4

Simple Lens Formula

U + D = V

or

100/u (cm) + D = 100/v (cm)

Where:

U = vergence of object at the lens

u = object position = 100/U (cm)

D = lens power

V = vergence of image rays

v = image position = 100/V (cm)

Vergence: The reciprocal of the distance from a reference point. U = 100/u, where u is measured in centimeters or U = 40/u where u is measured in inches.

**NOTE: Light travels from left to right unless otherwise stated.

**NOTE: Light never comes out of the eyes.


Question: If parallel light rays strike a +4.00D lens, where will the image be? (Figure 4)


Click on image to enlarge.


Answer: Parallel light has no vergence. Therefore, using the equation U + D = V

U = vergence of object at the lens = 0.00D

D = lens power = +4.00D

Vergence of image rays = V = 0.00D + (+4.00D) = +4.00D.

Converting to centimeters, 100cm/+4.00D = +25 cm to the right of the lens.



Question: An object is placed 25 cm in front of a refracting surface of power +10.0D. (Figure 5)
Click on image to enlarge.

• What is the object vergence?
  
  
  Use the equation U = 100/u where u = distance of object from lens = 25 cm
  
  U = object vergence = 100/-25 = -4.00D
  
  
• What is the image vergence?
  
  
  Use the equation U + D = V
  
  U = object vergence = -4.00D
  
  D = lens power = +10.00D
  
  V = image vergence= –4.00D +(+10.00D) = +6.00D
  
  
• Where is the image placed?

To find the location of the image, the image vergence (V) is converted into cm.

Use the equation, v = image position = 100/V where V = +6.00D

v = image position = 100/+6 = +16.66 cm to the right of the lens.

• Is the image real?
  

Yes, because its position is positive and to the right of the lens.



Question: An object is located 25 cm in front of a +5.00D lens.

• What is the vergence of the incident rays?
  
  Use the following equation to calculate the object vergence: U = 100/u
  
  Where u = object location in cm
  
  U = object vergence = 100/-25 = -4.00D
  
  
  
• What is the refracting vergence?
  
  Use the equation U + D = V where
  
  U = object vergence = -4.00D
  
  D = lens power = +5.00D
  
  V = refracting vergence of the image = -4.00D + (+5.00D) = +1.00D
  
  
  
• Where is the image located?
  
  Use the equation, v = image position = 100/V where V = +1.00D
  
  v = image position = 100/+1 = +100 cm to the right of the lens.
  
  
• Is the image real or virtual?
  
  Real
  

Question: Define a plus lens

Answer: Plus lenses always add vergence; define a focal point; and converge image rays to produce a real image of an object at infinity to the right of a plus lens.



Question: Define a minus lens

Answer: A minus lens always reduces vergence; defines a focal point; and diverges image rays to produce a virtual image of an object at infinity to the left of the minus lens.



Question: What is the focal length of a plus lens whose image is 20 cm behind the lens for an object that is 50 cm in front of the lens? (Figure 6)


Click on image to enlarge.

Answer: First, determine the lens power by using the equation U + D = V where

u = object distance = -50cm

v = image distance = +20cm

U = object vergence = 100/u = 100/-50 = -2.00D

D = lens power

V = image vergence = 100/v = 100/20 = +5.00D

Lens power = D = V – U = +5.00D – (-2.00D) = +7.00D

Therefore, the focal length (f) of the lens, in cm, is 100/D, f = 100/+7.00D = +14.29cm.



Question: An object is imaged 20cm behind a –15.00D lens. (Figure 7)

Click on image to enlarge.

• Is the object real or virtual?
  
  
  Virtual, because it is behind the lens.
  
  
• Where will the image be focused?
  
  
  Use the equation U + D = V where
  
  u = object distance = +20cm
  
  U = object vergence = 100/u = 100/+20 = +5.00D
  
  D = lens power = -15.00D
  
  V = image vergence = +5.00D + (-15.00D) = -10.00D
  
  image distance = v = 100/-10.00D = -10cm in front of the lens.
  
  
  
• Is the image real or virtual?
  
  
  Virtual, because it is in front of the lens.