Mirrors and lenses

By Levi Clancy for Student Reader on
updated

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TermUnitsOverview

Image Distance (di, q)

meters

di = ( do F ) / ( do - F )

Object Distance (do, P, S)

meters

Magnification (m)

unitless

Positive m → Upright image.
Negative m → Inverted image.
| m | 1 → Magnified image.

m = di / do = heighti / heighto (determine sign based on above rules)

| m | = | F | / ( | F | - do )

Center (C)
Radius (R)

meters

This is the center of curvature, referred to as the center or radius.

R = 2 F

Focal Length (F)

meters

F = ½ R

1 / F = 1 / do + 1 / di

di = ( do F ) / ( do - F )

Mirrors

MirrorTypeRFdodim|m|

Convex

Diverging

+

+

+

do = 2F → ∞ (no image)
do > F → + (real, incident side)
do = F → -∞ (no image)
do < F → - (virtual)

do > F → - (inverted)
do = F → ∞ (no image)
do < F → + (upright)

do > R → shrunk
d0 = R → same size
d0 < R → magnified
do = F → no image

Concave

Converging

-

-

+

- (virtual, behind mirror)

0 < m < 1 (upright)

|m| < 1 (shrunk)

TermDescription

Snell's Law

n1 sin θ1 = n2 sin θ2Where θ is the angle from the normal, the line perpendicular to the surface of the medium.
0° ≤ θ ≤ 90°
Note: if θ cannot be defined when you try using Snell's Law (ie, θ = sin-1 x where x > 1) then take θ to be 90°.

Depth Perception

Depthapparent = Depth ( n2 / n1 )

diverging convex mirror ray diagram upright virtual shrunk image

diverging convex mirror ray diagram inverted real shrunk image

diverging convex mirror ray diagram inverted real magnified image

diverging convex mirror ray diagram inverted real magnified image

Lenses

LensTypeFdimUpright

Thinner at edges
Thicker at middle

Diverging
Concave
Planoconcave
Convex-Concave (|R1|<|R2|)

-

- di (virtual → incident side)

0 < m < 1

Upright

Thicker at edges
Thinner at middle

Converging
Convex
Planoconvex
Convex-Concave (|R1|>|R2|)

+

do > |F| → + di (real)
do < |F| → - di (virtual)

|m| > 1

do > |F| → Inverted
do < |F| → Upright

TermDescription

Lenses

Convex or planoconvex are converging and have a real image (opposite the incident side, unlike a mirror). Concave or planoconcave are diverging and have a virtual image (on the incident side). When working with lenses, disregard the signs of the focal lengths, etc and simply draw out a diagram; based on the rules for virtual or real images, and not based on the signs, determine the locations of virtual or real images. Treat di as positive when dealing with equations; its positive or negative value is entirely relative. The sign convention for positive and negative for diverging and converging lens is confusing and arbitrary; use absolute values and depend on figures and rules, not signs.

Convex-Concave Lens

F-1 = ( n - 1 ) ( R1-1 - R2-1 )

R1 = Radius of curvature for incident side
R2 = Radius of curvature for refracting side
n = Index of refraction for lens material

| R1 | > | R2 | → Converging lens with positive focal length.
| R1 | < | R2 | → Diverging lens with negative focal length.

diverging concave lens ray diagram upright virtual image object behind center

diverging concave lens ray diagram upright virtual image object behind focal point

converging convex lens ray diagram inverted real shrunk image object behind center

converging convex lens ray diagram inverted real image object at center

converging convex lens ray diagram inverted magnified real image object behind focal point

converging convex lens ray diagram upright magnified virtual image object before focal point

Multiple lenses

The subscript 1 refers to the first lens. The subscript 2 refers to the second lens. The variable L refers to the length between the two lenses (always positive). Use absolute values for all focal lengths, and base whether the image is real or virtual based on the sign of the resulting di: a negative di is virtual (object side) and a positive di is real (refraction side).

Calculate di,1 = do | F1 | / ( do - | F1 | )

Calculate do,2 as such: for a real image1 then do,2 = | L - di,1 | and for a virtual image1 then do,2 = L + do,1.

Calculate di,2 = [ | F2 | do,2 ] / [ do,2 - | F2 | ]