CUET UG Physics Optics Refraction at Curved Surfaces Lens Makers Equation Lens Formula

Must-Know

• The Lens Maker’s Formula is $\frac{1}{f} = (n - 1)\left(\frac{1}{R_1} - \frac{1}{R_2}\right)$, where $f$ is focal length, $n$ is refractive index of lens material relative to surrounding medium, and $R_1$, $R_2$ are radii of curvature of the two surfaces.
• For a convex lens, $R_1$ is positive and $R_2$ is negative by sign convention (NCERT uses Cartesian sign convention). Example: If $R_1 = +20\,\text{cm}$, $R_2 = -20\,\text{cm}$, $n = 1.5$, then $f = 20\,\text{cm}$.
• For a concave lens, $R_1$ is negative and $R_2$ is positive. Example: $R_1 = -20\,\text{cm}$, $R_2 = +20\,\text{cm}$, $n = 1.5$ → $f = -20\,\text{cm}$.
• The refractive index $n$ in Lens Maker’s formula is $n = \frac{n_{\text{lens}}}{n_{\text{medium}}}$; if lens is in air, $n_{\text{medium}} = 1$, so $n = n_{\text{lens}}$.
• If both surfaces of a lens have equal radii and are convex, it is called an equiconvex lens; e.g., $R_1 = +R$, $R_2 = -R$.
• A plano-convex lens has one flat surface ($R = \infty$) and one curved surface; e.g., $R_1 = +30\,\text{cm}$, $R_2 = \infty$, $n = 1.5$ → $f = 60\,\text{cm}$.
• The Lens Formula is $\frac{1}{v} - \frac{1}{u} = \frac{1}{f}$, where $u$ is object distance, $v$ is image distance, and $f$ is focal length (all in meters).
• According to sign convention: object distance $u$ is always negative for real objects; $v$ is positive for real images and negative for virtual images.
• Focal length $f$ is positive for converging (convex) lenses and negative for diverging (concave) lenses.
• Power of a lens $P = \frac{1}{f}$ (in meters), measured in dioptres (D); e.g., $f = +0.5\,\text{m}$ → $P = +2\,\text{D}$.
• For a convex lens, when object is at infinity, image forms at focus; image is real, inverted, and highly diminished.
• For a convex lens, when object is beyond $2F$, image is between $F$ and $2F$, real, inverted, and diminished.
• For a convex lens, when object is at $2F$, image is at $2F$ on other side, real, inverted, same size.
• For a convex lens, when object is between $F$ and $2F$, image is beyond $2F$, real, inverted, magnified.
• For a convex lens, when object is at $F$, image is at infinity.
• For a convex lens, when object is between optical center and $F$, image is virtual, erect, magnified (used in magnifying glass).
• A concave lens always forms virtual, erect, and diminished images regardless of object position.
• Refraction at a spherical surface follows $\frac{n_2}{v} - \frac{n_1}{u} = \frac{n_2 - n_1}{R}$; this is the basis for deriving lens formulas.
• In the derivation of Lens Maker’s formula, the assumption is that the lens is thin, so the thickness is negligible compared to $R_1$ and $R_2$.
• The principal focus (F) of a lens is the point on the principal axis where parallel rays converge (convex) or appear to diverge from (concave) after refraction.

Difficulty Level

Intermediate — requires understanding of sign conventions, formula application, and image formation scenarios, but no advanced derivations or calculus.

Common CUET Traps

• Trap: Using incorrect signs for $R_1$ and $R_2$ in Lens Maker’s formula. Avoid: Remember: for a surface, if the center of curvature lies on the side where light is going, radius is positive; otherwise negative. Convex surface facing incident light → $R_1 > 0$.
• Trap: Applying $\frac{1}{f} = \frac{1}{v} + \frac{1}{u}$ instead of $\frac{1}{v} - \frac{1}{u} = \frac{1}{f}$. Avoid: Use standard NCERT form: $\frac{1}{v} - \frac{1}{u} = \frac{1}{f}$ with sign conventions applied correctly.
• Trap: Assuming power of lens changes when immersed in water without recalculating $f$. Avoid: Power depends on medium; recalculate $f$ using $n = \frac{n_{\text{lens}}}{n_{\text{water}}}$ in Lens Maker’s formula.

Practice MCQs

1. Question: A double convex lens has radii of curvature 20 cm each and refractive index 1.5. What is its focal length in air?
   A) 10 cm
   B) 20 cm
   C) 30 cm
   D) 40 cm
   Answer: B
   Explanation: Using $\frac{1}{f} = (1.5 - 1)\left(\frac{1}{20} - \frac{1}{-20}\right) = 0.5 \times \frac{2}{20} = 0.05$ → $f = 20\,\text{cm}$.
   Why others fail: Option A assumes only one surface contributes or miscalculates the term $\left(\frac{1}{R_1} - \frac{1}{R_2}\right)$.

2. Question: An object is placed at 30 cm from a convex lens of focal length 15 cm. Where is the image formed?
   A) 10 cm
   B) 15 cm
   C) 30 cm
   D) 45 cm
   Answer: C
   Explanation: Using $\frac{1}{v} = \frac{1}{f} + \frac{1}{u} = \frac{1}{15} + \frac{1}{-30} = \frac{1}{30}$ → $v = +30\,\text{cm}$.
   Why others fail: Option B arises if $u$ is taken as positive instead of negative.

3. Question: A plano-concave lens has radius of curvature 30 cm and refractive index 1.5. What is its focal length in air?
   A) –60 cm
   B) +60 cm
   C) –30 cm
   D) +30 cm
   Answer: A
   Explanation: $R_1 = \infty$, $R_2 = +30\,\text{cm}$ (since concave surface faces incident light), so $\frac{1}{f} = (1.5 - 1)\left(0 - \frac{1}{30}\right) = -\frac{1}{60}$ → $f = -60\,\text{cm}$.
   Why others fail: Option C results from incorrect sign of $R_2$ or miscalculating $(n-1)$.

4. Question: Two thin lenses of powers +3 D and –5 D are placed in contact. What is the focal length of the combination?
   A) –0.5 m
   B) +0.5 m
   C) –2 m
   D) +2 m
   Answer: A
   Explanation: $P_{\text{total}} = +3 + (-5) = -2\,\text{D}$ → $f = \frac{1}{-2} = -0.5\,\text{m}$.
   Why others fail: Option C arises if reciprocal is taken incorrectly (e.g., $f = -2\,\text{D}$ → $f = -2\,\text{m}$).

5. Question: A convex lens is immersed in water ($n = 1.33$). Its refractive index is 1.5 and radii are 20 cm each. What happens to its focal length?
   A) Remains same
   B) Decreases
   C) Increases
   D) Becomes infinite
   Answer: C
   Explanation: In water, $n = \frac{1.5}{1.33} \approx 1.128$, so $(n - 1)$ decreases → $\frac{1}{f}$ decreases → $f$ increases.
   Why others fail: Option B is tempting if one assumes higher medium density always increases focusing power, but relative refractive index matters.

Last-Minute Revision

• ⚠️ Lens Maker’s Formula: $\frac{1}{f} = (n - 1)\left(\frac{1}{R_1} - \frac{1}{R_2}\right)$ — verify from NCERT.
• ⚠️ Lens Formula: $\frac{1}{v} - \frac{1}{u} = \frac{1}{f}$ — not plus.
• ⚠️ $u$ is always negative for real objects.
• ⚠️ Convex lens: $f > 0$; Concave lens: $f < 0$.
• ⚠️ Power $P = \frac{1}{f(\text{in m})}$ in dioptres.
• ⚠️ For equiconvex lens: $R_1 = +R$, $R_2 = -R$.
• ⚠️ For equiconcave lens: $R_1 = -R$, $R_2 = +R$.
• ⚠️ Plano-convex: one $R = \infty$, other $R > 0$.
• ⚠️ Refractive index in formula is relative: $n = n_{\text{lens}} / n_{\text{medium}}$.
• ⚠️ Thin lens assumption: thickness ≈ 0.
• ⚠️ Convex lens forms real image when object outside $F$.
• ⚠️ Concave lens always forms virtual, erect, diminished image.
• ⚠️ Object at infinity → image at $f$ for convex lens.
• ⚠️ Magnifying glass: object between $O$ and $F$ of convex lens.
• ⚠️ Sign convention: distances measured against incident light are negative.
• ⚠️ $R > 0$ if center of curvature is on the right side of surface (in direction of light).
• ⚠️ Combination of lenses: $P = P_1 + P_2$, $f_{\text{eq}} = \frac{1}{P}$.
• ⚠️ When lens immersed in liquid, recalculate $n$ and $f$ — verify from NCERT.