Light - Part XIV

Lens formula, power of a lens and magnification
We can determine the focal length of a lens by using a certain formula. All of us have heard of power of spectacles, this is nothing but inverse of the focal length. Lenses, as you have seen in the earlier section, can produce images which are larger or smaller than the object; with the lens formula one can determine the magnification produced by a lens. But before we see the lens formula, first lets learn a bit about the sign conventions used for lens geometry.

Sign convention :  

1. All distances are measured from the optical centre O.
2. Distances measured in the direction of the incident light rays are taken as positive.
3. Distances measured in the direction opposite to that of the incident rays, is taken as negative.
4. Distance measured upward from the principle axis is taken as positive distance.
5. Distance measured downward from the principle axis is taken as negative distance.

Let u = distance of the object from O.
      v = distance of the image from O.

In a convex or concave lens, u is always negative, as BO distance is measured in the direction opposite to that of the light rays striking the lens. In a convex lens, v can sometimes be positive and sometimes be negative. For concave lens, v is always positive.

Unlike mirrors, since lenses have two foci, there may be a confusion as to which focal length is taken as positive or otherwise. Conventionally the focal length of concave lens is taken as negative and that of a convex lens is taken as positive.

Lens formula
The lens formula gives the relationship between the object distance u, the image distance v and the focal length f. The formula is written as

    1                       1                        1
      -                 =   
    v                       u                         f

Thus if we know two of the three quantities from u,v,f, we will be able to find the third quantity. The values of u,v and f have to be inserted with proper sign convention.

Linear magnification of lenses
We have seen earlier with different position of the object in front of the lenses, the image can be either magnified or diminished; the image can be either erect or inverted. The ratio of the height of the image to the height of the object is known as the magnification.

                     Height of the image
Magnification =  
                                 Height of the object

If we say that
 h₁ = Height of the object
 h₂ = Height of the image,

                                             h₂
then magnification     m =    
                                             h₁

From the sign convention mentioned earlier,  a positive sign of m indicates that the image is virtual and erect. A negative sign of m indicates that the image is real and inverted.

Magnification is called as the linear magnification, because we are calculating the linear height of the object and images.

Those of you familiar with similar triangles, will immediately understand that

               h₂                      v
 m =        =    
              h₁                       u

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