An angular magnification (magnifying power) of $30\text{X}$ is desired using an objective of focal length $1.25\ \text{cm}$ and an eyepiece of focal length $5\ \text{cm}$. How will you set up the compound microscope?

An angular magnification (magnifying power) of 30X is desired using an objective of focal length 1.25cm and an

Exercises · 9.26

30\text{X}

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Concept understanding — Microscope Magnification

Microscope Magnification

A compound microscope views very small, nearby objects using two lenses in sequence: the objective (near the object) and the eyepiece (near the eye). Its total magnifying power is the product of what each lens contributes.

How the two lenses work together

The object sits just beyond the focus of the short-focal-length objective ( $f_o$ ), which forms a real, inverted, magnified image inside the tube.
That real image falls just inside the focus of the eyepiece ( $f_e$ ), which acts as a simple magnifier, producing a large virtual, magnified final image for the eye.

Because each stage magnifies, the effects multiply:

M = m_o \times m_e

The objective's magnification

m_o = \frac{v_o}{u_o} \approx \frac{L}{f_o}

where $L$ is the tube length (roughly the distance between the objective's second focal point and the eyepiece's first focal point). The object sits close to $f_o$ , so this approximation holds for a well-designed microscope.

The eyepiece's magnification

The eyepiece behaves as a simple magnifier:

Final image at the near point ( $D = 25\ \text{cm}$ , largest magnification):

m_e = 1 + \frac{D}{f_e}

Final image at infinity (relaxed eye, "normal adjustment"):

m_e = \frac{D}{f_e}

Total magnifying power

M = \frac{L}{f_o}\left(1 + \frac{D}{f_e}\right)

Important

High magnification needs short $f_o$ and $f_e$ (both sit in denominators) and a large tube length $L$ — this is why a microscope objective is always a very short-focus lens.

Worked example

Objective $f_o = 1.0\ \text{cm}$ , eyepiece $f_e = 2.5\ \text{cm}$ , tube length $L = 20\ \text{cm}$ , near point $D = 25\ \text{cm}$ . Find $M$ with the final image at the near point.

m_o = \frac{20}{1.0} = 20, \qquad m_e = 1 + \frac{25}{2.5} = 11

$M = 20 \times 11 = 220$ …

Why this formula?

Microscope Magnification: Why the Formula Holds

Let's build the understanding from first principles — not just memorizing formulas, but seeing why they work.

1. What Does "Magnification" Mean in a Microscope?

A microscope creates a larger apparent image of a tiny object. The total magnification is the product of two stages:

Objective lens — creates a real, enlarged, inverted image of the specimen.
Eyepiece (ocular) — acts as a simple magnifier to view that real image.

So:

Total magnification = (magnification by objective) × (magnification by eyepiece)

2. The Key Formula

For a compound microscope in normal adjustment (final image at infinity, relaxed eye):

M = M_o \times M_e = \left( \frac{L}{f_o} \right) \times \left( \frac{D}{f_e} \right)

Where:

$f_o$ = focal length of objective
$f_e$ = focal length of eyepiece
$L$ = tube length (distance between second focal point of objective and first focal point of eyepiece)
$D$ = near point distance of the eye (usually 25 cm)

3. Derivation of Objective Magnification $M_o$

Step 1: How the objective works

The objective lens forms a real, inverted, enlarged image of the specimen. The specimen is placed just outside its focal point ( $f_o$ ).

Step 2: Using the lens formula

For a thin lens:

\frac{1}{v_o} - \frac{1}{u_o} = \frac{1}{f_o}

(Using sign convention: $u_o$ is negative, $v_o$ is positive)

Step 3: The tube length approximation

In a standard microscope, the specimen is placed very close to $f_o$ , so:

$u_o \approx -f_o$ (object just beyond focal point)
The image is formed at the first focal point of the eyepiece, which is at a distance $L$ from the second focal point of the objective.

Thus:

v_o \approx f_o + L

Step 4: Magnification formula

Lateral magnification by objective:

M_o = \frac{v_o}{|u_o|} \approx \frac{f_o + L}{f_o} = 1 + \frac{L}{f_o}

Since $L \gg f_o$ in practice, $1$ is negligible:

\boxed{M_o \approx \frac{L}{f_o}}

Why this makes sense: A shorter $f_o$ means the objective is more "powerful" — it bends light more sharply, creating a larger image at the fixed tube length.

4. Derivation of Eyepiece Magnification $M_e$

Step 1: The eyepiece as a simple magnifier

The eyepiece takes the real image from the objective and acts like a magnifying glass. For relaxed eye (final image at infinity), the real image must be placed at the focal point of the eyepiece.

Step 2: Angular magnification

Angular magnification is defined as:

M_e = \frac{\text{angle subtended by image at eye}}{\text{angle subtended by object at near point}}

For a simple magnifier with image at infinity:

M_e = \frac{D}{f_e}

Where $D = 25$ cm (standard near point). …

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Form the final image at the near point ( $25\ \text{cm}$ ): the eyepiece gives $m_e = 6$ , so the objective must give $m_o = 5$ . This places the object $1.5\ \text{cm}$ from the objective and separates the two lenses by about $11.67\ \text{cm}$ .

Given

$f_o = 1.25\ \text{cm}$ , $f_e = 5\ \text{cm}$ , desired magnifying power $M = 30$ , near point $D = 25\ \text{cm}$ .

Step 1 — Eyepiece magnification (image at near point)

m_e = 1 + \frac{D}{f_e} = 1 + \frac{25}{5} = 6

Step 2 — Objective magnification

Since $M = m_o \times m_e$ ,

m_o = \frac{M}{m_e} = \frac{30}{6} = 5

Step 3 — Object position at the objective

The objective forms a real, inverted image, so $|v_o| = 5|u_o|$ . Using $\dfrac{1}{v_o} - \dfrac{1}{u_o} = \dfrac{1}{f_o}$ with $u_o<0$ , $v_o>0$ :

\frac{1}{5|u_o|} + \frac{1}{|u_o|} = \frac{1}{1.25} \;\Rightarrow\; \frac{6}{5|u_o|} = \frac{1}{1.25}

|u_o| = \frac{6 \times 1.25}{5} = 1.5\ \text{cm}, \qquad v_o = 5 \times 1.5 = 7.5\ \text{cm}

The object sits $1.5\ \text{cm}$ from the objective, just beyond its focus $f_o = 1.25\ \text{cm}$ .

Step 4 — Eyepiece object distance …

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Here are the common mistakes students make when solving this compound microscope problem, along with how to avoid each.

Mistake 1: Confusing Magnification for the Final Image at Infinity vs. at Near Point

The error:

Students often blindly use the formula for angular magnification when the final image is at infinity ( $M = \frac{L}{f_o} \cdot \frac{D}{f_e}$ ) without checking the problem’s condition. Here, the desired magnification is 30X, but using the infinity formula gives a different tube length.

How to avoid:

Always check which case is implied. In most exam problems, unless stated otherwise, the final image is formed at the near point (25 cm). The correct formula for that case is:

M = \frac{L}{f_o} \left(1 + \frac{D}{f_e}\right)

where $D = 25\ \text{cm}$ (least distance of distinct vision).

For this problem, using $f_o = 1.25\ \text{cm}$ , $f_e = 5\ \text{cm}$ , and $M = 30$ , you solve for $L$ :

30 = \frac{L}{1.25} \left(1 + \frac{25}{5}\right) = \frac{L}{1.25} \times 6

\Rightarrow L = \frac{30 \times 1.25}{6} = 6.25\ \text{cm}

So the tube length is $6.25\ \text{cm}$ .

Mistake 2: Forgetting to Add the Eyepiece Focal Length to Get Total Microscope Length

The error:

Students stop after finding $L$ (distance between the second focal point of the objective and the first focal point of the eyepiece) and report that as the total length of the microscope.

How to avoid:

The total length of the microscope is the distance between the objective and the eyepiece. This is:

\text{Total length} = f_o + L + f_e

For this problem:

\text{Total length} = 1.25 + 6.25 + 5 = 12.5\ \text{cm}

Always draw a quick ray diagram to remind yourself: the objective’s second focal point and the eyepiece’s first focal point coincide — so the physical separation includes both focal lengths.

Mistake 3: Using the Wrong Sign Convention for Lens Formula

The error:

When verifying the image distances, students plug values into the lens formula without consistent sign convention, leading to negative distances that confuse them.

How to avoid:

Use the Cartesian sign convention (distances measured from the optical centre, positive in the direction of incident light). For the objective:

$u_o$ is negative (object is real, on left)
$f_o$ is positive (convex lens)
$v_o$ is positive (real image on right)

For the eyepiece:

$u_e$ is negative (object is real, on left)
$f_e$ is positive
$v_e$ is negative (final virtual image on left)

This consistency prevents sign errors.

--- …

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Q.An angular magnification (magnifying power) of $30\text{X}$ is desired using an objective of focal length $1.25\ \text{cm}$ and an eyepiece of focal length $5\ \text{cm}$ . How will you set up the compound microscope?

Concept understanding — Microscope Magnification

Microscope Magnification

How the two lenses work together

The objective's magnification

The eyepiece's magnification

Total magnifying power

Worked example

Why this formula?

Microscope Magnification: Why the Formula Holds

1. What Does "Magnification" Mean in a Microscope?

2. The Key Formula

3. Derivation of Objective Magnification $M_o$

Step 1: How the objective works

Step 2: Using the lens formula

Step 3: The tube length approximation

Step 4: Magnification formula

4. Derivation of Eyepiece Magnification $M_e$

Step 1: The eyepiece as a simple magnifier

Step 2: Angular magnification

Given

Step 1 — Eyepiece magnification (image at near point)

Step 2 — Objective magnification

Step 3 — Object position at the objective

Step 4 — Eyepiece object distance …

Method: Two-Lens Ray Diagram Approach for Compound Microscope Setup

Step 1: Recall the magnification formula

Step 2: Substitute known values

Step 3: Solve for tube length $L$

Step 4: Determine lens positions

Mistake 1: Confusing Magnification for the Final Image at Infinity vs. at Near Point

Mistake 2: Forgetting to Add the Eyepiece Focal Length to Get Total Microscope Length

Mistake 3: Using the Wrong Sign Convention for Lens Formula

Q.An angular magnification (magnifying power) of 30X30\text{X}30X is desired using an objective of focal length 1.25 cm1.25\ \text{cm}1.25 cm and an eyepiece of focal length 5 cm5\ \text{cm}5 cm. How will you set up the compound microscope?

Given

Step 1 — Eyepiece magnification (image at near point)

Step 2 — Objective magnification

Step 3 — Object position at the objective

Step 4 — Eyepiece object distance …

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Q.An angular magnification (magnifying power) of $30\text{X}$ is desired using an objective of focal length $1.25\ \text{cm}$ and an eyepiece of focal length $5\ \text{cm}$ . How will you set up the compound microscope?

Unlock everything free for 14 days