# Measuring The Focal Length of a Converging Lens

Lab 3 – Measuring Focal Length of Converging Lens

Introduction: The purpose of this experiment is to determine the focal length of a converging lens using 5 different methods. The methods used are described below:

1. Object at large distance; image nearly in focal plane
2. Bessel’s Method
3. Autocollimation
4. Abbe’s Method
5. Lensmaker’s Formula

Part 1: Using “Object at Large Distance” to Measure Focal Length

Procedure: An optical rail is mounted by a window with a plano-convex lens and an image screen attached. The lens is ‘pointed’ at an object that is far away outside the window. Since the object is so far away, the light rays pass through the lens nearly parallel. The screen is then adjusted to be an appropriate distance from the lens such that the image produced is the sharpest, and hence at the focal length of the lens.

The image and lens positions will be measured and then used to determine the focal length.

Setup:

Data and Calculations:

 Lens Position Image Position Distance Between Lens & Image (s’ and f) 98.575 115 16.425

Note: The lens position is taken from the center of the length, which is assumed to be half the total thickness of the lens. The lens is measured at 10.5 mm, therefore 5.25 mm was added to the original measurement where the fiducial mark indicates.

The formula for focal length is as follows: 1/f = 1/s’ + 1/s. s is assumed to be extremely large, allowing us to treat 1/s≈0 which leaves f≈s’

Observations: As the lens is moved through the focal position, the image becomes inverted.

Problem: To calculate how far the object must be for our measurement of the focal length to be within 1% of the image distance we must determine a 1% difference in f, which is f*.01=0.164 (cm). The term 1/s can not contribute more than 0.164 cm to the previous lens equation, 1/f = 1/s’ + 1/s. Therefore, solving for s leaves s>6.25 cm.

Conclusion: Based on the “object at large distance” experiment, we have determined the focal length to be around 16.425 cm.

Part 2: Bessel’s Method

Procedure: In this experiment, we will use an optical rail with a light source, a lens and an image screen attached. The light source and image screen will be at fixed positions on opposite ends of the rail. A converging lens will be placed somewhere in between the light and screen.

There are 2 positions of the lens that create a sharp image, these positions will be measured and analyzed. Using that information, we can determine the focal length of the lens.

Setup:

Data and Calculations:

 L d Focal Length (L^2-d^2)/(4*L) 90 40.4 17.966 100 52.8 18.030 110 64.5 18.045

Observations: There is some degree of uncertainty due to attempting to determine a ‘sharp’ image.

Conclusion: The average focal length is 18.014

Part 3: Autocollimation

Procedure: In this experiment, we will attempt to refract diverging light rays into a parallel path then reflect them back along the same path till they eventually converge to the point they began from. To do this, the object (light source) must be placed at exactly the focal point of the lens. The set up will consists of a light source on one end of an optical rail. The light will then shine through a lens and eventually reflect off a mirror and back through the lens and onto an image screen in the same plane as the point source.

The goal is to create an image at the same plane as the point source. This position will be the focal length of the lens.

Setup:

Data and Calculations:

 Object Position Lens Position Distance Between (d) 18.8 37.6 18.8

• The distance measured between the lens and object (s) is the same as the focal length. In this experiment, using the equation 1/f = 1/s’ + 1/s, the image distance (when focused) essentially approaches 0, leaving f=s.

Conclusion: The focal length is 18.8 cm.

Part 4: Abbe’s Method

Procedure: In this experiment, we will use Abbe’s method to determine the focal length of a lens. Abbe’s method consists of a similar setup as Bessel’s method. The difference is that we will measure the object distance and magnification to determine focal length.

The setup will consist of an optical rail with the light source (object) placed at one end and an image screen placed at the other end. In between we will place the lens. The lens and screen are adjusted until an observable image is formed. After the appropriate measurements are recorded, the lens and screen are adjusted again. Our goal is to create 2 different setups where the magnification of the image is approximately 2 in one case and ½ in another case.

Setup:

Data and Calculations:

 Lens Position s Image Position s’ Object Height Image Height Magnification m Focal Length (s2-s1)/[(1/m1)-(1/m2)] 34 71.3 0.35 0.35 1.00 25.9 85.3 0.35 0.82 -2.34 -14.132 57.5 83.1 0.35 0.17 -0.49 -19.363

Conclusion: The average focal length is 16.747 cm.

Part 5: Lensmaker’s Formula

Procedure: In this experiment, we will determine the focal length by using the known refractive index of a material and the radii of curvature, R. To measure R, we shall use a spherometer. Based on those measurements alone, we should be able to determine the focal length of the lens.

To conclude this section of the lab, we will also derive the formula of the spherometer.

Setup:

Data and Calculations:

 n (air) nL (lens) r h R = (r/2)*[(r/h+h/r) f = (1/h)+(r^2)/(12*h) 1 1.5187 23.12 2.85 95.20 15.981

Conclusion: The focal length is 15.981

Part 6: Summary of Focal Length

 Section Focal Length (cm) Uncertainty 1 16.425 .5 cm 2 18.014 .5 cm 3 18.800 1 cm 4 16.747 2-3 cm 5 15.981 1 cm AVG 17.193

The most difficult technique (in terms of accuracy) was probably Abbe’s method due to the difficulty in measuring magnifications of less than ideal “sharp” images.

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