Thin Lens Equation

Calculate focal length, object distance, or image distance using the lens formula.

Select the variable to solve for, enter the other two known values, and the calculator will find the unknown, along with image characteristics.

Practical Examples

Explore common scenarios to understand how the calculator works.

Real Image Formation

Converging Lens

An object is placed outside the focal length of a converging lens.

Scenario: Solving For di

Object Distance (do): 30 cm

Focal Length (f): 10 cm

Virtual Image (Magnifying Glass)

Converging Lens

An object is placed inside the focal length of a converging lens.

Scenario: Solving For di

Object Distance (do): 5 cm

Focal Length (f): 10 cm

Virtual Image Formation

Diverging Lens

A diverging lens always forms a virtual, upright, and minified image.

Scenario: Solving For di

Object Distance (do): 30 cm

Focal Length (f): -10 cm

Determining Lens Power

Finding Focal Length

An object at 20 cm creates a real image at 20 cm. This means the object is at 2F.

Scenario: Solving For f

Object Distance (do): 20 cm

Image Distance (di): 20 cm

Other Titles
Understanding the Thin Lens Equation: A Comprehensive Guide
An in-depth look at the principles of geometric optics, lens behavior, and image formation.

What is the Thin Lens Equation?

  • Core Principles of Geometric Optics
  • The Formula and Its Components
  • Sign Conventions: The Key to Correct Calculations
The Thin Lens Equation is a fundamental formula in optics that relates the focal length of a lens, the distance of an object from the lens, and the distance of the image formed by the lens. It assumes the lens is 'thin', meaning its thickness is negligible compared to its focal length and the object/image distances. This simplification is incredibly effective for analyzing a wide range of optical systems, from simple magnifying glasses to complex camera lenses.
The Formula: 1/f = 1/do + 1/di
Where 'f' is the focal length, 'do' is the object distance, and 'di' is the image distance. Understanding the sign conventions for each variable is critical for correct application.
Sign Conventions
Focal Length (f): Positive (+) for a converging (convex) lens, Negative (-) for a diverging (concave) lens.
Object Distance (do): Almost always positive (+), as objects are typically real and placed in front of the lens.
Image Distance (di): Positive (+) for a real image (formed on the opposite side of the lens from the object), Negative (-) for a virtual image (formed on the same side as the object).

Step-by-Step Guide to Using the Calculator

  • Selecting the Unknown Variable
  • Entering Known Values
  • Interpreting the Results
This calculator simplifies the process, but understanding each step ensures you interpret the results correctly.
1. Choose What to Solve For
Use the dropdown menu to select whether you want to calculate the 'Focal Length (f)', 'Object Distance (do)', or 'Image Distance (di)'. The form will adapt, disabling the input for your chosen variable.
2. Input the Known Quantities
Fill in the two active input fields. Remember to apply the correct sign conventions. For instance, if you are using a diverging lens, enter its focal length as a negative number.
3. Analyze the Output
After clicking 'Calculate', the tool provides not just the numerical answer but also key characteristics of the image. The sign of the calculated image distance tells you if the image is real or virtual, and the magnification value indicates if it's inverted, upright, larger, or smaller than the object.

Real-World Applications

  • Corrective Eyewear (Glasses and Contacts)
  • Photography and Camera Lenses
  • Telescopes and Microscopes
The thin lens equation is not just an academic exercise; it's the principle behind many technologies we use daily.
Vision Correction
Optometrists use these principles to prescribe lenses. A farsighted (hyperopic) eye is corrected with a converging lens, while a nearsighted (myopic) eye requires a diverging lens to move the focal point onto the retina.
Camera Systems
A camera lens focuses light from an object onto a sensor (or film). By changing the distance between the lens and the sensor (adjusting 'di'), you can bring objects at different distances ('do') into sharp focus.

Magnification and Image Characteristics

  • Calculating Magnification
  • Real vs. Virtual Images
  • Inverted vs. Upright Orientation
Beyond just finding distances, the lens equation helps us characterize the image.
The Magnification Formula: m = -di / do
Sign of m: A negative magnification means the image is inverted (upside down) relative to the object. A positive magnification means it is upright.
Magnitude of m: If |m| > 1, the image is magnified (larger). If |m| < 1, the image is minified (smaller). If |m| = 1, the image is the same size as the object.
Understanding Image Types
A real image is formed where light rays actually converge. It can be projected onto a screen (like in a cinema projector). It occurs when 'di' is positive. A virtual image is formed where light rays appear to diverge from. It cannot be projected onto a screen and must be viewed through the lens (like a magnifying glass). It occurs when 'di' is negative.

Common Scenarios and Special Cases

  • Object at Infinity
  • Object at the Focal Point
  • Object at Twice the Focal Length (2F)
Certain object placements lead to interesting and predictable outcomes.
Object at Infinity (do → ∞)
For a very distant object, the light rays arriving at the lens are essentially parallel. The lens focuses them at its focal point. In the equation, 1/∞ is 0, so the formula becomes 1/f = 1/di, meaning the image forms at the focal length (di = f).
Object at the Focal Point (do = f)
If you place an object at the focal point of a converging lens, the exiting rays become parallel and never converge to form an image. The image distance is infinite. This is why the calculator shows an error in this case, as 1/di = 1/f - 1/f = 0, leading to a division by zero to find 'di'.
Object at 2F
Placing an object at twice the focal length (do = 2f) from a converging lens results in a real, inverted image of the same size forming at the same distance on the other side (di = 2f).