Lenses, Mirrors & the Human Eye

Concepts • Concave and convex mirrors

– Focus

• Converging and diverging lenses– Lens equation

• Eye as an optical instrument• Far and near points• Corrective lenses

Lenses• Convex lens bulges

out –converges light• Concave lens caves

in –diverges light

Focus• Light goes through – focal

points on both sides – F and F’– Always a question which focal

point to choose when ray tracing

• Converging lens:– Parallel beam of light is

converged in 1 point – focal point F

– Real focus: f>0– Key for the focal point choice:

Rays must bend in

• Diverging lens:– Parallel beam of light seems to

be coming out of 1 point F– Virtual focus: f<0– Key for the focal point choice:

Rays must bend out

Ray tracing for converging lens

3 Easy rays:1. Parallel

through focus F

2. Through focus F’ parallel (reversible rays)

3. Through the center itself

LENS QUESTIONS

LENSAPPLET

) Object between F and lens

                                        

 

VirtualErectLarger than objectBehind the object on the same side of the lens

Image formed by a diverging lens

e) Object at F

                                        

 

Characteristics of the image regardless of object postionVirtualErectSmaller than objectBetween object and lens

Diverging lens

• Same rules, but remember to diverge (bend out)• Parallel projection through focus F• Projection through F’ parallel• Through the center goes through

Lens equation

• d0 – distance to object

• di – distance to image

• f –focus

f1

d1

d1

io

f1

P

• P – power of lens, in Dioptry (D=1/m)

• f must be in m

Sign convention for lenses and mirrors

d0>0

h0>0

di>0 – real image

Opposite side from O

di<0 - virtual image

Same side with O

hi>0 – upright image

hi<0 - inverted image

f>0 – concave mirror

f<0 – convex mirror

f>0 – converging lens

f<0 – diverging lens

o

i

o

i

dd

hh

m •hi>0di<0 – upright image is always virtual•hi<0di>0 – inverted image is always real

• Converging lens, concave mirror

• d0>2f – (real, inverted), smaller• 2f>d0>f – (real, inverted), larger• d0<f – (virtual, upright), larger

• Diverging lens, convex mirror

• Image is always

(virtual, upright), smaller.

Images in lenses and mirrors

System of lenses

• Image of the 1st lens of object for the 2nd lens.

Spherical mirrors

• Convex mirror bulges out – diverges light• Concave mirror caves in – converges light

Focus

• Parallel beam of light (e.g. from a very distant object) is converged in 1 point – focal point F

• Distance from the mirror to F is called focal distance, or focus

f =r/2

Ray tracing3 Easy rays:

1. Parallel through focus

2. Through focus parallel (reversible rays)

3. Through the center of curvature C itself

Magnification

• h0 – object height– h0>0 - always

• hi – image height– hi>0 – upright image– hi<0 – inverted image

• m=hi/h0 - magnificationo

i

o

i

d

d

h

hm

|m|>1 –image larger than object|m|<1 –image smaller than object

Mirror equation

• d0 – distance to object– d0>0 - always

• di – distance to image– di>0 – real image– di<0 – virtual image

fdd io

111

Convex mirror

• Virtual focus – parallel beam focuses behind the mirror:

f<0• Same rules for ray

tracing.

Sign convention for mirrors

d0>0

h0>0

di>0 – real image di<0 - virtual image

hi>0 – upright image hi<0 - inverted image

f>0 – concave mirror f<0 – convex mirror

o

i

o

i

d

d

h

hm •hi>0di<0 – upright image is always virtual

•hi<0di>0 – inverted image is always real

Images in curved mirrors

• Concave mirror

• d0>r – (real, inverted), smaller

• r>d0>f – (real, inverted), larger

• d0<f – (virtual, upright), larger

• Convex mirror• Image is always

(virtual, upright), smaller.

Eye as an optical instrument• Eye is a converging lens• Ciliary muscles are used to

adjust the focal distance.– f is variable

• Image is projected on retina – back plane.– di stays constant

• Image is real (light excites the nerve endings on retina) inverted (we see things upside-down) – di>0, hi<0

• Optic nerves send ~30 images per second to brain for analysis.

Far and near points for normal eye

• Relaxed normal eye is focused on objects at infinity – far point

f0=eye diameter =~2.0 cm

• Near point – the closest distance at which the eye can focus - for normal eye is ~25cm. Adjusted focus:

f1=1.85 cm

Corrective lenses• Nearsighted eye

– far point<infinity– diverging lens f<0 P<0

• Farsighted eye – near point > 25 cm – converging lens f>0 P>0

• Lens+eye = system of lenses

• Nearsighted eye Far point = 17 cm di =-0.17m

Need to correct far point

Object at “normal far point” =infinity

• Farsighted eye Near point = 70 cm di =-0.70m

Need to correct near point

Object at “normal near point” =25cm

• Corrective lenses create virtual, upright image (di<0 !) at the point where the eye can comfortably see

od mdo 25.0

• Converging lens - for farsighted

• d0>2f – (real, inverted), smaller• 2f>d0>f – (real, inverted), larger• d0<f – (virtual, upright), larger

• Diverging lens - for nearsighted

• Image is always (virtual, upright), smaller.

Images in lenses

Image in corrective lenses is always virtual and upright

di<0 and hi>0

Corrective lenses

• Nearsighted eye

Far point = 17cm

Near point = 12 cm

P-?

new near point -?

Diverging lens projects infinity to 17 cm from the eye

Real and virtual imageMirrors:I and O – same side

I and O –opposite sides IO M

IO L

IO M

IO L

Lenses:I and O –opposite sides

I and O – same side

Real, invertedlight goes through

Virtual, uprightlight does not go through

Real, invertedlight goes through

Virtual, uprightlight does not go through