Lenses
Original Scanned PDF – View Notes
Lens
The source divides lenses into two main classes:
Convex Lens
The scanned notes show three types of convex lens:
Concave Lens
The scanned notes show three types of concave lens:
Lens Formula
where f is focal length, u is object distance and v is image distance.
Convex Lens – Real Image
Page 2 derives the lens formula for a convex lens forming a real image.
The source sets:
From similar triangles ΔA′B′P and ΔABP:
A′B′/AB = A′P/AP = v/u … (i)
Also, from similar triangles ΔA′B′F and ΔMPF:
A′B′/MP = A′F/PF
Since MP = AB:
A′B′/AB = (v − f)/f … (ii)
From (i) and (ii):
v/u = (v − f)/f
vf = uv − uf
uv = vf + uf … (iii)
Dividing both sides by uvf:
1/f = 1/u + 1/v
Convex Lens – Virtual Image
Page 3 derives the same lens formula for the virtual image produced by a convex lens.
The source takes:
From similar triangles:
A′B′/AB = −v/u … (i)
Also:
A′B′/AB = (−v + f)/f … (ii)
Using (i) and (ii):
−v/u = (−v + f)/f
−vf = −uv + uf
uv = vf + uf … (iii)
Dividing by uvf:
1/f = 1/u + 1/v
Concave Lens – Virtual Image
Page 4 derives the lens formula for a concave lens forming a virtual image.
The source takes:
From the first pair of similar triangles:
A′B′/AB = −v/u … (i)
From the second pair:
A′B′/AB = (−f + v)/(−f) … (ii)
Using (i) and (ii):
−v/u = (−f + v)/(−f)
vf = −uf + uv
uv = vf + uf … (iii)
Dividing by uvf:
1/f = 1/u + 1/v
Linear Magnification
The source also defines linear magnification as the ratio of image distance to object distance and denotes it by m.
Combined Magnification of Coaxial Lenses
If N lenses are combined coaxially, the source writes the combined magnification as:
where m1, m2, …, mN are the magnifications produced by the N lenses.
Power of a Lens
If focal length is expressed in metres, the power P is:
The SI unit of power is m−1, called the dioptre (D).
Lens Maker’s Formula
Step 1 – Lens as a Small-Angled Prism
Consider a thin convex lens of focal length f. A ray XB parallel to the principal axis strikes the lens at point B at height h above the optical centre C. After refraction, it meets the principal axis at focus F, making an angle of deviation δ.
From the figure:
tan δ = h/f
For small δ, tan δ ≈ δ:
δ = h/f … (i)
The source treats the lens locally as a small-angle prism and uses:
δ = A(μ − 1) … (ii)
Using (i) and (ii):
h/f = A(μ − 1)
1/f = (μ − 1)A/h … (iii)
Step 2 – Expressing Prism Angle in Terms of R1 and R2
Let C1 and C2 be the centres of curvature and let α and β be the angles made by BC1 and BC2 with the principal axis.
CC1 = R1, CC2 = R2
A = α + β … (iv)
For small α:
tan α = h/R1 ≈ α
α = h/R1 … (v)
For small β:
tan β = h/R2 ≈ β
β = h/R2 … (vi)
Using (v) and (vi) in (iv):
A = h/R1 + h/R2
A/h = 1/R1 + 1/R2 … (vii)
Using (vii) in equation (iii):
1/f = (μ − 1)(1/R1 + 1/R2)
This is the lens maker’s formula exactly in the form written in the supplied notes.
Focal Length of Combined Lenses
Page 8 considers two thin lenses L1 and L2 of focal lengths f1 and f2 placed in contact.
Lens L1 forms an intermediate image I′ of object O. This image acts as a virtual object for lens L2, which forms the final image I.
For lens L1:
1/f1 = 1/u + 1/v′ … (i)
For lens L2:
1/f2 = −1/v′ + 1/v … (ii)
Adding (i) and (ii):
1/f1 + 1/f2 = 1/u + 1/v … (iii)
If F is the equivalent focal length of the combination:
1/F = 1/u + 1/v … (iv)
From (iii) and (iv):
1/F = 1/f1 + 1/f2
For N Lenses in Contact
Power of Combined Lenses
Using P = 1/f, the source obtains:
The source concludes that the reciprocal-focal-length relation gives the equivalent focal length and the power relation gives the combined power of lenses in contact.
Discussion
Share a helpful question, idea, or explanation with other students.