where:

• $f$ = focal length (+ for converging, - for diverging)
• $d_o$ = object distance (always positive)
• $d_i$ = image distance (+ for real, - for virtual)

Magnification: $m = -\frac{d_i}{d_o} = \frac{h_i}{h_o}$

💡 Same equations as mirrors! But sign conventions differ slightly.

Sign Conventions (Lenses)

Key difference from mirrors: Real image on opposite side of lens from object!

Ray Diagrams (Converging Lens)

Draw any 2 of these 3 rays:

1. Parallel ray → refracts through F on far side
2. Focal ray (through F on near side) → refracts parallel
3. Center ray (through lens center) → straight through (no bend)

Where rays intersect = image location!

Cases:

• $d_o > 2f$: Real, inverted, reduced (cameras)
• $d_o = 2f$: Real, inverted, same size
• $f < d_o < 2f$: Real, inverted, enlarged (projectors)
• $d_o < f$: Virtual, upright, enlarged (magnifying glass!)

Lensmaker's Equation

$\frac{1}{f} = (n-1)\left(\frac{1}{R_1} - \frac{1}{R_2}\right)$

where:

• n = index of refraction of lens
• $R_1$, $R_2$ = radii of curvature of surfaces

(Not commonly used in AP Physics 2, but good to know!)

Power of Lens

$P = \frac{1}{f}$

Unit: Diopter (D) = m⁻¹

Higher power → shorter f → more bending

Example: Eyeglass prescription "+2.00 D" means f = 0.50 m

Compound Lens Systems

Two lenses in combination:

Method 1 (Total power): $P_{total} = P_1 + P_2$

Method 2 (Step-by-step):

1. Find image from lens 1
2. Use that image as object for lens 2
3. Total magnification: $m_{total} = m_1 \times m_2$

Optical Instruments

Magnifying Glass:

• Single converging lens
• Object inside focal length ($d_o < f$)
• Virtual, upright, enlarged image
• Angular magnification: $M = \frac{25 \text{ cm}}{f}$

Compound Microscope:

• Objective lens (short f): Forms real, enlarged image
• Eyepiece lens: Acts as magnifying glass on that image
• Total magnification: $M = m_o \times M_e = -\frac{L}{f_o} \times \frac{25}{f_e}$
• Very high magnification (~100-1000×)

Telescope:

• Objective lens (long f): Forms real, reduced image of distant object
• Eyepiece: Magnifies that image
• Angular magnification: $M = -\frac{f_o}{f_e}$
• Larger objective → more light → see fainter objects

Camera:

• Converging lens
• Object far away ($d_o >> f$), so $d_i \approx f$
• Real, inverted, reduced image on film/sensor
• Adjustable f (zoom) and aperture (brightness)

Human Eye:

• Cornea + lens = converging system
• Lens changes shape (accommodation) to focus
• Image on retina (real, inverted, reduced)
• Brain flips it!

Nearsighted (myopia): Eyeball too long → use diverging lens Farsighted (hyperopia): Eyeball too short → use converging lens

Aberrations

Spherical aberration: Rays at edge focus differently

• Solution: Parabolic lens, smaller aperture

Chromatic aberration: Different colors focus at different points

• Solution: Achromatic doublet (two lenses)

Problem-Solving Strategy

1. Identify lens type: f > 0 (converging) or f < 0 (diverging)
2. Use thin lens equation: $1/f = 1/d_o + 1/d_i$
3. Find magnification: $m = -d_i/d_o$
4. Interpret signs:
   • $d_i > 0$: Real (opposite side)
   • $d_i < 0$: Virtual (same side)

Common Mistakes

❌ Confusing lens and mirror sign conventions (real image location!) ❌ Forgetting f is negative for diverging lens ❌ Using wrong magnification formula for instruments ❌ Thinking diverging lens can form real image (never!) ❌ Not checking calculator mode (degrees vs radians)

Part (a): Image distance

Thin lens equation: $\frac{1}{f} = \frac{1}{d_o} + \frac{1}{d_i}$ $\frac{1}{15} = \frac{1}{30} + \frac{1}{d_i}$ $\frac{1}{d_i} = \frac{1}{15} - \frac{1}{30} = \frac{2-1}{30} = \frac{1}{30}$ $d_i = 30 \text{ cm}$

Part (b): Magnification

$m = -\frac{d_i}{d_o} = -\frac{30}{30} = -1.0$

Part (c): Image description

• $d_i > 0$: Real (opposite side of lens)
• $m < 0$: Inverted
• $|m| = 1$: Same size as object
• Note: Object is at 2f, image also at 2f!

Answer:

• (a) d_i = 30 cm (opposite side)
• (b) m = -1.0
• (c) Real, inverted, same size