1. Thin Lens Equation
1 𝑢 + 1 𝑣 = 1 𝑓

2. Terminology
The focal length (f) of the lens is the distance from the centre of the lens to the focal point (F).
Convex lens
(converging) 
focal length (f) F
1 𝑢 + 1 𝑣 = 1 𝒇
Concave lens
(diverging)
Convex lens: f is positive
Concave lens: f is negative
focal length (f)   F F

3. Terminology
The distance (u) is the distance from the object (O) to the centre of the lens
Convex lens
(converging) 
focal length (f) F O u
1 𝒖 + 1 𝑣 = 1 𝑓
Concave lens
(diverging)
Convex lens: u is positive O
Concave lens: u is positive
focal length (f)   u F

4. Terminology
The distance (v) is the distance from the image (I) to the centre of the lens
Convex lens
(converging)  O u v
1 𝑢 + 1 𝒗 = 1 𝑓 I
Concave lens
(diverging)
For both convex & concave lenses:
v is positive if the image is on the opposite side of the lens to the object O I  
For both convex & concave lenses:
v is negative if the image is on the same side of the lens as the object u F v

5. Example 1
Calculate the position and magnification of an image formed by a convex lens of focal length 20cm, when a 5cm high object is positioned 30cm in front of the lens. 
30 cm v
20 cm O
v = ?
u = 30 cm
f = 20 cm
1 𝑢 + 1 𝒗 = 1 𝑓 I
∴𝑣= 1 𝑓 − 1 𝑢 −1
1 𝒗 = 1 𝑓 − 1 𝑢
𝑣= − −1
𝑀𝑎𝑔=− 𝑣 𝑢 =− =−2
= 60 cm
Magnification = x2 (inverted image)

6. Example 2
Calculate the position and magnification of an image formed by a convex lens of focal length 10cm, when a 5cm high object is positioned 5cm in front of the lens. I 
5 cm v
10 cm
v = ?
u = 5 cm
f = 10 cm O
1 𝑢 + 1 𝒗 = 1 𝑓
1 𝒗 = 1 𝑓 − 1 𝑢
𝑀𝑎𝑔=− 𝑣 𝑢 =− −10 5 =+2
∴𝑣= 1 𝑓 − 1 𝑢 −1
𝑣= − −1
Magnification = x2 (upright image)
= -10 cm

7. O I 
60 cm v
40 cm
Example 3
Calculate the position and magnification of an image formed by a concave lens of focal length 40cm, when a 10cm high object is positioned 60cm in front of the lens.
v = ?
u = 60 cm
f = -40 cm
𝑀𝑎𝑔=− 𝑣 𝑢 =− −24 60 =+0.4
Magnification = x0.4 (upright image)
1 𝑢 + 1 𝒗 = 1 𝑓
1 𝒗 = 1 𝑓 − 1 𝑢
∴𝑣= 1 𝑓 − 1 𝑢 −1
𝑣= 1 −40 − −1
= -24 cm

8. Example 4
Calculate the position and magnification of an image formed by a concave lens of focal length 40cm, when a 10cm high object is positioned 20cm in front of the lens. O I 
20 cm v
40 cm
v = ?
u = 20 cm
f = -40 cm
𝑀𝑎𝑔=− 𝑣 𝑢 =− − =+0.67
Magnification = x0.67 (upright image)
1 𝑢 + 1 𝒗 = 1 𝑓
1 𝒗 = 1 𝑓 − 1 𝑢
∴𝑣= 1 𝑓 − 1 𝑢 −1
𝑣= 1 −40 − −1
= cm