Lenses & Ray Diagrams for Telescopes

Astrophysics | AQA A-Level Physics

Key Definitions

Converging (convex) lens: A lens that brings parallel rays of light to a focus at the focal point on the principal axis.

Focal length: The distance from the centre of the lens to the focal point. A shorter focal length means a more powerful lens.

Real image: An image formed where light rays actually converge. It is always inverted and can be projected onto a screen.

Virtual image: An image formed where light rays appear to diverge from. It is always upright and cannot be projected onto a screen.

Converging lenses

A converging lensAlso called a convex lens. It is thicker in the middle than at the edges and bends parallel rays inward to meet at the focal point. brings parallel rays of light to a focus along the principal axis at the focal point.
The key part is that the more curved (thicker) the lens, the shorter the focal length, and the more powerful the lens.
In the Astrophysics module, only converging lenses are considered.

Three rules for constructing ray diagrams

We assume lenses are very thin, which simplifies the diagrams by reducing the amount the incident rays refract.
Rule 1: Rays passing through the principal axis will pass through the optical centre of the lens undeviated.
Rule 2: Rays that are parallel to the principal axis will be refracted and pass through the focal point $f$.
Rule 3: Rays passing through the focal point $f$ will emerge parallel to the principal axis.

Image formation by a converging lens

Images formed by lenses can be described by their nature (real or virtual), orientation (inverted or upright), and size (magnified, diminished, or same size).
Object distance greater than $2f$: the image forms between $f$ and $2f$. It is real, inverted, and diminished.
Object distance equal to $2f$: the image forms at $2f$. It is real, inverted, and the same size.
Object distance between $f$ and $2f$: the image forms beyond $2f$. It is real, inverted, and magnified.
Object distance less than $f$ (a magnifying glass): the image forms on the same side as the object. It is virtual, upright, and magnified.

The lens equation

The lens equation relates the focal length of a lens to the distances between the lens, the image, and the object:

$$\frac{1}{f} = \frac{1}{u} + \frac{1}{v}$$

Where $f$ = focal length (m), $u$ = object distance (m), $v$ = image distance (m).
Crucially, for a real image, $v$ is positive. For a virtual image, $v$ is negative.

Magnification

Magnification $M$ is defined as the ratio of image height to object height:

$$M = \frac{h_i}{h_o}$$

Using similar triangles from the ray diagram, magnification can also be written as:

$$M = \frac{v}{u}$$

Since magnification is a ratio, it has no units.

Common Mistake

Students often forget the sign convention for the lens equation. When the image is virtual (object closer than $f$), $v$ is negative. If your calculated $v$ comes out negative, that tells you the image is virtual and on the same side as the object. Do not just ignore the sign.