4.1 – Verifying the Lens Equation (Leaving Cert Physics): Revision Notes

4.1 – Verifying the Lens Equation

Introduction

The lens equation is one of the most fundamental relationships in optics. This equation, written as:

$\frac{1}{f} = \frac{1}{u} + \frac{1}{v}$

connects three crucial measurements when working with converging lenses:

• f = focal length of the lens (a fixed property)
• u = object distance (distance from object to lens)
• v = image distance (distance from lens to image)

This experiment allows you to verify this mathematical relationship experimentally by collecting real data and demonstrating that the equation holds true for different object positions.

Equipment required

To conduct this experiment successfully, you'll need:

• A converging lens of known focal length
• A ray box to provide illumination
• White cardboard screen for image projection
• Illuminated cross-threads (acts as your object)
• Metre stick for distance measurements
• Retort stand and clamp to hold the lens
• Sheet of white cardboard for creating the illuminated object

Experimental method

The experiment follows a systematic five-step process to collect accurate data:

Step 1: Estimate the focal length Begin by finding an approximate value for the focal length. Focus the lens on a distant object (such as a window several metres away) and project this image onto your screen. The distance from the image to the lens gives you an approximate focal length value.

Step 2: Set up the apparatus Arrange your equipment as shown in the experimental setup. The illuminated cross-threads should be positioned at a distance from the lens greater than the approximate focal length you found in Step 1.

Step 3: Achieve sharp focus Adjust the cardboard screen position until the image of the cross-threads appears in sharpest focus and is most clearly visible on the screen surface.

Step 4: Record measurements Carefully measure the distance from the cross-threads to the centre of the lens, then measure the distance from the real image on the cardboard screen to the centre of the lens. Record these values to three significant figures where possible.

Step 5: Repeat for different positions Change the value of u (object distance) and repeat Steps 3 and 4. Perform this at least four more times, choosing some values of u that give a diminished image and some that give a magnified image.

Data collection and recording

Your experimental data should be organised in a clear table format:

u (cm)	v (cm)	1/u	1/v	1/u + 1/v	f (cm)

For each pair of measurements, calculate the reciprocals and their sum. This systematic approach helps you verify the lens equation through your collected data.

Data analysis methods

Method 1: Direct calculation Complete your data table and observe that the values of 1/u + 1/v are approximately the same. When calculating the average value of f, you should find that it matches the known focal length of the lens (or the true value if unknown), thus verifying the mathematical model.

Calculate the average value of f and confirm that it matches the known focal length of the lens or your Step 1 measurement, thereby verifying the model.

Method 2: Graphical analysis For each pair of u and v values, calculate 1/u and 1/v. On graph paper, plot values of 1/u against corresponding values of 1/v. Draw the line of best fit through your data points and produce a straight line.

From your graph, read the value where the graph cuts the 1/u axis and where it cuts the 1/v axis. These intercept values both equal 1/f, providing another method to determine the focal length.

Graphical interpretation

The relationship between 1/u and 1/v produces a straight line when plotted, which provides powerful evidence for the lens equation.

Understanding the graph: The graph shows 1/u plotted against 1/v, creating a straight line with negative slope. From coordinate geometry principles, any straight line cuts the x-axis when y = 0.

• The line 1/u + 1/v = 1/f cuts the 1/u axis when 1/v = 0
• This gives 1/u = 1/f, so the value where the line cuts the 1/u axis is 1/f
• Similarly, when the line cuts the 1/v axis, the value is also 1/f

This graphical method provides an independent verification of the focal length value, strengthening your experimental conclusion.

Sources of experimental error

Several factors can affect the accuracy of your results:

a) Focus quality: Judging when the image is in sharpest focus on the screen can introduce random error in measuring v. Sharp focus requires careful observation and patience.

b) Distance measurements: Measuring distances from the optic centre of the lens can cause random error in measuring both u and v. If you only measure to the lens surface rather than its centre, this introduces systematic error.

c) Parallax error: Using the metre stick incorrectly can introduce systematic error. Always ensure you read the scale from the same angle to avoid this error, which would be random if viewed from different angles each time.

Practice questions

Consider these key questions to test your understanding:

1. If you could not find any clear image of the cross-threads on the screen, what is likely to be the problem and how would you rectify it?

2. Why is the value for v obtained more accurately than the value for u in this experiment?

3. List two precautions that should be taken when measuring distances in this experiment.