The focal length of a convex lens.

AS and A Level Science

Physics Full Laboratory Report (3)

Title:

The focal length of a convex lens.
Aim:

＊To learn the technique of locating real image by no-parallax method.

＊To study the real image formed by a convex lens.

＊To determine the focal length of a convex lens by graphical method.

＊To determine the focal length of a convex lens by conjugate foci method.

Apparatus:

1 convex lens on a holder (with focal length not greater than 15 cm)

1 1-meter ruler,

1 translucent screen,

1 plane mirror,

1 optical pin on a cork,

1 illuminated object (a lamp house with a letter ‘F’ on it)

For further investigation:

1 more convex lens of the same focal length as above.

Principles:

A convex lens (converging lens) is a circular glass plate convex on both surfaces. The non-uniform thickness causes bending of light towards the principle axis. In particular, a convex lens converges light from infinity parallel to the principal axis to a point, called the focus (F). Light rays from infinity not parallel to the principal axis are converged to the focal plane as shown in Fig. a.

When an object in placed at the placed at the focus F of the convex lens, the emerging light would be parallel to the principal axis. If a plane mirror id placed on the other side of the lens as shown in Fig. b, the rays reflected by the plane mirror would follow the original path and the final image would occur at the same position as the object. The distance between the object and the lens gives the focal length of the convex lens.

An object placed beyond F from the lens would produce a real image. In this experiment, you will capture the real image of an illuminated object on a screen. Suppose the object distance is u and the image distance is v. They are related by the lens formula: 1/f = 1/u + 1/v …… (1), all the quantities being positive.

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A plot of 1/v against 1/u will give a straight line that has a slope of -1 as shown in Fig. d. The intercepts on the horizontal and vertical axes are the same. By taking the reciprocal of the intercept, the focal length can be determined.

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Another method of determining the focal length bases on the symmetric property of the two foci. In Fig. e, Q is the image position of P when the lens is either at position 1 or 2. The difference in both cases is only in the image size. In this experiment, you will capture a sharp image at Q when the lens in located at position 1. The lens is moved towards the screen until another sharp image is captured again.

The displacement of the lens (a) and the separation between the object and the screen (d) are measured. The object distance when the lens is at position 1 is u = 1/2 (d –a), while the image distance is v = d – u = 1/2 (d + a). Putting into equation (1), we have 1/f = 1/ [1/2 (d-a)] + 1/ [1/2 (d + a)]. Hence, d² – a² = 4F．d ……(2). A graph of d² – a² against d should give a straight line as shown in Fig. f. The focal length of the lens can be determined from the slope of the graph.

Procedures:

Experiment 1: Capturing image of distant objects

The convex lens was held facing distant objects outside an open window. The screen on the other side of the lens was moved until the images of distant object were focused on the screen.
The distance between the lens and the screen was measured. This gave the focal length f of the lens.

Experiment 2: Locating the focus using no-parallax method

The optical pin was placed close to the convex lens. The vertical position of the pin was adjusted so that its tip was at the same level as the center of the lens.
The vertical plane mirror was placed at a distance of 10cm on the other side of the lens with the reflecting surface the lens. The mirror and the lens were made to be parallel.
The pin was moved away from the lens along the principal axis until the separation was about the focal length. The position of the pin was adjusted until it coincides exactly with its image. In this case, there was no relative motion between the image and the object pin when we moved our head sideways, i.e. they do not separate.
The pin was then at the focus of the lens. The focal length of the lens was then measured.
The plane mirror was moved to vary the separation between the mirror and the lens.

Experiment 3: Graphical method using an illuminated object

The lamp house was placed beginning with an object distance greater than twice the focal length (beyond 2F) of the convex lens. The position of the screen on the opposite side was adjusted so that a sharp diminished image was formed. The object distance u and the image distance v were measured.
Step 1 was repeated to capture 3 other diminished images and 4 other magnified images on the screen. The object distance u and image distance v were recorded in each case.
A graph of 1/v against 1/u was plotted. The best-fit straight line was drawn and it was produced to intersect both axes.
Intersects on both axes were found. The mean intercept was calculated. Hence, the focal length of the convex lens was determined.

Experiment 4: Measuring focal length by the lens displacement method using conjugate foci.

The lamp house and the convex lens were arranged so that a magnified sharp image was formed on the screen.
The meter ruler was placed alongside the lens and parallel to the principle axis of the lens. The position of the lens was recorded.
The lens was moved towards the screen until a diminished sharp image was formed on the screen. The positions of the lamp house or the screen should not be changed. The displacement (a) of the lens and the distance of the separation between the object and the image (d) was measured
The separation between the lamp house and the screen was changed and the experiment was repeated for at least five other times.
A graph of d² – a² against d was plotted. The best-fit straight line was drawn and the slope of the graph was measured. Using equation (2), the focal length of the convex lens was determined.

Results:

Experiment 1:

Focal length of the convex lens found was 10cm.

Experiment 2:

Focal length of the convex lens found was 10cm.

When the lens-mirror separation is changed, the image remains unchanged.

Experiment 3:

Table 1

Plot of 1/v against 1/u

Intercept on vertical axis = 0.098 cm-1

Intercept on horizontal axis = 0.072 cm-1

Mean of intercept = 0.085 cm-1

Focal length = 1/mean of intercept = 11.76 cm

% Error = (11.76 – 10)/10 X 100% = 17.6 %

Experiment 4:

Table 2

Plot of d2 - a2 against d

Slope of graph = (400 – 200)/(10-5) = 40

Focal length = slope /4 = 10m

% Error = 0%!!!

Precautions:

In experiment 1, be careful not to capture image of the window frames instead of distance object. Also, make sure the image distance was measured correctly.

In experiment 2, make sure the pin, the lens and the mirror are vertically placed. If you cannot see the image of the pin, the probable reasons are (a) you are too close to the pin,

less than the least distance of distinct vision,

(b) The image occurs behind you,

(d) The lens and the mirror are not parallel.

In experiment 3, the lens must be placed vertically. In using the no-parallax method, you shouldn’t only move your eyes, but also your head.

Adjust the vertical position of the pin so that there is no gap between the object and the image.

In experiment 4, the displacement of the lens and the separation between the object and the image needed to be correctly measured.

Sources of error:

There will be error when we plot the graph that will affect the calculated result.
The sharp image of bulb instead of the object letter F is form on the screen may affect the measurement of the focal length.

Discussions:

Actually the graphical methods are more reliable, as the experiment can repeat for many times and more data can be obtained. If the size of sample data increases, the result is more accurate. In using graphical method, we can also discard those obviously discrepant data.

Theoretically, experiment 3 is more reliable than experiment 4. Since there is only one personal judgment in each trial in experiment 3, but there are 2 of that in experiment 4.

If the lower half of the convex lens is covered, the image will be dimmer but still visible.

In further investigation, we found that if 2 lens of similar focal length are placed in contact with each other, the new focal length will be shortened. If the original focal lengths of 2 lenses are the same, then the resulted focal length will be half of that of 1 lens.