Ray diagram of a convex lens when the object is at 2F₁
The image is at 2F₂: real and inverted, same size.
The ray diagram
Drag the object or type values. Signs follow the New Cartesian convention: distances are measured from the pole or optical centre, along the incident light is positive, against it negative, and heights above the axis are positive. So u is negative for a real object.
Convex lens of focal length +10 cm. An object +2 cm tall at u = -20 cm: the image is at v = +20 cm, real, inverted, same size.
Show the working
- f = +10 cm, u = -20 cm (the object is in front, so u is negative), h = +2 cm.
- 1/v = 1/f + 1/u = 1/(+10) + 1/(-20) = 10/200 = 1/20
- v = +20 cm: positive, so the image is behind the lens and real.
- m = v/u = (+20)/(-20) = -1: negative, so inverted, and the same size.
- h′ = m × h = -1 × +2 = -2 cm
- Power P = 1/f = 1/(+0.1 m) = +10 D
Take f = 10 cm and the object at 20 cm
- f = +10 cm, u = -20 cm (the object is in front, so u is negative), h = +2 cm.
- 1/v = 1/f + 1/u = 1/(+10) + 1/(-20) = 10/200 = 1/20
- v = +20 cm: positive, so the image is behind the lens and real.
- m = v/u = (+20)/(-20) = -1: negative, so inverted, and the same size.
- h′ = m × h = -1 × +2 = -2 cm
- Power P = 1/f = 1/(+0.1 m) = +10 D
Image formation by a convex lens
| Object | Image | Size | Nature |
|---|---|---|---|
| At infinity | At the focus F₂ | Highly diminished, point-sized | Real and inverted |
| Beyond 2F₁ | Between F₂ and 2F₂ | Diminished | Real and inverted |
| At 2F₁ | At 2F₂ | Same size | Real and inverted |
| Between F₁ and 2F₁ | Beyond 2F₂ | Enlarged | Real and inverted |
| At the focus F₁ | At infinity | Infinitely large or highly enlarged | Real and inverted |
| Between F₁ and the optical centre O | On the same side of the lens as the object | Enlarged | Virtual and erect |
The rays to use
- A ray parallel to the principal axis passes through the focus F₂ on the other side after refraction.
- A ray through the optical centre O goes straight on without bending.
- A ray through the focus F₁ comes out parallel to the principal axis.
- Where the refracted rays meet is the top of the image.
In real life
- A same-size copy: the image is as big as the object, which is how the focal length is checked on an optical bench.
Quick answers
Where is the image when the object is at 2F₁ of a convex lens?
The image is at 2F₂: real and inverted, same size. With f = 10 cm and the object at 20 cm, v = +20 cm and m = -1.
Is this image real or virtual?
Real: the rays actually meet there, so the image can be caught on a screen. Real images formed by one mirror or lens are inverted.
Where is this used?
A same-size copy: the image is as big as the object, which is how the focal length is checked on an optical bench.