The  prism:  

The Apex angle of a prism is Φ. We derive in class the above prism equation.

The light is refracted when entering and when leaving the prism. The minimum angle of deviation is δmin. It occurs when the incidence angle θ1 is such that the refracted ray inside makes the same angle with the normal to the two prism faces as shown in the above Figure.

Knowing apex angle Φ and measuring the deviation angle δmin , one can determine the refractive index n.

(see your Lab 1). A hollow prism filled with liquid can be used to determine n of the liquid.

n =sin ! +"min

2!

"#

$

%&

sin !2!

"#$

%&

Image  forma0on  for  mirrors:  

Object O is at object distance o.

Reflected rays appear to originate at point I behind the mirror. I is called the image and i image distance.

Image  forma0on  -­‐  geomtrically:  Up  to  four  rays  can  be  used  to  construct  the  image  geometrically  –  they  are  called  the  principal  rays.  There  are  of  course  infinitely  many  rays  that  one  could  use  –  all  one  has  to  do  is  to  apply  the  law  of  reflec<on  correctly.  But  the  four  principal  rays  go  through  special  points  on  the  mirror  or  op<cal  axis.    

For  concave  mirrors  they  are:  

Image  forma0on  -­‐  geometrically:  Up  to  four  rays  can  be  used  to  construct  the  image  geometrically  –  they  are  called  the  principal  rays.  The  principal  rays  are  the  same  as  for  concave  mirrors:  parallel  ray,  ray  through  centre  (vertex),  through  focal  point  (F)  and  through  C  (middle  of  circle).  

For  convex  mirrors  they  are:  

The  lens  equa0on:  

1o+1i=2R=1f

In class we derive the mirror equation:

The equation connects object distance o, image distance i and focal length f. The focal length f is the distance from the mirror on the optical axis where the image is formed for parallel rays (o= ) coming in: 1o+1i=1!+1i=1i=1f!"! f = i

!

The magnification M

is defined as

Note sign convention for M, i and o !

M =imageheightobject !height

=h 'h= !

io!

For spherical mirrors this happens at f=R/2.

The  near-­‐axis  or  pararaxial  approxima0on  h,  δ  <<  i,  o,  R:  

The mirror equation we derived is derived strictly using simple geometric identities and uses an important simplification h<<R (more specifically h,δ << o, i, R). This means that the distance of the point where the optic is struck by the ray to the optical axis is small compared to the curvature radius of the mirror. This is referred to as the paraxial or near-axis approximation. This also means that α, β, γ << 1 rad and often requires that the apertures of mirrors (or lenses) are small.

If rays are impinging on the optic further away from the optical axis, nothing changes in the physics – just the geometric approximations do not hold any longer and therefore the equations become much more complex.

Reflective optics - mirrors The sign conventions are important to memorize and to apply correctly.

The sign conventions for mirrors:

•  > 0 : real object (in front of lens)

•  o < 0 : virtual object (behind lens)

i > 0 : real image (behind lens - note that real image for mirrors is in front of mirror)

i < 0 : virtual image (in front of lens – behind for mirror!)

M > 0 : image upright (M>1: magnified)

M < 0 : image inverted (0<M<1: demagnified)

f = R/2 > 0 : converging lens with convex surface (converging mirror with concave surface)

f = R/2 < 0 : diverging lens with concave surface (diverging mirror with convex surface)

Example:  

(1) Determine image distance and magnification for

(a) o = 15 cm, f = 10 cm

(b)  o = 20 cm, f = - 15 cm

(2) Draw the rays for each case.