Topic 1 Lenses and  Refracting Telescopes

In the exam you are expected to know about:

• Principal focus, focal length of converging lens

• power = 1/f

• Formation of images by a converging lens

• Ray diagrams

• Lens formula: 1/f = 1/u + 1/v

• Astronomical telescope consisting of two converging lenses

• Ray diagram to show the image formation in normal adjustment

• Angular magnification in normal adjustment

• M =      angle subtended by image at eye    

           angle subtended by object at  unaided eye    

   

• Focal lengths of the lenses

• M = fo/fe

Convex Lens

Lenses work by refracting light at a glass-air boundary.  Although refraction occurs at the boundary, we will treat all lenses as bending the rays at the lens axis.

The lens in the eye is a convex or converging lens.  This means that the lens makes rays of light come together, or converge.

The rays parallel to the principal axis are converged onto the principal focus.  The focal length is the distance between the lens axis and the principal focus (strictly speaking, the focal plane).

Thicker lenses bend light more, and are therefore described as more powerful.  Powerful lenses have short focal lengths.  The power of a lens is measured in dioptres (D) and is given by the formula:

Power =          1               

               focal length (m)

Question 1 The power a lens is +0.2 D  What is the focal length in metres?  ANSWER

The principal focus of a convex lens is called real.  The images made by convex lenses are in most cases real.  This means that the image can be projected onto a screen.  We will see later how images are made with ray diagrams.

Ray Diagrams

We can determine where an image lies in relation to the objects by using a ray diagram.  We can do this by using two simple rules:

• Draw a ray from the top of the image parallel to the principal axis.  This ray bends at the lens axis and goes through the principal focus.

• Draw a ray from the top of the lens through the centre of the lens.

Where the two rays meet, that is where the image is found.  The diagrams shows how we do a ray diagram step-by-step:

Step 1  Draw the ray parallel to the principal axis.

Step 2 Draw the refracted ray so that it passes through the principal focus.

Step 3 Draw a ray from the top of the object through the middle of the lens.  This ray is undeviated.

Step 4 Where the rays meet, that is where the image is.

It is a good idea to draw your ray diagrams on graph paper as the following ray diagrams are.  Be careful with your drawing; a small change in the angle of the undeviated ray can lead to quite a big change in the final position of the image.  And PLEASE... Be a good chap and use a sharp pencil.

Click here to look at a ray diagram done on graph paper

This diagram shows where an object is at a distance of greater than twice the focal length.  The image is inverted (upside down), real, and diminished (smaller).

Question 2 What is the image like if the object is at 2F?  ANSWER

Question 3 What is the image like if the object is between 2F and F?  ANSWER

Question 4 What is the image like if the object is at F?  ANSWER

Question 5 What is the image like if the object is less than F?  ANSWER

 The Lens Formula

Lens diagrams have the main disadvantage that there is uncertainty in precisely where the image is.  Therefore the use of the lens formula is better.  The lens formula is:

                                     [f - focal length; u - object distance; v - image distance]

Example

It does not matter if you work in cm, as long as you are consistent.  However if you are going to use dioptres you must work in metres.

The magnification is worked out using this simple formula:

Since v is in metres, and u is in metres, M has no units.

The convention for the equation is that real is positive.  For a concave lens, the focal length is negative, because the principal focus is virtual.  If the image position gives a negative value, then the image is virtual. 

Question 6  Find the position and size of a pound coin, 2.2 cm in diameter placed 20 cm from a converging lens of focal length 40 cm  ANSWER

The Telescope

In this section we will look at the refracting telescope works by bending light with lenses.  The objective lens makes a small real image of the object while the eyepiece lens acts as a magnifying glass.  The following factors are important in making a good quality instrument:

• Lens quality: bad lenses, bad image

• lens diameter: brightness and detail observed depend on the area.  A 12 cm lens can resolve detail nine times better than a 4 cm lens.

• Angular magnification.

The diagram shows the telescope when it is set up normally (normal adjustment).

Light from object A (blue lines) meets at the principal focus of the objective lens.  It then spreads out until it meets the eyepiece.  The eyepiece is set at the focal length away from its principal focus.  Parallel rays emerge from the eyepiece.

At the same time parallel rays from object B arrives at the objective at a small angle a to the axis.  The light is focused onto the focal plane.  It then passes through the eyepiece to emerge as parallel rays.  The angle of these parallel rays is b to the parallel rays from A.

The angular magnification can be worked out by the simple formula:

where a and b are small angles in radians.  

• The angle a is the angle subtended by the object to the unaided eye.

• The angle b is the angle subtended by the image to the eye.

The magnification can also be shown to be related to the focal lengths of the lenses by:

Question 7  In a telescope the eyepiece has a focal length of 2 cm and the objective has a focal length of 220 cm.  What is the magnification?  If the moon subtends an angle of 8.8 x 10^-3 rad to the naked eye, what would the angle be for the image of the moon observed through the telescope?  ANSWER

The telescope shown above makes an inverted image.  To make the image the right way up we need to put in a third lens at the principal focus of the objective lens, but we won't go into that at this point.

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