To determine the focal length of a concave lens by combining it with a convex lens of known focal length and measuring the focal length of the combination.

• Optical bench with uprights and holders
• A concave lens (focal length unknown)
• A convex lens (focal length known)
• An illuminated object (pin or cross-wire)
• A screen or viewing lens
• A half-meter scale
• A lens holder
• A screen holder

A concave lens always forms a virtual image of a real object. Therefore, direct measurement of its focal length using the lens formula is not convenient. One method to determine the focal length of a concave lens is to combine it with a convex lens of known focal length to form a lens system.

When a concave lens and a convex lens are placed in contact or close to each other, the effective focal length of the combination can be measured. Using the lens maker's formula, we can determine the focal length of the concave lens.

When light passes through two lenses in succession, the combined power is the algebraic sum of the individual powers:

Since power is the reciprocal of focal length (in meters), we can write:

Where:

• F is the focal length of the combination
• f₁ is the focal length of the convex lens
• f₂ is the focal length of the concave lens (to be determined)

For a concave lens, the focal length is negative. Rearranging the equation to solve for f₂:

The formula to calculate the focal length of the concave lens is:

Where:

• f₂ = Focal length of the concave lens (in cm)
• f₁ = Focal length of the convex lens (in cm)
• F = Focal length of the combination (in cm)

The focal length of the combination can be found using the lens formula:

Which gives:

Where:

• u = Object distance from the lens combination (in cm)
• v = Image distance from the lens combination (in cm)

1. Determining the focal length of the convex lens:
   • Place the convex lens on the optical bench.
   • Position the illuminated object (pin or cross-wire) at one end of the optical bench.
   • Move the screen until a sharp image of the object is formed.
   • Measure the distance between the object and the lens (u₁) and the distance between the lens and the image (v₁).
   • Calculate the focal length of the convex lens using the formula:
     \[\frac{1}{f_1} = \frac{1}{u_1} + \frac{1}{v_1}\]
   • Repeat this process 2-3 times and calculate the average focal length of the convex lens.
2. Determining the focal length of the combination:
   • Place the concave lens close to the convex lens (in contact or with minimal separation).
   • Adjust the position of the object.
   • Move the screen to get a sharp image.
   • Measure the distance between the object and the lens combination (u) and the distance between the lens combination and the image (v).
   • Calculate the focal length of the combination using the formula:
     \[F = \frac{u \times v}{u+v}\]
   • Repeat this process 4-5 times with different object positions and calculate the average focal length of the combination.
3. Calculating the focal length of the concave lens:
   • Use the formula:
     \[f_2 = \frac{F \times f_1}{f_1 - F}\]
     to calculate the focal length of the concave lens.

Table 1: Determination of focal length of convex lens

S.No.	Object distance u₁ (cm)	Image distance v₁ (cm)	Focal length f₁ = \(\frac{u_1 \times v_1}{u_1 + v_1}\) (cm)
1
2
3
Mean value of f₁

Table 2: Determination of focal length of lens combination

S.No.	Object distance u (cm)	Image distance v (cm)	Focal length F = \(\frac{u \times v}{u + v}\) (cm)
1
2
3
4
5
Mean value of F

Step 1: Calculate the mean focal length of the convex lens (f₁)

Step 2: Calculate the mean focal length of the lens combination (F)

Step 3: Calculate the focal length of the concave lens (f₂)

Since f₂ is negative for a concave lens, the focal length of the concave lens is -___ cm.

The focal length of the given concave lens is _______ cm.

Note: For a concave lens, the focal length should be negative. The negative sign indicates that the lens is diverging.

1. Ensure that the optical bench is placed on a level surface.
2. The optical elements (object, lenses, and screen) should be aligned along the same axis.
3. The convex lens should be positioned with its optical center on the principal axis.
4. The concave and convex lenses should be placed in contact or very close to each other when measuring the focal length of the combination.
5. The room should be sufficiently dark to see the image clearly on the screen.
6. Take readings for different positions of the object to minimize errors.
7. Always read the position of the optical elements from the same reference point on the scale.
8. Be careful not to touch the lens surfaces with fingers to avoid fingerprints and smudges.
9. Parallax error should be avoided while taking measurements.
10. The object should be well-illuminated for clear visibility.

1. Parallax errors: These occur when the positions of the optical elements are not accurately read from the scale.
2. Alignment errors: The optical elements might not be perfectly aligned along the same axis.
3. Lens imperfections: Real lenses may have imperfections that cause aberrations.
4. Measurement errors: Errors in reading the scale or marking the positions of the optical elements.
5. Human errors: Subjective judgment of when the image is sharply focused.
6. Lens thickness: The thin lens approximation may not be entirely valid for thick lenses.
7. Lens separation: If the lenses are not exactly in contact, there might be errors in the calculation.
8. Illumination issues: Poor illumination can make it difficult to locate the exact position of the sharp image.
9. Environmental factors: Temperature variations can affect the properties of the lenses.
10. Scale calibration: The scale used for measurements might not be accurately calibrated.