Calculate Magnification

Linear magnification describes the ratio of image size to object size. In geometrical optics, it can be calculated using different equivalent formulas.

What is Magnification?

Optical magnification (or magnification) is a central concept in optics that describes how much larger or smaller an image appears compared to the object. The linear magnification (V) is the ratio of image size to object size. A magnification of 2 means the image is twice as large as the object. A magnification of 0.5 means the image is half as large.

Sign of Magnification

The sign of magnification indicates the image orientation:

• V > 0 (positive): The image is upright (same orientation as object)
• V < 0 (negative): The image is inverted (upside down)

The magnitude |V| gives the size change, independent of orientation.

Size Classification by |V|

Magnification Formulas

1. Formula from Image Distance and Object Distance

The most fundamental magnification formula is based on the ratio of optical distances:

• s' – Image distance (distance from optical element to image)
• s – Object distance (distance from optical element to object)

Important: In sign convention, s and s' are sign-dependent. A real image has s' > 0, a virtual image has s' < 0.

2. Formula with Focal Length and Object Distance

Alternatively, magnification can be calculated from focal length and object distance:

• f – Focal length (focus distance)
• s – Object distance

This formula is useful when focal length and object position are known but the image isn't directly measured.

3. Connection: Lens Equation and Magnification

The lens equation connects distances and focal length:

By rearranging, one can show that both magnification formulas are equivalent:

Special Cases and Limiting Situations

Case 1: s = f (Object at Focus)

When the object is exactly at the focal point:

• The image recedes to infinity
• Magnification is infinite
• The exiting ray is parallel to the optical axis

Case 2: s < f (Object Between Lens and Focus)

For convex lenses: Magnification is upright and magnified.

• V > 0 (upright)
• |V| > 1 (magnified)
• The image is virtual
• This is how a magnifying glass works

Case 3: s = 2f (Twice Focal Length)

A special interesting case is when the object is at twice the focal length:

• The image is exactly the same size as the object (|V| = 1)
• The image is inverted (V < 0)
• The image also lies at twice the focal length on the other side

Case 4: s > 2f (Object Far Away)

• V < 0 (inverted)
• |V| < 1 (reduced)
• The image is real and closer to the focal point
• This is the principle of a camera or telescope

Application Examples in Various Optics

Magnifying Glass

• Simple convex lens with short focal length
• Object lies between lens and focal point (s < f)
• Typical magnification: 2× to 10×
• Image is virtual, upright, and magnified

Camera Lens

• Object is very far away (s ≫ f)
• V ≈ -f/s (very small, negative magnification)
• Image is real, inverted, and greatly reduced
• Forms scenes on a sensor

Projector

• Object (slide/film) lies just beyond focal length (s > f, but s near f)
• Typical magnification: 50× to 1000×
• Image is real, inverted, and greatly magnified
• Projected onto screen

Telescope

• Uses two lenses (objective + eyepiece)
• Effective magnification: V = f_ob / f_ok
• Typical magnification: 20× to 1000×
• Magnifies distant, faint objects

Microscope

• Object lies just beyond focal length (s ≈ f + small distance)
• Effective magnification: V = L × f_ob / (f_ob × f_ok)
• Typical magnification: 50× to 1500×
• Image is real, inverted, and extremely magnified

Numerical Examples

Example 1: Simple Refraction (V = s'/s)

• Given: s' = 20 cm, s = 10 cm
• V = 20/10 = 2 (magnification, real image, inverted if real)

Example 2: Magnifying Glass Effect (V = f/(f-s))

• Given: f = 5 cm, s = 3 cm (object before focal point)
• V = 5/(5-3) = 5/2 = 2.5 (upright, virtual magnification)

Example 3: Camera Effect (Reduction)

• Given: f = 50 mm, s = 5000 mm (object far away)
• V = 50/(50-5000) ≈ -0.01 (50× reduction, inverted)

Example 4: Special Case s = 2f

• Given: f = 10 cm, s = 20 cm (twice focal length)
• V = 10/(10-20) = 10/(-10) = -1 (identical size, inverted)

Sign Convention and Notation

In geometric optics, there are different sign conventions. The Cartesian convention is most common:

• Object side: Distances are positive in propagation direction
• Image side: Distances positive for real images (opposite side), negative for virtual images
• Focal length: Convex (converging) f > 0, concave (diverging) f < 0

Typical Magnifications in Practice

Frequently Asked Questions

Q: Can magnification be negative?

A: Yes. The sign indicates image orientation. Negative means inverted, positive means upright.

Q: What does a magnification of -0.5 mean?

A: The image is half as large as the object (|V| = 0.5 = reduction) and inverted (V < 0).

Q: Why is the image in a camera inverted?

A: Because the object is far away (s ≫ f), the image distance s' lies just beyond the focal point f. This results in V < 0 (negative), meaning inversion.

Q: How does a magnifying glass work?

A: The object is placed between the lens and focal point (s < f), producing a virtual, upright, magnified image (V > 1, positive).

Q: What happens at s = f?

A: The denominator f - s becomes zero, leading to V → ∞. The image recedes to infinity and becomes infinitely large.

Summary