Cutting a Lens: What Happens to its Focal Length and Power?
Master this crucial JEE Optics concept that appears in 90% of papers. Understand different cutting scenarios and their effects.
The Fundamental Concept
When a lens is cut, what happens to its focal length and power depends on how it's cut. The key principle is that focal length depends on the radius of curvature and refractive index, not on the size of the lens.
🎯 JEE Relevance
This concept appears in most JEE Optics sections. Understanding the 4 main cutting scenarios can help you secure easy marks in both JEE Main and Advanced.
🔍 Quick Navigation
1. Lens Maker's Formula - The Foundation
The Fundamental Equation
Where:
- $f$ = Focal length
- $\mu$ = Refractive index
- $R_1$ = Radius of first surface
- $R_2$ = Radius of second surface
Key Insight:
Focal length depends only on:
- Refractive index ($\mu$)
- Radii of curvature ($R_1$, $R_2$)
- Not on the size of lens
Power of a Lens
where $f$ is in meters and $P$ is in Diopters (D)
⚠️ Important Note
When we cut a lens, if the radii of curvature don't change, the focal length remains the same. However, the effective light-gathering area changes, which affects intensity but not focusing ability.
2. Case 1: Cutting into Two Equal Parts
Scenario: Symmetrical Cutting Through Principal Axis
What Changes?
Focal Length
Remains unchanged
Radii of curvature don't change
Power
Remains unchanged
Since $P = 1/f$ and $f$ is same
Intensity
Becomes half
Light gathering area reduces to half
Aperture
Reduced
Smaller effective diameter
Example Problem
A convex lens of focal length 20 cm is cut into two equal parts along a plane perpendicular to the principal axis. What is the focal length of each part?
Solution:
Since the cut is perpendicular to principal axis and through center, the radii of curvature remain unchanged.
Therefore, focal length of each part = 20 cm (unchanged)
Power of each part = $1/0.2 = 5D$ (unchanged)
3. Case 2: Cutting Along Principal Axis
Scenario: Cutting Parallel to Principal Axis
What Happens?
This case is more complex. The resulting pieces may have different focal lengths depending on which part you consider.
Central Part
Focal length unchanged
Radii remain same as original
Edge Parts
May become plano-convex or change type
One surface becomes flat
Example: Convex Lens Cut Vertically
A biconvex lens ($R_1 = R_2 = R$) is cut along the principal axis into two equal parts. What is the focal length of each part?
Original Lens:
$$\frac{1}{f} = (\mu - 1)\left(\frac{1}{R} - \frac{1}{-R}\right) = (\mu - 1)\left(\frac{2}{R}\right)$$
After Cutting: Each part becomes plano-convex
For plano-convex: $R_1 = R$, $R_2 = \infty$
$$\frac{1}{f'} = (\mu - 1)\left(\frac{1}{R} - \frac{1}{\infty}\right) = \frac{\mu - 1}{R}$$
Comparing: $f' = 2f$
Each part has focal length = twice the original
4. Case 3: Cutting Perpendicular to Principal Axis (Off-center)
Scenario: Asymmetrical Cutting
Key Principle
When cut perpendicular to principal axis but not through optical center:
Focal Length
Remains unchanged
Radii of curvature don't change
Power
Remains unchanged
Since focal length is same
Important: The optical center shifts in this case, but the focal length measured from the optical center remains the same.
Quick Reference Table
| Cutting Type | Focal Length | Power | Key Change |
|---|---|---|---|
| Through principal axis (perpendicular) | Unchanged | Unchanged | Intensity halves |
| Along principal axis | May change | May change | Lens type may change |
| Perpendicular but off-center | Unchanged | Unchanged | Optical center shifts |
| Into multiple pieces | Unchanged (each piece) | Unchanged (each piece) | Only intensity reduces |
❌ Common Misconceptions
Wrong! Focal length depends on curvature, not size. A small piece of a lens can have the same focal length as the original.
Wrong! In most cases, power remains unchanged. Only when the lens type changes (like biconvex to plano-convex) does power change.
Wrong! Reduced intensity means less light, but the focusing ability (power) remains the same.
Practice Problems
Test Your Understanding
Problem 1: A convex lens of focal length 30 cm is cut into two equal parts along the principal axis. What is the focal length of each part?
Problem 2: A biconvex lens (μ=1.5, R=20cm) is cut into two equal parts perpendicular to principal axis. Calculate the power of each part.
Problem 3: If a lens is cut into four equal parts by two perpendicular cuts through optical center, what happens to the focal length of each piece?
🎯 JEE Problem Solving Strategy
Step-by-Step Approach
- Identify the cutting type - How is the lens being cut?
- Check radii of curvature - Do they change?
- Apply lens maker's formula to both original and cut lens
- Compare the results to find the relationship
- Verify with logic - Does the answer make physical sense?
Quick Memory Aid
- Cut perpendicular → Focal length unchanged
- Cut parallel → May change focal length
- Through optical center → No change in focal length
- Changing lens type → Focal length changes
Key Takeaway
Focal length depends on curvature, not size. Most cutting doesn't change focal length unless the lens type itself changes.