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Thin Lens Equation Calculator

Created by Dominik Czernia, PhD
Reviewed by Bogna Szyk, Jack Bowater and Adena Benn
Last updated: Jan 18, 2024


The thin lens equation calculator will help you to analyze the optical properties of the simple lens. Keep reading to learn about the thin lens equation and understand how a lens can magnify the image of an object. Everything is about light, so be sure to check out the principles of light refraction too!

💡 If you need to calculate the angle of refraction, then head over to our Snell's law calculator.

Thin lens equation

If we place the object near the lens, we will get its image somewhere. The position, orientation, and size of this image depend on two things: the focal length of the lens (which is specific for the particular lens) and the position of the original object. We can predict what we will see with the following thin lens equation:

1/x + 1/y = 1/f,

where:

  • x – Distance between the object and the center of the lens;
  • y – Distance between the image and the center of the lens; and
  • f – Focal length of the lens expressed in length units.

There are two basic types of lenses. We can distinguish converging lenses, which have focal length f > 0, and diverging lenses for which focal length f < 0. It should also be noted that when the image distance is positive y > 0, then the image appears on the other side of the lens, and we call it a real image. On the other hand, when y < 0, then the image appears on the same side of the lens as the object, and we call it a virtual image.

If you're interested in computing interference in thin optical coatings, use our thin film optical coating calculator.

Magnification lens equation

In the advanced mode, you can compute the magnification of the created image too (see the mirror equation calculator). It can be easily estimated if we know the distance of object x and the distance of image y:

M = |y|/x

Remember that magnification must always be a positive number. That's why we have taken the absolute value of y, which generally may be both positive and negative.

Images in the converging lens

Let us consider five different situations for a converging lens (f > 0). You can check it with our thin lens equation calculator!

  • for x > 2f image is real (y > 0) and diminished (M < 1);
  • for x = 2f image is real (y > 0) and of the same size as the object (M = 1);
  • for 2f > x > f image is real (y > 0) and magnified (M > 1);
  • for x = f image doesn't appears (y -> Infinity);
  • for x < f image is virtual (y < 0) and magnified (M > 1).

We encourage you to check similar cases for the diverging lens, which has a negative focal length f < 0 with our calculator!

FAQ

How do I calculate the focal length of a lens using the lens formula?

To calculate the focal length of a lens using the lens formula, follow these instructions:

  1. Determine the distance of the object from the lens, i.e., u, and take the reciprocal of it.

  2. Find out the distance between the image and the lens, i.e., v, and take the reciprocal of it.

  3. Add the value obtained in Step 1 to that obtained in Step 2.

  4. Take the reciprocal of the value from Step 3, and you will get the focal length of the lens.

How do I find the magnification of a lens?

The magnification of a lens is the ratio of the size of the image to the size of the object. Hence, to find the magnification of a lens, take the ratio of the two. You can also calculate magnification by taking the ratio of the image-lens distance to the object-lens distance.

Is the thin lens formula different for different lenses?

No, the thin lens formula is not different for different lenses. The thin lens formula is the same for both convex and concave lenses.

What is the formula for the power of a lens?

The power (P) of a lens is the reciprocal of its focal length (f). Hence we can express the formula for the power of a lens as:

P = 1/f

Dominik Czernia, PhD
Object distance
in
Image distance
in
Focal length
in
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