Lecture 16 More on Mirrors and Lenses Last time we drew the following ray diagram for spherical mirrors h f object R h q p We found that the magnification of the image was q h M p h The yellow triangles on the previous slide are similar so we know that h h p R R q h q R q h p R p pq pR qR pq 1 1 1 1 R q p R This leads to the mirror equation 1 1 2 p q R For the spherical mirror it happens that f R 2 The general form of the mirror equation is 1 1 1 p q f Sign Conventions Clearly p is always a positive number As I drew the diagram q was also positive And the image was real not virtual like the one from a flat mirror But in some cases the mirror equation will tell us that q is negative This means the image is behind the mirror and is therefore virtual This happens when p f Note that the image will be upright in this case The equation also holds when R is negative This means the mirror is convex rather than concave Example Let s say you want a spherical mirror to form an upright image of your face enlarged by a factor of 5 when your face is 10cm from the mirror What should the radius of the mirror be We know that p 10cm and want q M 5 p q 50cm Note that the image is virtual We then use the mirror equation to find R 1 1 2 p q R 1 1 2 10cm 50cm R 40cm 2 2 500cm R 1000cm 2 25cm R 40cm Note that R is positive so the mirror must be concave Lenses Last time we saw how reflection from mirrors can form images of an object Refraction can also be used to form images Let s assume an object is in material with index of refraction n1 Some rays then enter material with larger index of refraction n2 Boundary between two materials is spherical Object n1 R n2 Image p q As with the spherical mirror we ll only consider paraxial rays rays that are almost parallel to the principal axis As shown on the previous page all paraxial rays starting from the object meet at the image position We can consider just one off axis ray to work out the geometry R n1 1 Object d 2 R p q n2 Image From Snell s Law we know that n1 sin 1 n2 sin 2 But since we ve made the assumption that all angles are small this becomes n1 1 n2 2 We also see that 1 90o 90o 180 o 1 2 180o 180 o 2 We can now substitute in for Snell s Law n1 n2 n1 n2 n2 n1 Looking at the triangles on our diagram and keeping in mind that all the angles are small we find that tan d d d tan tan p R q Plugging into the Snell s Law equation we find d d d n1 n2 n2 n1 p q R n1 n2 n2 n1 p q R This expression looks similar to the one for the spherical mirror Except that now the result depends both on the geometry and on properties of the material the indices of refraction This is because light is now traveling through the medium rather than reflecting from the surface As with the spherical mirror there are sign conventions One big difference images formed behind the surface are real while those in front are virtual Exactly the opposite of the case for mirrors Note Even though we assumed n2 n1 in deriving our equation it holds even if n2 n1 Special Case A Flat Surface If we take the limit of R goes to infinity we end up with a flat surface Our equation tells us that n1 n2 0 p q n2 q p n1 This means that the image is always on the same side of the surface as the real object It also explains why things appear closer than they really are when you look into water Thin Lenses We now move from a single refracting surface to the case where there are two such surfaces Example glass with two curved sides The key to understanding the formation of images by lenses is to realize that the image formed by the first surface the light crosses acts as the object for the second surface R2 R1 Object p2 Image formed by first surface p1 t q1 For now we ll assume that the glass is in air so the index of refraction outside is 1 Then we know that 1 n n 1 p1 q1 R1 The second surface also forms an image with n 1 1 n p2 q2 R2 Note that in this case the rays first go through the glass which is why we set n1 1 Since the image formed by the first surface acts as the object for the second surface we have p2 t q1 The signs are tricky here This is due to the convention that q is negative when the image is on the same side of the surface as the object Now we let t become very small Much less than either radius This is the definition of a thin lens Then we can say p2 q1 so n 1 1 n q1 q2 R2 We already knew that 1 n n 1 p1 q1 R1 We can sum the equations to find n 1 1 n n 1 1 n q1 q2 p1 q1 R1 R2 1 1 1 1 n 1 q2 p1 R1 R2 Since the combination of the two surfaces results in one image for the object we can drop the subscripts and just say 1 1 1 1 This is the lens maker s n 1 equation q p R1 R2 Focal length As with a spherical mirror we can define the focal length of the lens as the point at which the image of an object infinitely far away would appear 1 1 1 1 n 1 f R1 R2 1 1 1 n 1 f R1 R2 With this definition the lens maker s equation looks just like the mirror equation 1 1 1 q p f