11 physics EXPERIMENT NO. 2: Focal Length of a Convex Lens by Displacement Method

 

EXPERIMENT NO. 2: Focal Length of a Convex Lens by Displacement Method

Aim: To determine the focal length of a convex lens using the displacement method.

Principle (Displacement Method):

The displacement method is a technique used to accurately determine the focal length of a convex lens. It is based on the principle of reversibility of light and the property of a convex lens to form both real and virtual images.

If an object is placed at a distance u from a convex lens, and its real image is formed at a distance v from the lens, then the lens formula is:

 (ignoring sign convention for magnitudes)

In the displacement method, for a given fixed distance D between a real object and a screen, a convex lens can form a sharp, real, and inverted image on the screen at two different positions (L₁ and L₂), provided that D > 4f.

Let the distance between the object and the screen be D.

Let the distance of the first position of the lens from the object be u₁. Then the distance of the image from the lens will be v₁ = D - u₁.

For this position (L₁), the lens forms a real image.

Now, if the lens is moved to a second position (L₂), closer to the screen, such that the distance from the object is u₂, it forms another sharp, real image on the screen.

According to the principle of reversibility of light, the object and image positions can be interchanged. This means that if u₁ was the object distance for the first position, then for the second position, the object distance u₂ will be v₁ (the previous image distance), and the image distance v₂ will be u₁ (the previous object distance).

So, for the second position (L₂), let u₂ = D - v₁ = D - (D - u₁) = u₁. And v₂ = D - u₂.

However, the unique feature of the displacement method is that if u₁ and v₁ are the object and image distances for the first position, then for the second position, the object and image distances are v₁ and u₁ respectively.

Let's denote the object distance for L₁ as u and image distance as v.

Then u+v = D.

For the second position L₂, the object distance is v and the image distance is u.

Let x be the distance between the two positions of the lens.

So, from the diagram (see image below), we have:

u1 = u

v1 = v

u2 = v

v2 = u

The distance x between the two lens positions is given by:

x = u2 - u1 = v - u

We have two equations:

1. D = u + v

2. x = v - u

Adding (1) and (2): D + x = 2v   ;  v ={(D+x) /2}

Subtracting (2) from (1): D - x = 2u ; u = {(D-x) /2}

Substitute u and v into the lens formula 1/f =(1/v) -(1/u) 

 (using sign convention, u is negative)

 (magnitudes)    1/f =(1/v) + (1/u) 

 1/f = 1/{(D+x)/2} + 1/{(D-x)/2}

Therefore, the focal length f is given by:

Conditions for Image Formation:

For a real image to be formed at two positions of the lens for a fixed object-screen distance D, the condition D > 4f must be met.

Apparatus:

1. Optical Bench: A long scale with adjustable uprights (riders) to hold the lens, object, and screen.

2. Convex Lens: The lens whose focal length is to be determined.

3. Object Pin: A pin or a cross-wire object (often illuminated) mounted on an upright.

4. White Screen: A screen (e.g., ground glass or white cardboard) mounted on an upright.

5. Illuminating Lamp: (Optional, but useful for a clear image of a cross-wire object).

6. Metre Scale: For precise measurements.

Diagram:

Procedure:

1. Rough Focal Length: Obtain a rough estimate of the focal length f {rough} of the convex lens by forming a sharp image of a distant object (e.g., a tree outside the window) on a screen. The distance from the lens to the screen will be approximately f(rough).

2.  This helps in setting up the experiment.

3. Setup on Optical Bench: Place the object pin (O), the convex lens (L), and the white screen (S) upright on the optical bench.

4. Initial Separation: Fix the object pin at one end of the optical bench (e.g., 0 cm mark). Position the screen at a distance D from the object pin such that D is greater than 4f {rough} (e.g., if f {rough} = 15cm , set D to about 80cm or 90cm. Ensure the object and screen are vertically aligned.

5. First Image Position (L₁): Place the convex lens between the object and the screen. Adjust the position of the lens (moving it from the object towards the screen) until a sharp, clear, real, and inverted image of the object pin is formed on the screen. Let this position of the lens be L1. Note the position of L1 on the optical bench. This image will usually be magnified.

6. Second Image Position (L₂): Without disturbing the object pin or the screen, continue moving the lens further towards the screen. At another position, L2, a second sharp, clear, real, and inverted image will be formed on the screen. This image will typically be diminished. Note the position of L2 on the optical bench.

7. Measure Distances:

   • Measure the distance D between the object pin and the screen.

   • Measure the distance x between the two positions of the lens (L1 and L2)

   • x = L2 - L1.

8. Repeat: Repeat steps 3-6 by changing the distance D between the object and the screen at least three to five times. For each set, ensure that D > 4f

Observations:

• Rough focal length of the convex lens, f =......cm

• Least count of the optical bench = ......cm

Observation Table:

Sr. No.	Position of Object (O) (cm)	Position of Screen (S) (cm)	Distance D = (S - O) (cm)	Position of Lens (L₁) (cm)	Position of Lens (L₂) (cm)	Distance x = (L₂ - L₁) (cm)	Focal Length f=(D2−x2)​    /4D  (cm)
1.	(e.g., 0.0)	(e.g., 85.0)	85.0	(e.g., 15.0)	(e.g., 70.0)	55.0
2.	(e.g., 0.0)	(e.g., 90.0)	90.0	(e.g., 18.0)	(e.g., 72.0)	54.0
3.	(e.g., 0.0)	(e.g., 80.0)	80.0	(e.g., 12.0)	(e.g., 68.0)	56.0
Average							Avg. f

Example Calculations for Row 1:

Given:

D = 85.0 cm

x = 55.0 cm

Using the formula: f= (D2−x2)​ /4D

f = {(85.0)2 - (55.0)^2}/{4x85.0}

f = {7225 - 3025}/{340}

f = {4200}/{340}

f = 12.35 cm

Calculate f for each row in the table and then find the average value.

Result:

The focal length of the given convex lens, as determined by the displacement method, is [Average Calculated f] cm.

Precautions:

• The optical bench should be horizontal to ensure the optical axes of the lens, object, and screen are aligned.

• The object pin, the center of the lens, and the center of the screen should be at the same height.

• The image should be made as sharp as possible. This can be done by adjusting the lens slightly back and forth.

• The distance D between the object and the screen must be greater than four times the focal length (D > 4f) for two distinct images to be formed.

• The object pin should be brightly illuminated if not a self-luminous object.

• Parallax should be removed to ensure a sharp image by slightly moving the screen.

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