Wave Optics — JEE Mains Physics

Formulas (1)

• Speed, wavelength and frequency in a medium · Light in a medium
  $$f = f_{0}, \qquad v = \frac{c}{\mu}, \qquad \lambda = \frac{\lambda_{0}}{\mu}, \qquad \frac{\lambda_{1}}{\lambda_{2}} = \frac{v_{1}}{v_{2}} = \frac{\mu_{2}}{\mu_{1}}$$

Reference tables (1)

Watch out for (6)

• The wave travels along the normal
  → Wavefronts, rays and Huygens' principle
• A prism does not curve a plane wave
  → Wavefronts, rays and Huygens' principle
• Photons, not waves
  → Wavefronts, rays and Huygens' principle
• Frequency never changes at a boundary
  → Speed, wavelength and frequency in a medium
• The μ ratio is upside down
  → Speed, wavelength and frequency in a medium
• Go through the vacuum value
  → Speed, wavelength and frequency in a medium

Formulas (3)

• Resultant intensity of two beams · Two coherent beams
  $$I = I_{1} + I_{2} + 2\sqrt{I_{1}I_{2}}\cos\phi, \qquad A^{2} = A_{1}^{2} + A_{2}^{2} + 2A_{1}A_{2}\cos\phi$$
• Brightest and darkest fringes · Fringe contrast
  $$\frac{I_{\max}}{I_{\min}} = \left(\frac{\sqrt{I_{1}} + \sqrt{I_{2}}}{\sqrt{I_{1}} - \sqrt{I_{2}}}\right)^{2} = \left(\frac{r + 1}{r - 1}\right)^{2}$$
• Optical path and thin films · Optical path and films
  $$\Delta = (\mu - 1)t, \qquad \Delta\phi = \frac{2\pi}{\lambda}(\mu - 1)t, \qquad 2\mu t = n\lambda \ \text{or}\ \left(n - \tfrac{1}{2}\right)\lambda$$

Watch out for (9)

• Incoherent means no cross term
  → Resultant intensity of two beams
• Add amplitudes, not intensities
  → Resultant intensity of two beams
• Watch the sign of cos φ
  → Resultant intensity of two beams
• Take the square root first
  → Brightest and darkest fringes
• Width ratio: intensity or amplitude?
  → Brightest and darkest fringes
• Only equal beams give true darkness
  → Brightest and darkest fringes
• Count the reversals first
  → Optical path and thin films
• Transmission is the opposite of reflection
  → Optical path and thin films
• The extra path is (μ − 1)t
  → Optical path and thin films

Formulas (3)

• Fringe width and what it depends on · Fringe width
  $$\beta = \frac{\lambda D}{d}, \qquad \theta = \frac{\lambda}{d}, \qquad \Delta\beta = \frac{\lambda\,\Delta D}{d}$$
• The apparatus in a liquid · Fringes in a liquid
  $$\beta' = \frac{\beta}{\mu} = \frac{\lambda D}{\mu d}, \qquad \theta' = \frac{\lambda}{\mu d}$$
• Fringe positions and two wavelengths · Positions and coincidence
  $$y_{\text{bright}} = \frac{n\lambda D}{d}, \qquad y_{\text{dark}} = \left(n - \tfrac{1}{2}\right)\frac{\lambda D}{d}, \qquad n_{1}\lambda_{1} = n_{2}\lambda_{2}$$

Watch out for (9)

• Angular width ignores the screen
  → Fringe width and what it depends on
• β falls as d rises
  → Fringe width and what it depends on
• A percentage change is not a ratio
  → Fringe width and what it depends on
• Divide by μ, do not multiply
  → The apparatus in a liquid
• The given wavelength is the one in air
  → The apparatus in a liquid
• Angles shrink too
  → The apparatus in a liquid
• The ratio flips
  → Fringe positions and two wavelengths
• Dark fringes are half a width short
  → Fringe positions and two wavelengths
• Reduce the ratio fully
  → Fringe positions and two wavelengths

Formulas (3)

• Path difference from the geometry · Path difference
  $$\Delta x = \frac{yd}{D}, \qquad \Delta x_{\text{opposite a slit}} = \frac{d^{2}}{2D}, \qquad \sqrt{D^{2} + d^{2}} - D \approx \frac{d^{2}}{2D}$$
• Intensity from path and phase · Double-slit intensity
  $$I = 4I_{0}\cos^{2}\frac{\phi}{2} = I_{\max}\cos^{2}\frac{\pi\,\Delta x}{\lambda}, \qquad \phi = \frac{2\pi}{\lambda}\Delta x$$
• A thin sheet over one slit · Sheet shift
  $$\Delta y = \frac{(\mu - 1)tD}{d} = \frac{(\mu - 1)t}{\lambda}\,\beta$$

Watch out for (9)

• Opposite a slit is y = d/2
  → Path difference from the geometry
• yd/D needs a far screen
  → Path difference from the geometry
• Know which I₀ the question means
  → Path difference from the geometry
• Half the phase inside the cosine
  → Intensity from path and phase
• I₀ for one slit means I_max = 4I₀
  → Intensity from path and phase
• Path is not phase
  → Intensity from path and phase
• Towards the covered slit
  → A thin sheet over one slit
• Use μ − 1, not μ
  → A thin sheet over one slit
• The fringes do not narrow
  → A thin sheet over one slit

Formulas (3)

• Dark fringes of a single slit · Single-slit minima
  $$a\sin\theta = n\lambda, \qquad y_{n} = \frac{n\lambda D}{a}, \qquad y_{3} - y_{1} = \frac{2\lambda D}{a}$$
• Width of the central maximum · Central maximum
  $$\theta_{\text{width}} = \frac{2\lambda}{a}, \qquad W = \frac{2\lambda D}{a}\ \left(\text{or } \frac{2\lambda f}{a}\right), \qquad N = \frac{2d}{a}$$
• Resolving power of a telescope and a microscope · Resolution
  $$\Delta\theta = \frac{1.22\lambda}{D}, \qquad \text{RP}_{\text{microscope}} = \frac{2\mu\sin\theta}{1.22\lambda}$$

Watch out for (9)

• nλ is dark for a single slit
  → Dark fringes of a single slit
• First to third is two steps
  → Dark fringes of a single slit
• Angle of one minimum or the angle between two?
  → Dark fringes of a single slit
• Twice λD/a, not λD/a
  → Width of the central maximum
• a, not d
  → Width of the central maximum
• Large angles need the sine
  → Width of the central maximum
• Limit and power are opposites
  → Resolving power of a telescope and a microscope
• Keep the 1.22
  → Resolving power of a telescope and a microscope
• Oil helps a microscope, not a telescope
  → Resolving power of a telescope and a microscope

Formulas (3)

• One polaroid after another: Malus' law · Malus' law
  $$I_{1} = \frac{I_{0}}{2}, \qquad I_{2} = I_{1}\cos^{2}\theta$$
• Chains of polaroids · A sheet between crossed polaroids
  $$I = \frac{I_{0}}{2}\cos^{2}\theta\,\sin^{2}\theta = \frac{I_{0}}{8}\sin^{2}2\theta, \qquad I_{n} = \frac{I_{0}}{2^{n}}\ (45^{\circ}\ \text{steps})$$
• Polarisation by reflection: Brewster's law · Brewster's law
  $$\tan i_{B} = \frac{\mu_{2}}{\mu_{1}}, \qquad i_{B} + r = 90^{\circ}$$

Watch out for (9)

• The first sheet halves, it does not use cos²
  → One polaroid after another: Malus' law
• Polarised light is not halved
  → One polaroid after another: Malus' law
• Cos squared, not cos
  → One polaroid after another: Malus' law
• Angle to the previous sheet
  → Chains of polaroids
• Count the halving once
  → Chains of polaroids
• Two angles can give the same output
  → Chains of polaroids
• Tangent, not sine
  → Polarisation by reflection: Brewster's law
• Refraction is the complement
  → Polarisation by reflection: Brewster's law
• Reverse direction, reverse ratio
  → Polarisation by reflection: Brewster's law