Lenses

Introduction:

Determining the focal length of a lens is a simple procedure that may be done by focusing parallel light rays that may be provided by the sun.  In this experiment, we will measure the focal length of thin converging lens and verify the relationship of the thin lens equation:


Steps:

The focal length is first found by measuring the actual distance from the lens the point of focus.  With limited sunlight on the day of this experiment, we were forced to improvise on our technique of measuring the focal length.  We used to lasers pointed into the lens.  The resulting intersection would be the focal point.

Improvised method of measuring converging lens focal length.

The measured focal length was

f = 5.0 ± 1.0 cm
The reason for our high uncertainty is that the intersection of the lasers was only apparent within a range of 1 cm. The experiment we ran varied the object distance at multiples of the focal length.
Object height is a constant as measured from the light box used.  Image distance and height were measured.

A screen is used for the projected image when the object distance is fairly small.

A ruler is used to measure image height at greater object distances.

The data collected is tabulated below.


d0 (± .1 cm) di (± .1 cm) h0 (± .1 cm) hi (± .1cm) M               
5f = 25.000         7.000           1.800         0.400 0.23 ± 0.07
4f = 20.000         7.300           1.800         0.600 0.34 ± 0.07
3f = 15.000         8.200           1.800         0.700 0.39 ± 0.08
2f = 10.000       10.500           1.800         1.800 1.0   ± 0.1
1.5f =7.500       18.700           1.800         4.100 2.3   ± 0.2



Calculating fmax and fmin from the given data we find
fmax = ( 1/(5f +.1) + 1/(7.0+.1) )^-1   = 5.53 cm
fmin  = ( 1/(2f -.1) + 1/(10.5-.1) )^-1   = 5.07 cm
At an object distance of .5f, the image is now virtual as seen behind the the lens.

A virtual image created by standing our object  ahead of our focal point

The lens equation describes an inverse relation ship between d0 and di.  We expect our di vs d0 to behave in this way.

Object distance vs Image distance (1)

Inverse Image Distance vs Negative Inverse Object Distance (2)

The characteristics of graph 2 are
Slope = .9475
y-intercept = .184


Questions/Conclusions:


Rearranging the thin lens equation we find

The focal length f is the constant y-intercept.  Our slope is approximately 1, as expected from the above equation. 

Equating 1/f = .184

the result is f = 5.43cm
This is the average taken from our experimental data.

The uncertainty cam be take to be (fmax-fmin)/2 = .2cm

f = 5.4 ± .2 cm

From our original measurement of f = 5.0 ± 1.0 cm, we are within uncertainty.

Thus, we obtain an error of 8.6% from our experimental measurements.

The table above shows that image distance and image height decrease as a function of object distance.  Our graphed data illustrated this as well.