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Wave Optics formulas

12 formulas and 12 common traps for MHT-CET Physics Wave Optics, grouped by subtopic.

Full notes with worked examples

Wavefronts, Huygens' Principle and Coherent Sources

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Wavefronts and the Wave Theory

Wavelength in a medium

λmedium=λvacuumμ,N=μtλ\lambda_{\text{medium}} = \frac{\lambda_{\text{vacuum}}}{\mu}, \qquad N = \frac{\mu t}{\lambda}

Coherent Sources

Condition for a steady pattern

ν1=ν2,ϕ1−ϕ2=constant\nu_1 = \nu_2, \quad \phi_1 - \phi_2 = \text{constant}

Common traps

Crediting Huygens with corpuscles

'Different colours are due to different sizes of the corpuscles' is Newton's picture, and the statement a 'NOT correct for Huygens' question wants.

Expecting red and blue fringes side by side

Each filter passes a different frequency, so there is no pattern at all — not separate red and blue fringe systems, which the options offer.

Young's Double Slit: Fringe Width, Fringe Positions and Shifts

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Fringe width

β=λDd\beta = \frac{\lambda D}{d}

Positions of Bright and Dark Fringes

Fringe positions

ybright=nβ,ydark=(n−12)βy_{\text{bright}} = n\beta, \qquad y_{\text{dark}} = \left(n - \tfrac{1}{2}\right)\beta

A Transparent Sheet Over One Slit

Fringe shift

Δy=(μ−1)tDd,N=(μ−1)tλ\Delta y = \frac{(\mu - 1)tD}{d}, \qquad N = \frac{(\mu - 1)t}{\lambda}

Common traps

Multiplying the medium's μ

In water the WAVELENGTH is divided by μ, so the fringe width is divided by μ too. Multiplying gives wider fringes, the wrong way round.

Placing the nth dark fringe at nβ

The first dark fringe is only HALF a fringe width out. The 4th dark is at 3.5β3.5\beta, not 4β4\beta — and the difference is always one of the options.

Using μt instead of (μ − 1)t

The sheet REPLACES a thickness t of air, so the extra path is (μ−1)t(\mu - 1)t. Using μt\mu t makes the shift about three times too big.

Intensity in an Interference Pattern

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Intensity From the Phase Difference

Two equal coherent sources

I=4I0cos⁡2ϕ2,ϕ=2πλ ΔxI = 4I_0\cos^2\frac{\phi}{2}, \qquad \phi = \frac{2\pi}{\lambda}\,\Delta x

Unequal Sources: Maxima, Minima and Contrast

Maxima and minima

Imax⁡Imin⁡=(I1+I2I1−I2)2\frac{I_{\max}}{I_{\min}} = \left(\frac{\sqrt{I_1} + \sqrt{I_2}}{\sqrt{I_1} - \sqrt{I_2}}\right)^2

Common traps

Forgetting to halve the phase

I∝cos⁡2ϕ2I \propto \cos^2\frac{\phi}{2}. At λ4\frac{\lambda}{4}, ϕ=90∘\phi = 90^\circ and the answer is 12\frac{1}{2}, not cos⁡290∘=0\cos^2 90^\circ = 0.

Taking the intensity ratio as the max–min ratio

Sources in the ratio 9 : 1 do NOT give fringes in the ratio 9 : 1. Take square roots (3 and 1), add and subtract, then square: 4 : 1.

Single-Slit Diffraction and Resolving Power

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Minima and the Central Maximum

Single-slit minima

asin⁡θ=nλ,Wcentral=2λDaa\sin\theta = n\lambda, \qquad W_{\text{central}} = \frac{2\lambda D}{a}

Secondary maxima

asin⁡θ=(n+12)λa\sin\theta = \left(n + \tfrac{1}{2}\right)\lambda

Resolving Power

Angular resolution

Δθ=1.22λD\Delta\theta = \frac{1.22\lambda}{D}

Common traps

Half-width or full width

λDa\frac{\lambda D}{a} is the distance from the centre to the first minimum. 'Between the first minima on either side' is twice that, 2λDa\frac{2\lambda D}{a}.

Using nλ for a maximum

In SINGLE-slit diffraction asin⁡θ=nλa\sin\theta = n\lambda gives the MINIMA — the opposite of the double-slit rule. Secondary maxima need the extra half.

A longer wavelength sees finer detail

Resolving power goes as 1λ\frac{1}{\lambda}: longer waves blur more. 'Increase the wavelength' is the planted wrong way to improve a microscope.

Polarisation: Malus' Law and Brewster's Law

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Malus' Law and Chains of Polaroids

Malus' law

I=Iincos⁡2θI = I_{\text{in}}\cos^2\theta

Brewster's Angle

Brewster's law

tan⁡ip=μ=cv\tan i_p = \mu = \frac{c}{v}

Common traps

Measuring every angle from the first polaroid

Malus' law uses the angle between a polaroid and the ONE BEFORE it. At 60° then 90° from the first, the second step is only 30°: the factor is cos⁡230∘\cos^2 30^\circ, not cos⁡290∘=0\cos^2 90^\circ = 0.

Using sin instead of tan

Brewster's law is tan⁡ip=μ\tan i_p = \mu; sin⁡ic=1μ\sin i_c = \frac{1}{\mu} is the critical angle. The two get swapped in the options.

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