Find the focal length of a given convex lens.

                      L1                                       L2

                         u                        l                    u

                       Illuminated object

         

                 d

                   Screen

   Following is the derivative to find the focal length of the lens: -

When the lens is at position L2: -

1. The object distance is u and the image distance (v) is u + l

                                             (v = u + l)

2. The total distance d = 2u + l.

                 

So according to the formula,

                               

     

So, y = l 2/d, x = d and m = 1

as, c = – 4f

So f can be found by dividing the y – intercept by 4.

 

Below is the expected graph when a graph of l 2/d against d is plotted.

           l 2/d  

                0

                  d / cm

           4f

Variables

The dependent variable is the distance l, the distance between the two positions of the lens.

The independent variable is the distance d, the distance between the screen and the illuminated object.

The controlled variable is the intensity of the illuminated object and the focal length of the lens

Procedure

1. Place the illuminated object at a distance of 50 cm from the screen and set up the apparatus as shown in the figure above.
2. Now place the lens somewhere close to the screen and see in which position of the lens does the illuminated object comes into focus on the screen.
3. Mark this position as L1 and record its distance from the illuminated object.
4. Now place the lens somewhere close to the illuminated object where the object comes into focus on the screen.
5. Mark this position as L2 and record its distance from the illuminated object.
6. Repeat the experiment for values ranging from 50 cm to 100 cm.
7. Tabulate six sets of the values of d, L1, L2, l and l 2/d.
8. Plot a graph of l 2/d against d.
9. From the graph, find the y- intercept of the graph and calculate the focal length.

Tabulation

Calculation

                

     

Therefore,

   The y-intercept, c = 4f 

                              f = c/4

As seen in the graph c = 40 cm

So, f = 40/4

         = 10 cm 

Precautions

1. Make sure the lens is upright. The vertical plane perpendicular to its principal axis should be parallel to the screen.
2. Object and image distances should be measured along a line parallel to the principal axis.
3. The illuminated object should be placed in the region of the optical centre.
4. Check the focal length of the lens using the distant object method to ensure that the object distances used would be suitable.

Evaluation

The focal length of the lens is found out to be 10 cm which is very accurate to the actual value. Therefore the hypothesis stated in the beginning is proven to be correct and the focal length of the lens is found out. The actual value told is 10.4 cm so the absolute error and percentage error can be found out: -

Absolute error:

Absolute error = actual result – observed result

                        = 10.4 – 10.0

                        = 0.4

Relative error:

Relative error =

                         

                       =  

                       

                      = 0.0385

Percentage error:

 

Percentage error = relative error × 100

                           = 0.0385 × 100

                           = 3.85 %