Transcript
Reflection at a Spherical Surface Sign Rule (#4) for the radius of curvature of a spherical surface: radius of curvature R is + when the center of curvature C is on the same side as the outgoing light (concave) and – otherwise.
concave V
The
(CV is called the optical axis.)
convex
V
Reflection at a Spherical Surface From PBC ,
From CBP ',
Eliminating between these 2 Eqs: 2
Reflection at a Spherical Surface Relating the angles to the physical distances, we have: tan
h s
tan
tan
h R
h s '
Reflection at a Spherical Surface Paraxial Assumption: We consider only rays which are nearly parallel to the optical axis and close to it. These rays are called paraxial rays. With this approximation, the angles , , and will be small and can be neglected with respect to s, s’, and R and h tan s
h tan s'
tan
h R
Combining with 2 and eliminating h, we then arrive at,
1 1 2 s s' R
(object-image relation, spherical mirror)
Focal Point and Focal Length For an object (stars) very far away from the mirror ( s ) , the incoming rays can be considered to be parallel.
What will happen to these rays?
1 1 2 s' R
R s' 2
Focal Point and Focal Length All parallel rays from ( s ) will converge to the same image point at s’ = R/2. This special point is called the focal point F and the distance from the vertex of the mirror to F is call the focal length f, R f 2
(focal length of a spherical mirror)
Focal Point and Focal Length Let consider the reverse situation, light rays are emanating from the focal point F, where will the image be? R f
2 1 2 R s' R
2
1 0 or s'
s'
Thus, as expected, the situation is time reversed, rays starting out from F will be reflected out toward infinity as parallel rays.
Spherical Aberration Recall that this is only an approximation. The focal point is a sharp point only if we consider paraxial rays. For non-paraxial rays, they do not necessary converge to a precise point. The blurring of the focal point in an actual spherical mirror is called spherical aberration.
Parabolic vs. Circular Mirrors With aberrations only paraxial rays fall on Focus
No aberrations all parallel rays fall on Focus
Lateral Magnification of a Spherical Mirror Following rays #1 and #2, we can form the following two similar triangles: beige and blue. 1 2
Lateral Magnification of a Spherical Mirror
The two similar triangles gives,
y y' s s'
Substituting this into the definition for m,
m
y ' s ' y s
Convex Spherical Mirrors The geometric optics formulas for a convex mirror are the same as for a concave mirror except that R and f are negative.
Convex Spherical Mirrors With R and f negative, parallel rays falling upon a convex mirror will diverge as if emanating from a virtual focal point F behind the mirror. Similar to the concave mirror, rays aiming toward this virtual focal point will be reflected back toward infinity as parallel rays. convex
concave
Convex Spherical Mirrors With R and f negative, parallel rays falling upon a convex mirror will diverge as if emanating from a virtual focal point F behind the mirror. Similar to the concave mirror, rays aiming toward this virtual focal point will be reflected back toward infinity as parallel rays. convex
concave
Summary for Spherical Mirrors The following are valid for both concave and convex spherical mirrors if we follow the proper sign conventions.
1 1 1 s s' f R f 2 s' m s
(object-image relation, spherical mirror)
(focal length, spherical mirror)
(lateral magnification, spherical mirror)
Note: these equations agree with results for a flat mirror if we take R .
Sign Rules for Spherical Mirrors 1.
Object Distance:
2.
Image Distance:
3.
s’ is + if the image is on the same side as the outgoing light and is – otherwise.
Object/Image Height:
4.
s is + if the object is on the same side as the incoming light (for both reflecting and refracting surfaces) and s is – otherwise.
y (y’) is + if the image (object) is erect or upright. It is – if it is inverted.
Radius of Curvature and Focal Length:
R and f is + when the center of curvature C is on the same side as the outgoing light and – otherwise.
Geometric Methods: Rays Tracing Principal rays for concave mirror
Geometric Methods: Rays Tracing Principal rays for convex mirror
(might need to extrapolate lines to intersect at image point)
Example 34.4: Concave Mirror P>C>F
P=C
Example 34.4: Concave Mirror P=F
P < F< C
Example 34.4: Concave Mirror
Mirrors & Thin Lens Applet (by Fu-Kwun Hwang) http://www.physics.metu.edu.tr/~bucurgat/ntnujava/Lens/lens_e.html
This applet is for both mirrors and thin lens. Use the drop down menu to choose.
Demo with Two Circular Mirrors forming a real image here
Dsp mirror
Ddouble mirror
Refraction at a Spherical Surface 2 1
• • •
Ray 1 from P going through V (normal to the interface) will not suffer any deflection. Ray 2 from P going toward B will be refracted into nb according to Snell’s law. Image will form at P’ where these two rays converge.
Refraction at a Spherical Surface At B, Snell’s law gives,
2 1
na sin a nb sin b
From PBC ,
a
From P ' BC ,
b b
From trigonometry, we also have,
tan
h s
tan
h tan R
h s '
Refraction at a Spherical Surface Again, consider only paraxial rays so that the incident angles are small, we can use the small angle approximations: sin ~ tan ~ . With this, Snell’s law becomes: na a nbb Substituting a and b from previous slide, we have,
na ( ) nb ( ) na na nb nb na nb (nb na ) With the small angle approximations, the trig relations reduce to,
h tan s
tan
h s'
tan
h R
Refraction at a Spherical Surface Substituting these expressions for , and into Eq. common factor h, we then have,
na nb nb na s s' R (object-image relationship, spherical refracting surface)
and eliminating the
Refraction at a Spherical Surface To calculate the lateral magnification m, we consider the following rays: 1 2
• •
Ray 1 from Q going toward C (along the normal to the interface) will not suffer any deflection. Ray 2 from Q going toward V will be refracted into nb according to Snell’s law.
Refraction at a Spherical Surface 1 2
From geometry, we have the following relations,
tan a y s From Snell’s law, we have,
tan b y ' s '
na sin a nb sin b
Refraction at a Spherical Surface Using the small angle approximation again (sin ~ tan ), the Snell’s law can be rewritten as, na sin a nb sin b
y y' na nb s s' sin tan
Substituting these into the definition for lateral magnification, we have, m
n s' y' m a nb s y
(lateral magnification, spherical refracting surface)
Refraction at a Flat Surface For a flat surface, we have R . Then, the Object-Image relation can be reduced simply as, na nb nb na 0 s s'
na n b s s'
na s ' 1 nb s
virtual
Combing this with our result for lateral magnification, we have, m 1 (upright) so that, the image is unmagnified and upright.
Example 34.5 & 34.6
Images formed by a spherical surface can be real or virtual depending on na, nb, s, and R.
Ex 34.5:
na nb nb na s s' R
1.00 1.52 0 .5 2 8.00cm s' 2.00cm
s ' 11.3cm
Example 34.5 & 34.6
Images formed by a spherical surface can be real or virtual depending on na, nb, s, and R.
Ex 34.6:
na nb nb na s s' R
1.33 1.52 0 .1 9 8.00cm s' 2.00cm
s ' 21.3cm
