Ray Optics — JEE Mains Physics Formulas and Traps

Plane and Spherical Mirrors

Reflection at a plane mirror

\delta = 180^{\circ} - 2i, \qquad \hat b = \hat a - 2(\hat a\cdot\hat n)\hat n, \qquad n = \frac{360^{\circ}}{\theta} - 1

The mirror formula and magnification

Mirror formula

\frac{1}{v} + \frac{1}{u} = \frac{1}{f}, \qquad f = \frac{R}{2}, \qquad m = -\frac{v}{u} = \frac{f}{f - u}

Image speed for a moving object

Image speed along the axis

\frac{dv}{dt} = -m^{2}\frac{du}{dt}, \qquad m = \frac{f}{f - u}

Common traps

Deviation is not the angle of reflection

A ray reflected at 35° to the normal is turned through 180° − 70° = 110°, not 35° or 70°. Deviation is measured from the ray's original direction.

Moving the mirror is not moving the object

With the object fixed, a mirror moved by d shifts the image by 2d. With the mirror fixed, an object moved by d shifts the image by d. Read which one moves.

A plane-mirror image is erect

Lateral inversion swaps left and right; it does not turn the image upside down. The image is virtual, erect and the same size, so m = +1.

m = −v/u for a mirror, v/u for a lens

The mirror formula has a plus sign and its magnification a minus sign. Using the lens form m = v/u for a mirror makes every real image come out erect.

An erect, smaller image means a convex mirror

For a real object, a concave mirror never gives an erect diminished image. An erect image smaller than the object can come only from a convex mirror, with f > 0.

A mirror's focal length does not depend on the medium

f = R/2 contains no refractive index. Dipping a mirror in water changes nothing; dipping a lens in water does.

Two positions, two kinds of image

When the same image size appears at two object positions, one image is real and one virtual. Use m = −k for one and m = +k for the other, never the same sign twice.

Along the axis, speed scales with m², not m

Differentiating the mirror formula gives dv/dt = −m² du/dt. Using m alone gives an image speed that is too large for a diminishing convex mirror.

A long rod is not a short object

The rule that the image length is m² times the rod's length holds only when the rod is short compared with its distance from F. For a long rod, image each end.

Use the speed relative to the mirror

When the mirror rides on one car and the object is another car, du/dt is the rate at which the gap closes: their relative speed.

Refraction at Plane Surfaces and Apparent Depth

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Snell's law and the parallel slab

Snell's law and lateral shift

n_1\sin i = n_2\sin r, \qquad n = \frac{c}{v}, \qquad d = \frac{t\sin(i - r)}{\cos r}

Apparent depth and the normal shift

Apparent depth

d_{\text{app}} = \frac{d}{\mu}, \qquad \text{shift} = d\left(1 - \frac{1}{\mu}\right), \qquad d_{\text{app}} = \sum \frac{d_i}{\mu_i}

Common traps

Angles are measured from the normal

A ray 'at 30° with the surface' has an angle of incidence of 60°. Putting 30° into Snell's law gives a wrong index, and that wrong value is usually one of the options.

A slab shifts a ray but does not turn it

The emergent ray is parallel to the incident one; only a sideways shift remains. The bending i − r at the first face is undone at the second.

Refractive index is not mass density

Refractive index measures how much light slows down, n = c/v. A medium with a larger mass density does not, for that reason, have a proportionally larger refractive index.

The shift is not the apparent depth

A question may give the shift d(1 − 1/μ) or the apparent depth d/μ. With μ = 4/3 they are d/4 and 3d/4. Read which one is stated before solving.

Never average the indices of a stack

For layered liquids, divide each layer's own thickness by its own μ and add the results. A single averaged μ gives the wrong apparent depth.

Looking up multiplies, looking down divides

An object in water seen from air appears at d/μ. An object in air seen from under water appears at μd, farther away than it is.

Critical Angle and Total Internal Reflection

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Critical angle

\sin C = \frac{n_{\text{rarer}}}{n_{\text{denser}}} = \frac{v_{\text{denser}}}{v_{\text{rarer}}}, \qquad n = \sqrt{\mu_r\varepsilon_r}

Total internal reflection in tanks, blocks and prisms

Circle of light and a coated face

r = h\tan C = \frac{h}{\sqrt{\mu^{2} - 1}}, \qquad \sin C = \frac{n_2}{n}\ (\text{coated face})

Common traps

No total reflection from rarer to denser

Light entering a denser medium bends towards the normal and always gets through. Total internal reflection needs the light to start in the denser medium.

The slower medium is the denser one

Given speeds, the medium where light is slower has the larger refractive index. The critical angle has the slower speed on top: sin C = v(slow)/v(fast).

At i = C the light is not yet trapped

At exactly the critical angle the ray grazes along the surface. Total internal reflection needs an angle of incidence greater than C, so the condition is a strict inequality.

The circle's radius uses tan C

The edge ray leaves the lamp at C to the vertical, so it travels h tan C sideways before reaching the surface. Using sin C gives a circle that is too small.

A coating raises the critical angle

With a film of index n₂ on a face, sin C = n₂/n, which is larger than 1/n. A ray that was totally reflected in air may escape through the coated face.

A 'minimum index for total reflection' may be a maximum

Total reflection at a coated face gets harder as n₂ rises. The boundary value is the largest n₂ that still works; check the direction of the inequality before choosing.

Refraction at a Spherical Surface and the Lens-Maker's Formula

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Refraction at a single spherical surface

Single spherical surface

\frac{\mu_2}{v} - \frac{\mu_1}{u} = \frac{\mu_2 - \mu_1}{R}, \qquad m = \frac{\mu_1 v}{\mu_2 u}

Lens-maker's formula

\frac{1}{f} = (\mu - 1)\left(\frac{1}{R_1} - \frac{1}{R_2}\right), \qquad P = \frac{1}{f}

Common traps

Each distance carries its own index

The image distance is divided into μ₂ and the object distance into μ₁. Writing 1/v − 1/u, as for a lens, ignores the two media.

The sign of R depends on where the centre is

A concave face met from outside has its centre on the incoming side, so R is negative. Using +R for every curved face can turn a virtual image into a real one.

The magnification has the indices too

For one surface m = μ₁v/(μ₂u), not v/u. Leaving out the indices gives a wrong image height.

R₂ of a biconvex lens is negative

Writing both radii as positive in 1/R₁ − 1/R₂ subtracts the curvatures instead of adding them, and f comes out far too long.

A flat face has 1/R = 0, not R = 0

A plane surface has an infinite radius. Its term 1/R vanishes, so a plano-convex lens has f = R/(μ − 1).

μ − 1 is for a lens in air

In another medium the factor becomes μ(lens)/μ(medium) − 1. Using μ − 1 for a lens in water gives a focal length that is far too short.

Thin Lens Formula and Lens Combinations

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Thin lens formula and magnification

Thin lens formula

\frac{1}{v} - \frac{1}{u} = \frac{1}{f}, \qquad m = \frac{v}{u} = \frac{f}{f + u}, \qquad P = \frac{1}{f}

Combinations of lenses

Lenses in combination

P = P_1 + P_2 - dP_1P_2, \qquad m = m_1 m_2

Common traps

The lens formula has a minus sign

A lens uses 1/v − 1/u = 1/f and a mirror 1/v + 1/u = 1/f. Swapping them gives the wrong sign, or a wrong size altogether.

A concave lens never forms a real image of a real object

With f < 0 and u < 0, 1/v = 1/f + 1/u is negative: the image is always virtual, erect and inside F. A statement placing it at a real point on the far side is false.

A long or slanted object needs two magnifications

Heights across the axis scale by m; lengths along it scale by about m² only when the object is short. For anything longer, find the image of each end.

A virtual object has u > 0

When the first lens's image lies beyond the second lens, the object for the second lens is on its outgoing side. Taking u as negative there puts the final image on the wrong side.

Measure from the next lens

Before using the second lens, subtract the separation. The first image's distance from lens 1 is not its distance from lens 2.

Powers add only in contact

For lenses a distance d apart, P = P₁ + P₂ − dP₁P₂. Adding the powers alone ignores the gap.

Lenses in a Medium, Cut Lenses and Silvered Lenses

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A lens in a liquid

Lens in a medium

\frac{1}{f_m} = \left(\frac{\mu_l}{\mu_m} - 1\right)\left(\frac{1}{R_1} - \frac{1}{R_2}\right), \qquad \frac{f_m}{f_{\text{air}}} = \frac{\mu_l - 1}{\mu_l/\mu_m - 1}

A silvered lens as a mirror

Silvered lens

P = 2P_L + P_M, \qquad F = \frac{1}{P}, \qquad F = \frac{f_L}{2}\ (\text{plane face silvered})

Focal length of the pieces of a cut lens

How the lens is cut	Each piece is	Focal length of a piece	Power of a piece
Along a plane containing the principal axis	Half of the same lens, both curved faces kept	$f$	$P$
Across, perpendicular to the axis, through the centre	A plano-convex lens	$2f$	$P/2$
Along the axis, then one half across it	A plano-convex quarter	$2f$	$P/2$
Half the lens covered, not cut	The whole lens, with less light	$f$	$P$ The image is complete, only dimmer.
Two plano-convex halves put back together	The original lens	$f$	$P$

For an equiconvex lens of focal length f and power P. Only a cut across the axis changes the focal length.

Common traps

The radii stay; only the factor changes

Immersing a lens does not change its shape. Find the curvature factor from the air data, then multiply it by the new μ(lens)/μ(medium) − 1.

A convex lens can diverge

If the liquid is optically denser than the glass, μ(lens)/μ(medium) − 1 is negative and the converging lens becomes a diverging one.

Use the ratio of the factors

The focal length in water is neither f(air) × μ(water) nor f(air)/μ(water). It follows from the ratio of the two (relative index − 1) factors.

A smaller lens is not a weaker lens

Cutting along the axis halves the size, not the power. The focal length depends only on the curvatures and on μ.

Across the axis, the power halves

Each plano-convex piece has one curved face instead of two, so its power is half and its focal length twice the original.

The lens counts twice

Light crosses the lens on the way in and again on the way out, so the lens's power enters as 2P(lens), not P(lens).

A silvered plane face adds no power

A plane mirror has zero power, so with the flat face silvered the system's focal length is just half the lens's.

The silvered curved face is concave from inside

Light inside the glass sees a silvered convex surface as a concave mirror of radius R, with power 2/R.

Prisms: Minimum Deviation, Grazing Emergence and Dispersion

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Prism and minimum deviation

r_1 + r_2 = A, \qquad \delta = i + e - A, \qquad \mu = \frac{\sin\frac{A + \delta_m}{2}}{\sin\frac{A}{2}}

Grazing emergence from a prism

Grazing emergence

\sin r_2 = \frac{1}{\mu}\ (e = 90^{\circ}), \qquad r_1 = A - r_2, \qquad \sin i = \mu\sin r_1

Thin prisms and dispersion

Thin prism and dispersion

\delta = (\mu - 1)A, \qquad (\mu_1 - 1)A_1 = (\mu_2 - 1)A_2, \qquad \omega = \frac{\mu_v - \mu_r}{\mu_y - 1}

Common traps

Use A/2 inside, not A

At minimum deviation each internal angle is A/2. Putting A into Snell's law at the first face doubles the refraction angle.

Through a prism the deviation is i + e − A

i − r is the bending at one face only. A prism bends the ray at both faces, and i + e − A adds the two.

Two angles of incidence give the same deviation

Except at the minimum, each deviation occurs for a pair of values of i, the ray and its reverse. The graph of deviation against i is a curve with one minimum, not a straight line.

Grazing emergence fixes r₂, not i

e = 90° sets the angle inside the second face to the critical angle. The first-face angles then follow from r₁ = A − r₂ and Snell's law.

A coating changes the critical angle at that face only

A film of index n₂ on the exit face makes sin C' = n₂/μ there. The entry face still has the air critical angle.

The prisms face opposite ways

Dispersion without deviation needs the two prisms placed opposite ways. The condition equates the sizes of their mean deviations, (μ₁ − 1)A₁ = (μ₂ − 1)A₂.

δ = (μ − 1)A is for thin prisms only

It holds for small prism angles and small angles of incidence. For a 60° prism use the minimum-deviation formula instead.

Red bends least

Red has the longest wavelength and the smallest refractive index in glass. It is deviated least, focuses farthest from a lens, and sits on top of the primary rainbow.

Optical Instruments and the Eye

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Magnification of microscopes and telescopes

Instrument	First element	Magnification, final image at infinity	Distance between the elements
Simple microscope	One short-focus convex lens	$D/f$	Only one lens
Compound microscope	Short-focus objective lens	$\dfrac{L}{f_o}\cdot\dfrac{D}{f_e}$	Set by the tube length L
Refracting telescope	Long-focus objective lens	$f_o/f_e$	$f_o + f_e$
Reflecting telescope	Concave mirror, $f_o = R/2$	$f_o/f_e$	A secondary mirror folds the light to an eyepiece outside the tube
Beam expander	Convex lens of focal length $f_1$	Beam width multiplied by $f_2/f_1$	$f_1 + f_2$

D = 25 cm, the least distance of distinct vision. Normal adjustment puts the final image at infinity.

Resolving power and defects of vision

Defect	What goes wrong	Correcting lens	How to find the lens
Myopia (short sight)	Far point closer than infinity	Concave	$f = -(\text{far-point distance})$
Hypermetropia (long sight)	Near point farther than 25 cm	Convex	It images an object at 25 cm onto the near point
Presbyopia	The near point recedes with age as focusing weakens	Convex for reading, often in a bifocal	As for hypermetropia
Astigmatism	Unequal curvature of the cornea; lines in one direction blur	Cylindrical	Shaped to correct the faulty plane only

A reading glass is tested at the near point: an object at 25 cm must appear at the person's own near point.

Common traps

A telescope's M is fₒ/fₑ, not fₑ/fₒ

The objective has the long focal length. Inverting the ratio gives a magnification below 1, which is never an option for a working telescope.

A wider objective does not raise the magnification

A larger aperture lets in more light and resolves more detail, but in normal adjustment M depends only on the focal lengths.

Normal adjustment uses D/fₑ, not 1 + D/fₑ

The extra 1 appears only when the final image is at the near point. Read where the final image is before choosing the formula.

Resolving power is not magnification

A stronger eyepiece enlarges the blur too. Only a wider aperture, a shorter wavelength or, in a microscope, a denser medium in front of the objective resolves finer detail.

A reading glass forms a virtual image

It images an object at 25 cm onto the person's near point, on the same side as the object. So v is negative and farther from the lens than 25 cm.

Blurred is not the same as distorted

Distant objects that look blurred point to myopia. Lines that look uneven or distorted point to astigmatism, which needs a cylindrical lens.

Ray Optics formulas

Plane and Spherical Mirrors

The mirror formula and magnification

Image speed for a moving object

Common traps

Refraction at Plane Surfaces and Apparent Depth

Snell's law and the parallel slab

Apparent depth and the normal shift

Common traps

Critical Angle and Total Internal Reflection

Total internal reflection in tanks, blocks and prisms

Common traps

Refraction at a Spherical Surface and the Lens-Maker's Formula

Refraction at a single spherical surface

Common traps

Thin Lens Formula and Lens Combinations

Thin lens formula and magnification

Combinations of lenses

Common traps

Lenses in a Medium, Cut Lenses and Silvered Lenses

A lens in a liquid

A silvered lens as a mirror

Focal length of the pieces of a cut lens

Common traps

Prisms: Minimum Deviation, Grazing Emergence and Dispersion

Grazing emergence from a prism

Thin prisms and dispersion

Common traps

Optical Instruments and the Eye

Magnification of microscopes and telescopes

Resolving power and defects of vision

Common traps

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