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Thin Lens Equation & Magnification Calculator

The Gaussian Thin Lens Equation relates an object’s distance from a lens, the lens focal length, and the resulting image location (1/f = 1/d_o + 1/d_i).

Positive (+) for converging convex lenses; negative (-) for diverging concave lenses.

Distance from the object to the optical center of the lens in centimeters.

Physical height of the source object in centimeters.

Calculated Result
15.00 cm

Image Distance (dᵢ)

Image Distance

15.00 cm

Magnification

-0.50×

Optical Power

10.00 D

Image Type

Real, Inverted

Calculation Breakdown

  1. Optical Power in Diopters10.00 D (Converging Convex)P=100f(cm)P = \frac{100}{f(\text{cm})}
  2. Gaussian Thin Lens Equation for Image Distanced_i = 15.00 cm1f=1do+1di  ⟹  di=f⋅dodo−f\frac{1}{f} = \frac{1}{d_o} + \frac{1}{d_i} \implies d_i = \frac{f \cdot d_o}{d_o - f}
  3. Linear Lateral Magnificationm = -0.500×m=−didom = -\frac{d_i}{d_o}
  4. Image Dimensions & Geometryh_i = 2.50 cm (Inverted, Real)hi=m⋅hoh_i = m \cdot h_o

Image Distance vs Object Distance (for f = 10 cm Convex Lens)

Interactive visualization based on your current inputs

Image Distance (cm)
0.07.515233015 cm20 cm30 cm50 cm100 cmObject Distance (cm)Image Distance (cm)

What Is the Thin Lens Equation & Magnification Calculator?

The Thin Lens Equation is the central mathematical formula in geometric optics describing how lenses refract light to form images.

It provides the optical foundation for eyeglasses, cameras, microscopes, telescopes, and projectors.

How Does the Thin Lens Equation & Magnification Calculator Work?

Under the paraxial ray approximation (rays close to the principal optical axis), light refraction across two curved spherical surfaces collapses into a single thin-lens interface.

Ray tracing confirms that parallel rays pass through the focal point, center rays continue undeflected, and rays through the near focal point exit parallel.

Thin Lens Equation & Magnification Calculator Formula & Variables

The core mathematical equation utilized by this calculator is expressed as:

1f=1do+1di  ⟺  di=f⋅dodo−f,m=−dido\frac{1}{f} = \frac{1}{d_o} + \frac{1}{d_i} \iff d_i = \frac{f \cdot d_o}{d_o - f}, \quad m = -\frac{d_i}{d_o}

Gaussian paraxial lens equation. Real images form on the opposite side of the lens (d_i > 0); virtual images form on the same side as the object (d_i < 0).

How to Use the Thin Lens Equation & Magnification Calculator

  1. Input the focal length in centimeters (positive for convex, negative for concave).
  2. Enter the distance of the object from the lens.
  3. Optionally supply object height to compute physical image dimensions.

Step-by-Step Example Calculation

Camera Lens Projection

Input Values:

focalLength:10
objectDistance:30
objectHeight:5
Worked Steps: An object placed 30 cm from a +10 cm focal lens projects a real, inverted image at 15 cm with -0.5× magnification (2.5 cm tall).

Understanding Your Result

Image Distance (d_i): Location of the focal plane relative to the lens.

Magnification (m): Scale factor (m > 1 enlarged, m < 1 diminished).

Image Nature: Classifies whether image is real vs virtual and inverted vs upright.

Factors That Affect the Result

  • Focal Length: Shorter focal lengths bend light more sharply, creating higher diopter power.
  • Object Distance: Moving closer to the focal point dramatically shifts image position toward infinity.

When Should You Use This Calculator?

  • Photography lens selection and sensor crop focus distance calculations.
  • Optometry eyeglass power modeling and telescope objective design.

Assumptions & Limitations

  • Assumes lens thickness is negligible compared to radii of curvature.
  • Neglects chromatic and spherical aberrations.

Frequently Asked Questions

Calculation Accuracy & Reference Note

Standard paraxial geometric optics formulation with 100% theoretical precision.

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