Astrophysics Tutorial 1 -  Lenses and  Refracting Telescopes

Contents

Convex Lens

Ray Diagrams

The Lens Formula

The Telescope

Brightness of the Image

 

Introduction

The night skies have fascinated people for many thousands of years.  The earliest scientific observations were of the movements of stars and planets.  The term planet comes from the Greek word "planein" - to wander.  They were wandering stars.  The wandering movement is now explained by the idea of the planets orbiting the sun.  Our fascination with the stars and planets has not diminished even though we know a lot more about the Universe now than we did before.

 

Astrophysics is about the application of physics principles to explain how the Universe works.  Here are some terms that astronomers and astrophysicists use to describe bodies in the universe:

New discoveries are being made all the time.  Current theories about the way the Universe works are being constantly tested and update.  Some are being re-written.

 

  Do NOT use the term "astrology" to describe any aspect of astronomy or astrophysics. 

  Astrology is a system of superstitions that claims to be able to predict events in the future by the movements of stars.  Columns are found in the popular tabloid newspapers that claim that "today is a good day for financial speculation".

The publicity for A-level Physics for the college I used to work for was marred by the inclusion of "astrology" as an option.  I complained several times to the marketing department.  Despite the involvement of the Principal, the howler remained. 

 

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).  The focal length is given the code F.

 

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 powers (in dioptres) of several thin lenses add up so that:

 

Ptot = P1 + P2 + P3 + ...

 

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.

 

 

Principal focus, not principle focus.  (Principal means main, or chief; principle means rule.)

 

 

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:

Where the two rays meet, that is where the image is found.  In this example, we have placed the object between F and 2F.  The diagrams show 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.  The image is inverted (upside down), real, and magnified (bigger).

 

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 (m); u - object distance (m); v - image distance (m)]

 

 

Worked Example

An object of height 1.60 cm is placed 50 cm from a converging (convex) lens of focal length 10 cm.  What is the position of the image?

Answer

Substitute:

 

 

v = (0.080 cm-1)-1 = 12.5 cm

 

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.

 

Worked example

What is the magnification in the example above?  What is the size of the image?

Answer

M = 12.5 cm 50 cm = 0.25

 

Image height = 1.60 cm 0.25 = 0.40 cm = 4.0 mm

 

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 an old pound coin, 2.2 cm in diameter placed 20 cm from a converging lens of focal length 40 cm.

What are the properties of the image? 

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:

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

 

 

 

Click HERE for a presentation on how to draw the ray diagram.

 

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 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.

 

A big problem with the refracting telescope is chromatic aberration.  Different colours refract by different amounts.  You can see this in poor quality telescopes, in the form of spectra around the edges of the lenses.

 

 

Brightness of the Image

When you look at a star with a telescope, the star seems brighter.  This is because the area of the telescope lens is much bigger than the area of your pupil.  Brightness is due to the intensity of the light, or power per square metre.  The brightness goes up proportionally with the area.  If your eye is 1 cm in diameter, a telescope with a lens 5 cm in diameter will give an image that is 25 times brighter.  If the lens is 15 cm in diameter, the intensity will be 152 = 225 times more.