**(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 10^{3}
=
25 000 µm (1 mm = 1 x 10^{3} 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 10^{3}
µm

Therefore
**image magnification** = image size / real image size =
35
x 10^{3}
/ 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 10^{3} = 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 10^{3}
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 10^{6}
µm

1 m
≡
1.0 x 10^{9} nm, 1 mm
≡
1.0 x 10^{3} µm, 1 cm
≡
1.0 x 10^{4} µm, 1 m
≡
1.0 x 10^{6} 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 13/10^{6})

1 cm
≡ 10 mm
≡ 10 x 1 x 10^{3} = 1.0 x 10^{4}
µm, so diameter = 13
÷
10^{4} =
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 10**^{4}
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 10**^{1}
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 10**^{4}
nm (1 m
≡ 1000 m,
1 m
≡ 10^{9} 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 10^{3}
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 10^{6}

therefore image size =
6.0 x 10^{-3} x 10^{6} =
__6.0 x 10__^{3}__
__**
µm**

**
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.

**(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
****
µm**^{2}

**
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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