UK GCSE level age ~14-16, ~US grades 9-10 Biology revision notes re-edit 20/05/2023 [SEARCH]

Microscopy: 4. Examples of numerical calculations in microscopy - magnifying power of a microscope

Doc Brown's Biology exam study revision notes

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There are various sections to work through,

after 1 they can be read and studied in any order.

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(4) 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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