, that is clearly observed, using a microscope.

Most cell dimensions e.g. diameter of cell are measured in micrometers (µm, 10^-6 m), but the size of even smaller  subcellular structure or viruses are often measured in the smaller unit nanometre (nm, 10^-9 m)

You need to be able to use lots of equivalents - length conversion factors e.g.

based on nanometre nm, micrometre µm, millimetre mm, centimetre cm, metre m length units

1 mm ≡ 0.1 cm ≡ 1/1000 m = 0.001 m = 1.0 x 10^-3 m ≡ 1000 or 1 x 10³  µm

1 µm ≡ 1/1000 mm ≡ 1.0 x 10^-3 mm ≡ 1 x 10^-6 m ≡ 1000 nm (1 nm = 1.0 x 10^-9)

1 nm ≡ 1/1000 µm = 0.001 µm = 1.0 x 10^-3 µm

1 µm ≡ 10^-6 m, 1 nm ≡ 1.0 x 10^-9 m, 1 cm ≡ 1.0 x 10^-2 m, 1 m ≡ 1.0 x 10⁶ µm

1 m ≡ 1.0 x 10⁹ nm, 1 mm ≡ 1.0 x 10³ µm, 1 cm ≡ 1.0 x 10⁴ µm, 1 m ≡ 1.0 x 10⁶ µm

(You need to use the above for scale conversions to solve the problems given below)

 

Q1 A white blood cell has a diameter of 13.0 µm (13 micrometres)

Express the diameter of the white blood cell in various units...

13.0 µm ≡ 13 x 1.0 x 10^-6 m = 1.3 x 10^-5 m  (you get the same answer from 13/10⁶)

1 cm ≡ 10 mm ≡ 10 x 1 x 10³ = 1.0 x 10⁴ µm, so diameter = 13 ÷ 10⁴ = 1.3 x 10^-3 cm

1 mm ≡ 1000 µm, so 13 µm ≡ 13 ÷ 1000 = 0.013 mm = 1.3 x 10^-2 mm

1 µm ≡ 1000 nm, so 13.0 µm ≡ 13  x 1000 = 13,000 = 1.3 x 10⁴ nm

Just try working them out in your own way and see if you agree with my answers

 

Q2 A red blood cell has a diameter of 0.012 mm.

Give the red blood cell diameter in metres, millimetres, micrometres and nanometres all written in standard form.

0.012 ÷ 1000 = 1.2 x 10^-5 m  (1 m  ≡ 1000 mm)

0.012 mm = 1.2 x 10^-2 mm  (just changed to standard form)

0.012 mm ≡ 0.012 x 1000 = 1.2 x 10¹ µm  (1 mm  ≡ 1000 µm)

0.012 mm ≡ 0.012 ÷ 1000 = 1.2 x 10^-5 m, 1.2 x 10^-5 ÷ 10^-9 = 1.2 x 10⁴ nm  (1 m  ≡ 1000 m, 1 m  ≡ 10⁹ nm)

Again, just try working them out in your own way and see if you agree with my answers

 

Q3 Suppose the real length of a cell is 150 µm.

What is the size of the cell image with a microscope magnification of X 50? (give you answer in µm and mm)

image size = magnification x object size, 1 mm ≡ 1000 µm

image size = 50 x 150 = 7500 µm and 7500 ÷ 1000 = 7.5 mm

 

Q4 The image of a red blood cell is 7.0 mm under a microscope magnification of 1000 X.

What is the real size of the red blood cell? (give your answer in micrometres and nanometres)

1 mm ≡ 1000 µm, so 7.0 mm ≡ 7.0 x 1000 = 7000 µm.

object size = image size ÷ magnification

real object size = 7000 ÷ 1000 = 7.0 µm

1 µm ≡ 1 x 10^-6 m, 1 nm ≡ 1 x 10^-9 m, so 1 µm ≡ 10^-6 ÷ 10^-9 = 1000 nm

Therefore cell diameter = 7.0 x 1000 = 7.0 x 10³ nm

 

Q5 What magnification is needed to produce a 10 mm image of a cell from a specimen cell of diameter 125 µm?

magnification = image size ÷ actual size

1 mm ≡ 1000 µm, so 10 mm ≡ 10,000 µm

magnification needed = 10,000 ÷ 125 = 80 X

 

Q6 If a cell has diameter of 0.000090 m what is its diameter in ...

(a) standard form?

move the decimal place 5 place to the right and multiply the answer by x 10^-5 giving 9.0 x 10^-5 m

(b) micrometres (µm)?

1 µm = 10^-6 m, so move the decimal point forward 6 places to give 90 µm

 

Q7 A specimen is 60 µm wide.

Calculate the width of the image in mm under a magnification of x200.

image size = magnification x real size

image size = 200 x 60 = 12000 µm

1000 µm = 1 mm (1 µm = 1 x 10^-6 m, 1 mm = 1 x 10^-3 m, 10^-3/10^-6 = 1000)

12000/1000 = 12 mm

 

Q8 A cell is 3 x 10^-5 m wide.

Calculate the width of the image under a microscope magnification of x 200.

Quote you answer in (a) metres and (b) micrometers, written in standard form.

image size = magnification x real object size

(a) image size = 200 x 3 x 10^-5 = 600 x 10^-5 = 6.0 x 10^-3 m

(b) Since 1 µm ≡ 1.0 x 10^-6 m, micrometers = metres x 10⁶

therefore image size = 6.0 x 10^-3 x 10⁶ = 6.0 x 10³ µm

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Keywords, phrases and learning objectives for Part 4A

Know how to calculate the magnification of a microscope in biological microscopy.

Understand examples of numerical scaled magnification calculations in microscopy based on the magnifying power of a microscope e.g. ratio of image size to object size and total magnification calculations for a microscope.

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(4B) More on microscope magnification, measuring the size of a cell using a graticule & stage micrometer

See also (4) Examples of numerical calculations in microscopy - magnification and magnifying power of a microscope

As already mentioned, the size of structures is important in biological sciences.

Accurate measurements can be essential but even estimates can be good enough and quicker to obtain.

In Part 4. Ex 2. you were shown how to estimate the size of an onion cell - diagram below.

I've reworked the diagram to give the idea of the circular field of view when observing a specimen under the microscope.

Diagram of the microscope field of view of onion cells using a relatively high magnification.

Now cells tend to vary a little bit in size, so you would want an average value.

However, in the above field of view, there are only two cells across the diameter of the field of view giving a limited, and therefore less accurate, estimation of the average width of the onion cell.

The more cells you can clearly see and count in a row across the diameter of the field of view, the more accurate is your estimate cell size.

A better image to estimate the size of an individual cell.

Average size of a single cell = diameter of field of view (d) divided by the number of cells (n) in a row across the diameter of the field of view.

In this case: average cell size = d / n = d / 9 (with appropriate length units)

Suppose the field of view of the above cells was ~0.20 mm, what would the average width of the cell be?

0.20 mm ≡ 0.20 x 10^-3 m ≡  2.0 x 10^-4 m ≡ 2.0 x 10^-4 / 10^-6 = ~200 µm

average cell size = 200 / 9 = ~22 µm (2 sf)

Accurate measurement of cell size

In order to make accurate measurements of cell size you need to be able to calibrate the microscope.

Both the eyepiece and the field of view of the microscope stage need an accurate scale that can be focussed as well as the image of the specimen being examined under the microscope.

(i) The graticule

A graticule is a thin piece of glass/plastic onto which an accurate scale has been draw.

The graticule is positioned into the eyepiece of the microscope.

(ii) The stage micrometer

A stage micrometer is a microscope slide on which an accurate scale has been etched.

The stage micrometer is placed onto the microscope stage.

The microscope procedure using the graticule and micrometer

You place a stage micrometer on the stage of the microscope.

You line up one of the scale divisions of the eyepiece graticule with specific point on the stage micrometer.

You count the number of divisions on the eyepiece graticule that correspond with a specific measurement on stage micrometer.

You calculate the distance in micrometers of one division on the eyepiece graticule.

Comparing the eyepiece graticule and stage micrometer scales

Diagram to show the positioning of the eyepiece graticule and stage micrometer scales in a microscope

You use the stage micrometer scale to calibrate the eyepiece graticule scale.

On the above diagram I've drawn two thin vertical lines to match up the scales of the eyepiece graticule and stage micrometer.

The stage micrometer is marked in 50 µm divisions.

The eyepiece graticule is marked as 100 arbitrary units (a.u.).

From the two vertical lines we can now calibrate the arbitrary graticule scale.

As you can see from the diagram: 64 - 35 = 29 a.u. ≡ 50 µm

Therefore each a.u. on the graticule scale = 50/29 = 1.72  µm

In this case the field of view is about 200 µm (0.20 mm)

Once the eyepiece graticule is calibrated, the stage micrometer can be removed from the stage and replaced with a specimen microscope slide for examination.

How to use the eyepiece graticule scale to measure cell size.

Diagram showing the eyepiece graticule superimposed on the microscope specimen image.

From above we have a calibration of each a.u. on the graticule scale = 1.72 µm

Looking at the last diagram, the middle seven cells are measure from 7 to 90 a.u. on the graticule scale.

Therefore the average cell width is (90 - 7) / 7 = 11.87 a.u.

Using the conversion factor from above: average cell width = 11.86 x 1.72 = 6.96 = 20  µm (2 sf)

You can also pick out an individual cell and measure its size using the calibrated eyepiece graticule scale.

e.g. the 4th cell from the left: width = 43 - 30 = 13 a.u. on the graticule scale.

Therefore using the calibration factor: width of cell = 13 x 1.72 = 22 µm (2 sf)

Some other calculations based on the same cell diagram from above

(i) The diameter of the nucleus

The nucleus is about 3 a.u. wide on the eyepiece graticule.

This equates to 3 x 1.72 = 5.1 µm  (2 sf)

OR you can make a less accurate visual estimate from the image.

On average the diameter of the nucleus is about a 1/4 of the diameter of the cell.

It looks as if the cell diameter equals about 4 diameters of the cell.

The average width of the cells was measured to be 20 µm

Therefore the average diameter of the nucleus is 20/4 = 5 µm  (1 sf)

(ii) The area of a cell

Most cells are roughly a square or rectangle in shape.

Suppose one of the cells in the diagram is 20 µm wide and 22 µm in length.

Area = length x width = 20 x 22 = 400 µm²

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Keywords, phrases and learning objectives for Part 4B

Be able to do numerical calculations from measurements using calibrated scales from microscope slide measurements.

Know the measurements involve using an eyepiece graticule with a micrometer scale, from which you can estimate-calculate the size of objects under examination and the magnifying power of the microscope.

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