calculate magnifications of images and actual sizes of specimens from drawings, photomicrographs and electron micrographs (scanning and transmission)

The Microscope in Cell Studies

Learning Objective

Calculate the magnification of images and determine the actual size of specimens from drawings, photomicrographs and electron micrographs (scanning and transmission).

1. Types of Microscopes Used in Cell Biology

• Light (optical) microscope – up to \overline{2000}× total magnification.
• Scanning Electron Microscope (SEM) – provides a 3‑D surface view; magnification up to \overline{1} 000 000×.
• Transmission Electron Microscope (TEM) – provides a 2‑D internal view; magnification up to \overline{10} 000 000×.

2. Key Concepts of Magnification

The total magnification (\$M\$) of an image is the product of the objective magnification (\$mo\$) and the eyepiece magnification (\$me\$):

\$\$

M = mo \times me

\$\$

For electron microscopes the “objective” is the electron lens system, and the “eyepiece” is replaced by a camera or screen. The same principle applies, but the magnification is usually given directly by the instrument.

3. Calculating Actual Size from an Image

If the size of the image on a screen or paper is known, the actual size of the specimen (\$S\$) can be found from:

\$\$

S = \frac{I}{M}

\$\$

where \$I\$ is the measured image dimension (e.g., in mm) and \$M\$ is the total magnification.

4. Practical Steps for Different Media

4.1 Hand‑drawn Microscopy Sketches

1. Measure the length of the drawn feature with a ruler (in mm). Call this \$I_{draw}\$.
2. Determine the magnification used for the sketch (often given on the diagram). If not given, calculate \$M\$ from the known size of a reference structure.
3. Apply \$S = I_{draw}/M\$ to obtain the real size.

4.2 Photomicrographs (Digital Images)

1. Open the image in a viewer that can display pixel dimensions.
2. Measure the feature in pixels (\$I{px}\$) and note the scale bar length in pixels (\$L{px}\$) and its real length (\$L_{real}\$, usually given in µm).
3. Calculate the pixel‑to‑real conversion factor: \$k = \dfrac{L{real}}{L{px}}\$ (µm per pixel).
4. Actual size: \$S = I_{px} \times k\$.

4.3 Scanning Electron Micrographs (SEM)

1. Identify the scale bar on the image (e.g., 5 µm).
2. Measure the scale bar length on the printed or screen image (\$L_{px}\$).
3. Conversion factor: \$k = \dfrac{L{real}}{L{px}}\$.
4. Measure the feature of interest in pixels (\$I{px}\$) and compute \$S = I{px} \times k\$.

4.4 Transmission Electron Micrographs (TEM)

1. Locate the scale bar (often in nm).
2. Measure its length in pixels (\$L{px}\$) and note the real length (\$L{real}\$).
3. Conversion factor: \$k = \dfrac{L{real}}{L{px}}\$ (nm per pixel).
4. Measure the structure of interest (\$I{px}\$) and calculate \$S = I{px} \times k\$.

5. Worked Examples

Example 1 – Hand‑drawn Light‑Microscope Sketch

A student draws a plant cell with a nucleus that measures 12 mm on the sketch. The sketch was made at 400× total magnification.

\$\$

S = \frac{12\ \text{mm}}{400} = 0.030\ \text{mm} = 30\ \mu\text{m}

\$\$

The actual diameter of the nucleus is therefore 30 µm.

Example 2 – Photomicrograph of a Bacterial Cell

Image details:

• Scale bar: 2 µm = 150 px
• Measured bacterial length: 450 px

Conversion factor:

\$\$

k = \frac{2\ \mu\text{m}}{150\ \text{px}} = 0.0133\ \mu\text{m/px}

\$\$

Actual length:

\$\$

S = 450\ \text{px} \times 0.0133\ \mu\text{m/px} = 6.0\ \mu\text{m}

\$\$

Example 3 – SEM of a Pollen Grain

Conversion factor:

\$\$

k = \frac{10\ \mu\text{m}}{200\ \text{px}} = 0.05\ \mu\text{m/px}

\$\$

Actual diameter:

\$\$

S = 800\ \text{px} \times 0.05\ \mu\text{m/px} = 40\ \mu\text{m}

\$\$

Example 4 – TEM of a Mitochondrion

Image details:

• Scale bar: 100 nm = 250 px
• Measured width of mitochondrial cristae: 120 px

Conversion factor:

\$\$

k = \frac{100\ \text{nm}}{250\ \text{px}} = 0.40\ \text{nm/px}

\$\$

Actual width:

\$\$

S = 120\ \text{px} \times 0.40\ \text{nm/px} = 48\ \text{nm}

\$\$

6. Summary of Formulas