MCAT Exam > MCAT Notes > Formula Sheet: Thin Lenses
Formula Sheet: Thin Lenses
Thin Lens Fundamentals
Sign Conventions
- Focal length (f): Positive for converging lenses, negative for diverging lenses
- Object distance (do): Positive when object is on the same side as incident light (real object), negative for virtual object
- Image distance (di): Positive for real images (opposite side from object), negative for virtual images (same side as object)
- Height/magnification: Positive for upright images, negative for inverted images
- Radius of curvature (R): Positive if center of curvature is on the side opposite to incident light, negative otherwise
Thin Lens Equation
\[\frac{1}{f} = \frac{1}{d_o} + \frac{1}{d_i}\]- f = focal length (meters)
- do = object distance from lens (meters)
- di = image distance from lens (meters)
- Condition: Applies to thin lenses in the paraxial approximation (rays close to and nearly parallel to optical axis)
Magnification
Linear/Lateral Magnification
\[m = -\frac{d_i}{d_o} = \frac{h_i}{h_o}\]- m = magnification (dimensionless)
- hi = image height (meters)
- ho = object height (meters)
- |m| > 1: Magnified (enlarged) image
- |m| <> Reduced (diminished) image
- m > 0: Upright image
- m <> Inverted image
Lens Power
Power Definition
\[P = \frac{1}{f}\]- P = power of lens (diopters, D)
- f = focal length in meters
- Unit: 1 diopter (D) = 1 m-1
- Note: Converging lenses have positive power; diverging lenses have negative power
Combined Lens Power
\[P_{total} = P_1 + P_2 + P_3 + ...\] \[\frac{1}{f_{total}} = \frac{1}{f_1} + \frac{1}{f_2} + \frac{1}{f_3} + ...\]- Condition: Lenses are in contact or separated by negligible distance
- Note: Powers add algebraically (accounting for sign)
Lensmaker's Equation
\[\frac{1}{f} = (n - 1)\left(\frac{1}{R_1} - \frac{1}{R_2}\right)\]- n = refractive index of lens material (dimensionless)
- R1 = radius of curvature of first surface (meters)
- R2 = radius of curvature of second surface (meters)
- Assumption: Lens is in air (nsurrounding ≈ 1) and is thin
- Note: Sign of R depends on surface curvature direction relative to incident light
Types of Lenses and Image Characteristics
Converging Lens (Convex, f > 0)
- Object beyond 2f: Real, inverted, diminished image between f and 2f on opposite side
- Object at 2f: Real, inverted, same-size image at 2f on opposite side
- Object between f and 2f: Real, inverted, magnified image beyond 2f on opposite side
- Object at f: Image at infinity (no image formed)
- Object inside f: Virtual, upright, magnified image on same side as object
Diverging Lens (Concave, f <>
- All object positions: Virtual, upright, diminished image on same side as object
- Image location: Always between focal point and lens
- Note: Diverging lenses always produce virtual images for real objects
Special Cases and Relationships
Object at Focal Point
- When do = f, image distance di → ∞
- Rays emerge parallel after passing through lens
Parallel Rays
- Parallel rays entering a converging lens converge at the focal point
- Parallel rays entering a diverging lens appear to diverge from the focal point
Real vs Virtual Images
- Real image: di > 0; can be projected on screen; inverted (m <>
- Virtual image: di < 0;="" cannot="" be="" projected;="" upright="" (m=""> 0)
Ray Diagrams - Principal Rays
Three Principal Rays for Ray Tracing
- Parallel ray: Enters parallel to optical axis → passes through (or appears to come from) focal point on opposite side
- Focal ray: Passes through (or aims toward) focal point on object side → emerges parallel to optical axis
- Central ray: Passes through center of lens → continues straight without bending (for thin lens)
- Note: Any two rays are sufficient to locate image; third ray provides verification
- Image location: Where rays (or their extensions) intersect
Multiple Lens Systems
Two-Lens System Approach
- Use thin lens equation for first lens to find di1
- Calculate object distance for second lens: do2 = separation distance - di1
- Use thin lens equation for second lens to find final image distance di2
- Total magnification: mtotal = m1 × m2
Total Magnification
\[m_{total} = m_1 \times m_2 \times m_3 \times ...\]- Note: Signs must be carefully tracked for proper image orientation
Applications and Special Instruments
Simple Magnifier
\[m = \frac{25 \text{ cm}}{f}\]- 25 cm = near point distance (standard viewing distance for relaxed eye)
- f = focal length of magnifying lens (in cm)
- Condition: Image formed at infinity for relaxed viewing
Compound Microscope
\[m_{total} = m_o \times m_e = -\frac{L \times 25 \text{ cm}}{f_o \times f_e}\]- mo = magnification of objective lens
- me = magnification of eyepiece
- L = tube length (distance between objective and eyepiece focal points)
- fo = focal length of objective
- fe = focal length of eyepiece
- Note: Negative sign indicates inverted final image
Refracting Telescope (Astronomical)
\[m = -\frac{f_o}{f_e}\]- fo = focal length of objective lens
- fe = focal length of eyepiece
- Note: Produces inverted image; |fo| >> |fe| for high magnification
Human Eye as Optical System
Eye Accommodation
- Near point: Closest distance for clear vision (typically 25 cm for young adults)
- Far point: Farthest distance for clear vision (infinity for normal eye)
- Power range: Eye changes lens power to focus on objects at different distances
Vision Correction
- Myopia (nearsightedness): Corrected with diverging lens (negative power)
- Hyperopia (farsightedness): Corrected with converging lens (positive power)
- Note: Corrective lens compensates for eye's focusing error
Key Problem-Solving Tips
- Always establish sign convention before solving problems
- Check image characteristics: Use sign of di and m to determine if image is real/virtual and upright/inverted
- For multiple lens systems: Treat sequentially; output of first lens becomes input for second
- Virtual objects: Can occur when image from first lens would form beyond second lens; do becomes negative
- Verify reasonableness: Check if calculated values make physical sense with lens type
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Apr 19, 2026
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