Search formulas...⌘K

Optics

Thin Lens Equation

The thin-lens equation relates object distance, image distance, and focal length for paraxial rays.

\frac{1}{d _{o}} + \frac{1}{d _{i}} = \frac{1}{f}

Study tools

Save, share, export, or report this formula.

Source & review

Difficulty

Intermediate

Read time

5 min

Prerequisites

Basic algebra, consistent units, and the stated model assumptions

Review statusNeeds review

Source

OpenStax, College Physics 2e, §25.6 (Paul Peter Urone and Roger Hinrichs)

OpenFormulaGraph equation reference; FormuLab adaptation details and license are provided below.

Adapted from OpenFormulaGraph, under CC BY 4.0.

FormuLab adapted wording, notation, applicability conditions and topic placement, and expanded the numerical check into a worked example.

Definition

Understanding the core concept

The thin-lens equation relates object distance, image distance, and focal length for paraxial rays. Signed distances distinguish real and virtual objects or images.

Variables & Units

Understanding each component

Symbol	Meaning	Units
$d_{o}$	Length (object distance)	$m$
$d_{i}$	Length (image distance)	$m$
$f$	Length (focal length)	$m$

Real-World Applications

Where this formula is used in practice

Image formation

Find image distance for an ideal thin lens.

Lens models

Compare focal lengths and object positions.

Worked Example

Step-by-step calculation with real numbers

Problem

An ideal converging thin lens has object distance 30 m and focal length +10 m. Find its image distance.

Given

d_{o} = 30 m

f = 10 m

Solution

\frac{1}{d _{o}} + \frac{1}{d _{i}} = \frac{1}{f}

d_{i} = \frac{1}{( \frac{1}{f} - \frac{1}{d _{o}} )}

d_{i} = \frac{1}{( \frac{1}{( 10 m )} - \frac{1}{( 30 m )} )}

d_{i} \approx 15 m

Final Answer

d_{i} \approx 15 m

Formula Guide

Info

DifficultyIntermediate

Read Time5 min

PrerequisitesBasic algebra, consistent units, and the stated model assumptions

Tools

Source & review

StatusNeeds review

Last reviewedPending

Source

OpenStax, College Physics 2e, §25.6 (Paul Peter Urone and Roger Hinrichs)

OpenFormulaGraph equation reference; FormuLab adaptation details and license are provided below.

Adapted from OpenFormulaGraph, under CC BY 4.0.

FormuLab adapted wording, notation, applicability conditions and topic placement, and expanded the numerical check into a worked example.

Assumptions

• Use a thin lens in the same surrounding medium on both sides with small ray angles.
• For a real object, d_o > 0; a real image has d_i > 0, a virtual image d_i < 0, and a converging lens f > 0.

Languages

FormuLab is a study reference. Verify formulas, units, assumptions, and course-specific conventions before relying on them.

Browse by category

See all

Mathematics Physics Fluid Mechanics Electrical Computer Science Engineering

Thin Lens Equation

Image formation

Lens models

Problem

Given

Solution

Final Answer

Contents

Info

Tools

Source & review

Languages

Browse by category

Product

Company

Legal