Option G: Electromagnetic Waves G2: Optical Instruments.

Option G: Electromagnetic Waves

G2: Optical Instruments

Lenses

Different types of lenses:

A convex lens converges light rays to a point... the focal point (or principle focus).

A concave lens diverges rays so that they appear to come from a common focus...

Real and Virtual Images

If the rays reaching your eye come from the image itself, it is a real image.

If they only appear to come from a point but never pass through that point, it is a virtual image.

Technical Terms

Principle Axis: A line perpendicular to the plane of the lens, passing through the optical centre of the lens.

Focal Point (or Principle Focus): Point on the principle axis at which refracted rays meet if rays parallel to the principle axis are incident upon the lens.

Focal Length: Distance from the centre of the lens to the principle focus.

Linear Magnification: Ratio of the height of the image to the height of the object

Drawing Ray Diagrams

We are only interested in convex lenses. If you know the position of the object then draw two/three rays from the top of the object...

- one parallel to the principle axis then refracting and passing through the principle focus.

- one through the optical centre of the lens. This will not be deviated.

- One passing through the near focus then refracting parallel to the principle axis.

The three rays cross at the top of the image. The object “feet” will always be on the principle axis.

E.g.

I

FO C

Exercise

Sketch ray diagrams for a convex lens where the object is at distances u from the optical centre...

a. u > 2F

b. u = 2F

c. 2F > u > F

d. u < F

In each case state whether the nature of the image is...

• real or virtual

• bigger or smaller than the object

• further or closer to the lens than the object

• upright or inverted

Images from www.antonine-education.co.uk

Subtitle

Text

Object at Infinity

In this special case, rays approaching the lens from a point on the object can be considered to be parallel.

Nature of image: Real; inverted; at F; smaller than object.

IF

O C

From infinity

Linear Magnification, m

However the triangles formed by the image and object are similar. Thus...

I

O

θθ

u v

u = object distancev = image distance

m = height of imageheight of object

m = - v u

Note: as we will see later, v can be positive or negative depending on whether it is real or virtual.

Power of a Lens

Opticians refer to the power of a lens instead of the focal length.

The power of a convex lens is given by...

... where focal length is in metres and power is in dioptre (D).

One dioptre (1D) is the power of a lens with focal length one metre.

Power = 1focal length

The Thin Lens Formula

We can determine the position of an image in two ways...

1. By constructing scale ray diagrams (estimate)

2. By using the thin lens formula:

1 1 1f u v

= +f = focal lengthu = object distancev = image distance

Sign Convention

When using the thin lens formula we use the convention...

“ Real is positive, virtual is negative”

i.e. If the image distance v is negative, it must be a virtual image.

Q1. A lens has a focal length of 20cm. An object is placed 40cm from the lens, determine

a. the power of the lens

b. the image distance

c. the linear magnification of the image

d. whether the image is real or virtual

Q2. A lens has a focal length of 30cm. An object is placed 20cm in front of the lens. Determine

a. the power of the lens

b. the image distance

c. the linear magnification of the image

d. whether the image is real or virtual

End of G2 part A

Option G2, part 2

Microscope, Telescope,

& Aberration

The Human Eye

The lens of your eye can be pulled or squashed to change how thick it is and thus enable it to focus on objects at different distances.

Near point and far point

Near point: The closest distance an object can be to your eye and still be focused on the retina. Usually about 25cm.

Far point: The furthest distance an object can be from your eye and still be focused on the retina. Usually infinity.

Short sightedness (Myopia)

For a normal eye...

But if the eyeball is too long or the lens too powerful...

So the short sighted person’s far point is nearer than infinity...

Q. How can this be corrected?

The Magnifying Glass (Simple Microscope)

Normally the largest we can see an object is when it is at the near point. However we can use a lens in such a way as to put the image at the near point, making it seem bigger.

F’

hi

ho θi

D

Near point

D = distance to near point = - 25 cm

F

Without a magnifying glass the closest the object can be focused is at the near point. It will appear normal size:

θo

D

ho

Near point

Angular Magnification, M

This is given by

θi = angle subtended at the eye by the image formed when using the lens

θo = angle subtended at the unaided eye i.e. when not using the lens.

M = θi

θo

From the previous two diagrams and using the small angle approximation, we can write the following expressions for the angles θo and θi:

so...

So in this case angular magnification and linear magnification are the same.

θo = ho

Dθi = hi

D

M = θi = hi D θo D ho

M = m = hi

ho

Magnifying Glass / Image at infinity

(Use the ‘Lenses’ simulation to work out the position of the object when the upright image is largest i.e. at infinity)

For the image to be at infinity, the object must be at the focal point.

In this case... Also...

so...

θi = ho

fθo = ho

D

M = θi = ho D θo f ho

M = m = D f

Where D = 25cm

Magnifying Glass / Image at Near Point

It can be shown that

M = m = D + 1 f

Where D = 25cm

The Compound Microscope

- The objective lens forms a real image of the object. This acts as the object for the eyepiece lens.

- The eyepiece lens forms a virtual image of this real image.

- The greatest angular magnification is formed when the virtual image is at the near point of your eye.

hi

Near point Fo

ho Fo

D

The Astronomical Telescope

- Again, the objective lens produces a real image which acts as the object for the eyepiece lens. A virtual image of this is then seen.

- The focal planes of the two lens coincide although the objective has a much longer focal length than the eyepiece.

AB is the common focal plane

M = fo

fe

Here the angular magnification is given by...

Spherical Aberration

This problem is due to the spherical shape of lens surfaces.

It occurs because rays that are incident upon the lens far from the principle axis have a different focal length to those incident close to the principle axis:

An idealised convex lens

Convex lens showing spherical aberration.

The effect can be reduced by reducing the aperture of the lens, i.e. reducing the diameter of the hole in front of the lens. This is called ‘stopping down’.

In effect this cuts out the shorter focal length rays coming from points furthest from the principle axis.

Spherical aberration results in uneven focus from the middle to the outside of an image...

Stopping down can cause barrel distortion at the image corners:

Out of focus

In focus

Chromatic Aberration

Each wavelength of light refracts different amounts. This results in each part of the visible spectrum having a slightly different focal length for a particular lens:

As a result the edges of an image may appear coloured:

Chromatic aberrations can be eliminated for two colours (and reduced for all) by an achromatic doublet. This consists of a diverging lens stuck to the converging lens:

The two lenses create equal but opposite amounts of dispersion so the two colours recombine at the focal point.

Note: other colours are still focused at different points but the difference has been decreased (see the green ray below):