Refraction through Lenses (Lens Maker's Formula & Lens Formula)

PHYRefraction through Lenses

Refraction at the two spherical surfaces of a thin lens is combined into two master equations: the Lens Maker's Formula, which links focal length to the surface radii and the relative refractive index, and the Thin Lens Formula, which relates object distance, image distance and focal length. Together they let you design a lens (find $R$ or $\mu$ ), locate any image, find magnification and power, and combine lenses in contact -- the bread-and-butter of ISC ray-optics numericals.

This topic is heavily examined because almost every derived result (power, combinations, silvered lenses, behaviour in a liquid) flows from these two formulas plus a careful Cartesian sign convention.

Lens Maker's Formula (thin lens in air)

\dfrac{1}{f} = (\mu - 1)\left(\dfrac{1}{R_1} - \dfrac{1}{R_2}\right)

f

= focal length,

\mu

= refractive index of the lens material relative to the surroundings,

R_1, R_2

= radii of curvature of the first and second surfaces met by the light (signed by the Cartesian convention). For a thin lens in a medium of index

\mu_m

, replace

(\mu-1)

\left(\dfrac{\mu}{\mu_m}-1\right)

, where

\mu

is the lens index.

Thin Lens Formula

\dfrac{1}{v} - \dfrac{1}{u} = \dfrac{1}{f}

u

= object distance,

v

= image distance,

f

= focal length, all measured from the optic centre with the Cartesian sign convention (distances measured against the incident light are negative); for a converging lens

f > 0

, for a diverging lens

f < 0

Linear magnification (lens)

m = \dfrac{v}{u} = \dfrac{h_i}{h_o}

m

= magnification,

h_o, h_i

= object and image heights; for a lens

m = v/u

(note: no minus sign, unlike a mirror).

m > 0

means an erect (virtual) image,

m < 0

means an inverted (real) image, and

|m| > 1

means enlarged.

Power and lenses in contact

P = \dfrac{1}{f}, \qquad \dfrac{1}{F} = \dfrac{1}{f_1} + \dfrac{1}{f_2} \ \Rightarrow\ P = P_1 + P_2

P

= power in dioptres (D) with

f

in metres;

F

= combined focal length of two thin lenses placed in contact. Powers (and reciprocal focal lengths) add algebraically with sign.

Silvered lens (equivalent mirror)

\dfrac{1}{F} = \dfrac{2}{f_l} + \dfrac{1}{f_m}

F

= focal length of the equivalent mirror,

f_l

= focal length of the lens part (light passes through it twice, hence the factor 2),

f_m

= focal length of the reflecting silvered surface treated as a mirror; a plane silvered face gives

\dfrac{1}{f_m}=0

. After finding

F

, use the mirror formula

\dfrac{1}{v}+\dfrac{1}{u}=\dfrac{1}{F}

The lens formula assumes a thin lens, paraxial (near-axis, small-angle) rays, and the same medium on both sides; thickness is neglected, so both refractions are treated as occurring at the single optic centre.
Sign of the radii: for a biconvex lens the first surface is convex to the incoming light ( $R_1 > 0$ ) and the second is concave to it ( $R_2 < 0$ ), so $\left(\dfrac{1}{R_1} - \dfrac{1}{R_2}\right)$ is positive and $f$ comes out positive. For an equiconvex (equal radii) lens $R_1 = R$ , $R_2 = -R$ , giving $\dfrac{1}{f} = (\mu-1)\dfrac{2}{R}$ .
A converging lens has positive focal length and positive power; a diverging lens has negative focal length and negative power. A power of $+5\,\text{D}$ means $f = +0.20\,\text{m} = +20\,\text{cm}$ .
Immersing a lens in a denser medium changes the factor from $(\mu-1)$ to $\left(\dfrac{\mu}{\mu_m}-1\right)$ , which weakens, and can even reverse, the lens: if $\mu_m > \mu$ a converging lens turns diverging (its $f$ changes sign).
For lenses in contact, focal lengths combine reciprocally while powers add directly with their signs; a strong converging lens plus a weaker diverging lens can give either net sign depending on the magnitudes.
Magnification rule of thumb: a single convex lens gives a virtual, erect, magnified image ( $m > +1$ ) only when the object is inside the focus ( $|u| < f$ ); for an object beyond the focus the image is real and inverted ( $m < 0$ ).
For a silvered lens, first find the equivalent-mirror focal length $F$ , then apply the MIRROR formula (with the $+$ sign) -- not the lens formula -- to locate the image, since the system finally reflects light back.

Worked example · 2 marks

The objective of a telescope is two thin lenses in contact, of powers

+2.0\,\text{D}

and

-1.5\,\text{D}

. Calculate the focal length of the objective.

Powers of thin lenses in contact simply add: $P = P_1 + P_2$ .
$P = (+2.0) + (-1.5) = +0.5\,\text{D}$ .
Focal length is the reciprocal of the power, in metres: $F = \dfrac{1}{P} = \dfrac{1}{0.5}$ .

Answer

F = 2\,\text{m} = 200\,\text{cm}

. The combination is still converging because the positive power wins.

Where the marks go

Sign-convention slips on the radii: students often plug both $R$ values in as positive in the Lens Maker's Formula. You must assign $R_1$ and $R_2$ by the Cartesian convention (a radius is positive when the centre of curvature lies on the outgoing-light side); a biconvex lens needs $R_1 > 0$ and $R_2 < 0$ .
Confusing $\mu$ with $(\mu - 1)$ : the focal length depends on $(\mu - 1)$ , not $\mu$ . Using $\mu = 1.5$ directly instead of $0.5$ makes $f$ come out far too short.
Forgetting that the lens formula uses a MINUS sign, $\dfrac{1}{v} - \dfrac{1}{u} = \dfrac{1}{f}$ , while the mirror formula uses PLUS, $\dfrac{1}{v} + \dfrac{1}{u} = \dfrac{1}{f}$ -- and mixing them up in silvered-lens or combination problems. Also recall that for a lens $m = v/u$ but for a mirror $m = -v/u$ .
In silvered-lens problems, forgetting the factor of 2 on the lens term (light traverses the lens twice) and treating a plane silvered face as having a finite mirror focal length instead of $\dfrac{1}{f_m}=0$ .

How the board asks it

Numericallens maker formula and thin lens formula2 mkAsked 2026
A biconvex lens has surfaces of radii of curvature $20\,\text{cm}$ and $30\,\text{cm}$ and is made of glass of refractive index $1.5$ . An object is placed $40\,\text{cm}$ in front of it. Calculate the focal length of the lens, and the position and magnification of the image.
Derive / provelens maker formula
Derive the lens maker's formula $\frac{1}{f}=(\mu-1)\left(\frac{1}{R_1}-\frac{1}{R_2}\right)$ for a thin biconvex lens by considering refraction at its two spherical surfaces, stating the assumptions made.
Numericalpower and lenses in contact2 mkAsked 2025 · 2026
A convex lens of power $+5\,\text{D}$ is placed in contact with a concave lens of power $-2\,\text{D}$ . Calculate the power and focal length of the combination, and state its nature.
Give reasonsbehaviour in a liquid1 mkAsked 2025 · 2026
A convex lens of glass $(\mu=1.5)$ behaves as a diverging lens when immersed in a certain liquid. Give reasons for this, and state the condition on the refractive index $\mu_m$ of the liquid for this to happen.
Numericalsilvered lens as equivalent mirror
A plano-convex lens of radius of curvature $20\,\text{cm}$ and $\mu=1.5$ has its plane face silvered. Find the focal length of the equivalent mirror, and locate the image of an object placed $30\,\text{cm}$ from the lens.
Diagram / graphimage formation by a convex lens
Draw a labelled ray diagram showing the formation of the image by a convex lens when the object is placed between the optic centre and the focus, and state the nature of the image formed.
Define / statewhat the focal length of a lens actually depends on5 mkAsked 2023
A passage on converging and diverging lenses is followed by: state any two factors on which the focal length of a lens depends, and state when a convex lens behaves as a diverging one.
Give reasonsthe no-parallax method for finding $f$ of a convex lens3 mkAsked 2024
For each instruction given to a student measuring the focal length of a convex lens by the no-parallax method, state whether the response is correct or incorrect, with a reason.

Practise this topic

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Written for Sublevo. Question text quoted anywhere in these notes is the Council’s and carries its year and paper; the board’s own diagrams are not reproduced.