Thin Lens Formula
Reference for the thin lens equation 1/f = 1/do + 1/di.
Covers magnification, real vs virtual images, and converging/diverging lens uses.
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The Formula
1/f = 1/dₒ + 1/dᵢ
The thin lens formula connects the focal length of a lens to the distances of the object and its image. It works for both converging (convex) and diverging (concave) lenses.
Variables
| Symbol | Meaning |
|---|---|
| f | Focal length of the lens (positive for convex, negative for concave) |
| dₒ | Object distance — distance from the object to the lens |
| dᵢ | Image distance — distance from the lens to the image |
Example 1
An object is 30 cm from a lens with focal length 10 cm. Where is the image?
1/10 = 1/30 + 1/dᵢ
1/dᵢ = 1/10 - 1/30 = 3/30 - 1/30 = 2/30
dᵢ = 15 cm (real image, on the opposite side of the lens)
Example 2
An object is 20 cm from a diverging lens with f = -15 cm. Where is the image?
1/(-15) = 1/20 + 1/dᵢ
1/dᵢ = -1/15 - 1/20 = -4/60 - 3/60 = -7/60
dᵢ = -8.57 cm (virtual image, same side as the object)
When to Use It
Use the thin lens formula when:
- Designing cameras, telescopes, or microscopes
- Finding where an image forms for a given lens and object position
- Calculating the focal length needed for a specific magnification
- Understanding how eyeglasses correct vision
Key Notes
- The thin lens equation: 1/f = 1/d_o + 1/d_i: f is the focal length, d_o is the object distance from the lens, and d_i is the image distance on the other side. All distances are measured from the center of the lens.
- Sign conventions vary: The real-is-positive convention treats distances in the direction of light travel as positive. The Cartesian convention uses the direction of incident light as positive. Always state which convention you're using.
- Converging vs diverging lenses: Converging (convex) lenses have a positive focal length and can form real, inverted images. Diverging (concave) lenses have a negative focal length and always form virtual, upright images.
- Magnification: m = −d_i / d_o: A negative magnification means the image is inverted. |m| > 1 means the image is larger than the object; |m| < 1 means it is smaller.
- Thin lens limitation: The formula assumes the lens thickness is negligible compared to d_o and d_i. For thick lenses, compound optical systems, or fisheye lenses, matrix (ray transfer) methods are needed.