Lenses — Revision Notes

⚡ 30-Second Revision

Lens Formula: — $rac{1}{v} - \frac{1}{u} = \frac{1}{f}$
Magnification: — $m = \frac{h'}{h} = \frac{v}{u}$
Lens Maker's Formula: — rac{1}{f} = (\frac{n_2}{n_1} - 1) left( \frac{1}{R_1} - \frac{1}{R_2} \right)
Power of Lens: — $P = \frac{1}{f \text{ (in meters)}}$ (Unit: Dioptre, D)
Combination of Lenses (in contact): — $P_{eq} = P_1 + P_2$, $rac{1}{F_{eq}} = \frac{1}{f_1} + \frac{1}{f_2}$
Convex Lens: — Converging, $f$ positive, forms real/virtual images.
Concave Lens: — Diverging, $f$ negative, always forms virtual, erect, diminished images.
Sign Convention: — Cartesian system (light from left, distances from optical centre, right is positive, left is negative, up is positive, down is negative).

2-Minute Revision

Lenses are transparent devices that refract light to form images. The two main types are convex (converging) and concave (diverging). Convex lenses are thicker in the middle, have a positive focal length, and can form both real and virtual images.

Concave lenses are thinner in the middle, have a negative focal length, and always form virtual, erect, and diminished images. The lens formula $rac{1}{v} - \frac{1}{u} = \frac{1}{f}$ relates object distance ($u$), image distance ($v$), and focal length ($f$), with strict adherence to Cartesian sign conventions.

Magnification $m = \frac{v}{u}$ tells us the image size and orientation. The Lens Maker's Formula rac{1}{f} = (\frac{n_{lens}}{n_{medium}} - 1) left( \frac{1}{R_1} - \frac{1}{R_2} \right) explains how focal length depends on the lens material and surrounding medium.

Power of a lens $P = 1/f$ (in meters) is measured in dioptres (D), with positive power for convex and negative for concave lenses. For lenses in contact, powers simply add: $P_{eq} = P_1 + P_2$. Remember that a convex lens can act as a concave lens if immersed in a denser medium.

5-Minute Revision

Lenses are crucial optical components that bend light through refraction. We primarily deal with thin lenses, where thickness is negligible. The two main types are convex (converging) and concave (diverging).

Convex lenses are thicker at the center, have a positive focal length ($f > 0$), and converge parallel rays to a real focus. They can form real, inverted images (object beyond F) or virtual, erect, magnified images (object between F and O).

Concave lenses are thinner at the center, have a negative focal length ($f < 0$), and diverge parallel rays, which appear to come from a virtual focus. They always form virtual, erect, and diminished images.

Key Formulas:

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Lens Formula: — $rac{1}{v} - \frac{1}{u} = \frac{1}{f}$. Remember $u$ is always negative for real objects. $v$ positive means real image (opposite side), $v$ negative means virtual image (same side).
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Magnification: — $m = \frac{h'}{h} = \frac{v}{u}$. $m$ positive means erect image, $m$ negative means inverted. $|m|>1$ is magnified, $|m|<1$ is diminished.
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Lens Maker's Formula: — rac{1}{f} = (\frac{n_{lens}}{n_{medium}} - 1) left( \frac{1}{R_1} - \frac{1}{R_2} \right). This is vital for understanding how focal length changes when a lens is placed in a different medium. If $n_{medium} > n_{lens}$, a convex lens becomes diverging.
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Power of a Lens: — $P = \frac{1}{f}$ (where $f$ is in meters). Unit is Dioptre (D). Convex lenses have positive power, concave lenses have negative power. Higher power means stronger bending.
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Combination of Lenses: — For lenses in contact, $P_{eq} = P_1 + P_2 + dots$ and $rac{1}{F_{eq}} = \frac{1}{f_1} + \frac{1}{f_2} + dots$. For two lenses separated by distance $d$, $rac{1}{F_{eq}} = \frac{1}{f_1} + \frac{1}{f_2} - \frac{d}{f_1 f_2}$.

Example: An object is placed $20,\text{cm}$ from a convex lens ($f = +15,\text{cm}$). Find image position and magnification. $u = -20,\text{cm}$, $f = +15,\text{cm}$. $rac{1}{v} - \frac{1}{-20} = \frac{1}{15} implies \frac{1}{v} = \frac{1}{15} - \frac{1}{20} = \frac{4-3}{60} = \frac{1}{60}$. So, $v = +60,\text{cm}$ (real image). $m = \frac{v}{u} = \frac{+60}{-20} = -3$ (inverted, magnified).

Prelims Revision Notes

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Lens Types:

* Convex (Converging): Thicker middle, positive focal length ($f > 0$). Forms real/virtual images. * Concave (Diverging): Thinner middle, negative focal length ($f < 0$). Always forms virtual, erect, diminished images.

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Sign Conventions (Cartesian):

* Light from left. Optical centre is origin. * Distances right of O: positive. Distances left of O: negative. * Heights above principal axis: positive. Heights below: negative. * $u$ (object distance) is always negative for real objects.

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Lens Formula: — $rac{1}{v} - \frac{1}{u} = \frac{1}{f}$

* $v > 0 implies$ Real image (opposite side of object). * $v < 0 implies$ Virtual image (same side as object).

1
Magnification: — $m = \frac{h'}{h} = \frac{v}{u}$

* $m > 0 implies$ Erect image. * $m < 0 implies$ Inverted image. * $|m| > 1 implies$ Magnified. $|m| < 1 implies$ Diminished. $|m| = 1 implies$ Same size.

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Image Formation by Convex Lens:

* Object at $infty$: Real, inverted, highly diminished, at $F_2$. * Object beyond $2F_1$: Real, inverted, diminished, between $F_2$ and $2F_2$. * Object at $2F_1$: Real, inverted, same size, at $2F_2$. * Object between $F_1$ and $2F_1$: Real, inverted, magnified, beyond $2F_2$. * Object at $F_1$: Real, inverted, highly magnified, at $infty$. * Object between $F_1$ and O: Virtual, erect, magnified, same side as object.

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Image Formation by Concave Lens:

* Object at $infty$: Virtual, erect, highly diminished, at $F_1$. * Object anywhere between $infty$ and O: Virtual, erect, diminished, between $F_1$ and O.

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Lens Maker's Formula: — rac{1}{f} = (\frac{n_{lens}}{n_{medium}} - 1) left( \frac{1}{R_1} - \frac{1}{R_2} \right)

* If $n_{medium} > n_{lens}$, a convex lens acts as a concave lens (and vice versa).

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Power of a Lens: — $P = \frac{1}{f \text{ (in meters)}}$. Unit: Dioptre (D).

* Convex lens: $P > 0$. Concave lens: $P < 0$.

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Combination of Lenses:

* In contact: $P_{eq} = P_1 + P_2 + dots$, or $rac{1}{F_{eq}} = \frac{1}{f_1} + \frac{1}{f_2} + dots$. * **Separated by $d$:** $rac{1}{F_{eq}} = \frac{1}{f_1} + \frac{1}{f_2} - \frac{d}{f_1 f_2}$.

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Aberrations: — Chromatic (color dispersion), Spherical (different focal points for marginal/paraxial rays).

Vyyuha Quick Recall

Convex Positive Focal Length Real Images (mostly), Concave Negative Focal Length Virtual Erect Diminished (always).