Ray diagram of a convex lens when the object is between 2F₁ and F₁
The image is beyond 2F₂: real and inverted, enlarged.
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 = -15 cm: the image is at v = +30 cm, real, inverted, enlarged.
Show the working
- f = +10 cm, u = -15 cm (the object is in front, so u is negative), h = +2 cm.
- 1/v = 1/f + 1/u = 1/(+10) + 1/(-15) = 5/150 = 1/30
- v = +30 cm: positive, so the image is behind the lens and real.
- m = v/u = (+30)/(-15) = -2: negative, so inverted, and enlarged.
- h′ = m × h = -2 × +2 = -4 cm
- Power P = 1/f = 1/(+0.1 m) = +10 D
Take f = 10 cm and the object at 15 cm
- f = +10 cm, u = -15 cm (the object is in front, so u is negative), h = +2 cm.
- 1/v = 1/f + 1/u = 1/(+10) + 1/(-15) = 5/150 = 1/30
- v = +30 cm: positive, so the image is behind the lens and real.
- m = v/u = (+30)/(-15) = -2: negative, so inverted, and enlarged.
- h′ = m × h = -2 × +2 = -4 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 projector: a slide just beyond F₁ gives a large, real image on a distant screen.
Quick answers
Where is the image when the object is between 2F₁ and F₁ of a convex lens?
The image is beyond 2F₂: real and inverted, enlarged. With f = 10 cm and the object at 15 cm, v = +30 cm and m = -2.
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 projector: a slide just beyond F₁ gives a large, real image on a distant screen.