Optics — Complete Formula Sheet (Ray & Wave Optics)
Concise formulas, units, and quick memory tips for JEE / Class 12 revision. MathJax enabled.
Quick facts & constants
\(c = 3.00\times10^8\ \mathrm{m\,s^{-1}}\)
\(n = \dfrac{c}{v}\) — dimensionless
\(\lambda = \dfrac{v}{f} = \dfrac{c}{nf}\)
Geometrical (Ray) Optics — Core formulas
Mirrors & Thin lenses
\[ \dfrac{1}{f} = \dfrac{1}{v} + \dfrac{1}{u} \] where \(f\) = focal length, \(u\) = object distance, \(v\) = image distance. (Sign convention needed.)
Magnification (linear):
\(m = \dfrac{h_i}{h_o} = -\dfrac{v}{u}\)
Radius \(R\) and focal length: \(f = \dfrac{R}{2}\) (spherical mirror).
Lens power:
\(P = \dfrac{1}{f\ (\text{in meters})}\) dioptre (D).
Lens-maker & refractive formulas
\[ \dfrac{1}{f} = (n-1)\left(\dfrac{1}{R_1} - \dfrac{1}{R_2}\right) \] where \(R_1\) & \(R_2\) are radii of curvature (signs per convention), \(n\) refractive index of lens medium.
Refraction (Snell's law):
\(n_1\sin\theta_1 = n_2\sin\theta_2\)
Total internal reflection: occurs when \(n_1 > n_2\) and \(\theta_1 > \theta_c\), where \(\sin\theta_c = \dfrac{n_2}{n_1}\).
Lens combinations & optical instruments
\[ \dfrac{1}{f_\text{eq}} = \dfrac{1}{f_1} + \dfrac{1}{f_2} \quad\text{or}\quad P_\text{eq} = P_1 + P_2 \]
Microscope (approximate):
Total magnification \(M \approx M_\text{objective}\times M_\text{eyepiece}\), with \(M_\text{objective} \approx \dfrac{L}{f_o}\) and \(M_\text{eyepiece} \approx \dfrac{D}{f_e}\) where \(L\) is tube length, \(D\approx 25\ \mathrm{cm}\) near point.
Astronomical telescope (in normal adjustment):
\(M = -\dfrac{f_o}{f_e}\) (objective focal length \(f_o\), eyepiece \(f_e\)).
Wave Optics — Interference, Diffraction & Resolution
Interference (Young's double slit)
\[ \beta = \dfrac{\lambda D}{d} \] where \(\lambda\) wavelength, \(D\) screen distance, \(d\) slit separation.
Path difference for maxima/minima:
Maxima: \(\delta = m\lambda\) (constructive), Minima: \(\delta = (m+\tfrac{1}{2})\lambda\) (destructive).
Diffraction (single-slit)
\[ a\sin\theta = m\lambda \quad (m = \pm1, \pm2, \dots) \] where \(a\) is slit width.
First minima (small angle): \(\theta \approx \dfrac{\lambda}{a}\).
Resolving power
\[ \theta_\text{min} \approx 1.22\dfrac{\lambda}{D} \] where \(D\) is aperture diameter.
Resolving power \(= \dfrac{1}{\theta_\text{min}}\).
Polarisation
\[ I = I_0\cos^2\theta \] where \(\theta\) is angle between transmission axes, \(I_0\) initial intensity.
Brewster's angle:
\(\tan\theta_B = \dfrac{n_2}{n_1}\) — reflected light is fully polarized at this angle.
\(n_1\sin\theta_1 = n_2\sin\theta_2\)
Refraction at a curved surface:
\[
\frac{n_2}{v} - \frac{n_1}{u} = \frac{n_2 - n_1}{R}
\]
where \(R\) = radius of curvature of refracting surface.
Shift in apparent depth:
\[
\text{Apparent depth} = \frac{\text{Real depth}}{n}
\]
Lens combinations & Optical instruments
\[ \frac{1}{F} = \frac{1}{f_1} + \frac{1}{f_2} \]
Separated by distance \(d\):
\[
\frac{1}{F} = \frac{1}{f_1} + \frac{1}{f_2} - \frac{d}{f_1 f_2}
\]
Angular magnification:
\[ M = 1 + \frac{D}{f} \] where \(D = 25\,\text{cm}\) = least distance of distinct vision.
\[ M = m_o \, m_e = \left(\frac{v_o}{u_o}\right)\left(1+\frac{D}{f_e}\right) \]
\[ M = -\frac{f_o}{f_e} \] Negative sign → inverted image.
Final image at infinity → relaxed eye.
Wave Optics — Interference & Diffraction
Interference (Young's Double Slit)
\[ \Delta x = n\lambda \quad (n = 0,1,2,...) \]
Destructive:
\[
\Delta x = \left(n + \frac{1}{2}\right)\lambda
\]
Fringe width:
\[
\beta = \frac{\lambda D}{d}
\]
where \(D\) = distance to screen, \(d\) = slit separation.
Shift due to thin film of thickness \(t\):
\[
\Delta x = \frac{2t(n-1)}{d}
\]
Diffraction (Single-Slit)
\[ a\sin\theta = n\lambda, \quad n = 1,2,3... \]
Angular width of central maximum:
\[
\Delta\theta = \frac{2\lambda}{a}
\]
Resolving Power
\[ \theta_{\min} = 1.22 \frac{\lambda}{D} \] where \(D\) = diameter of aperture.
Resolving power of telescope:
\[
\text{RP} = \frac{1}{\theta_{\min}}
\]
Resolving power of a grating:
\[
\text{RP} = nN
\]
where \(n\) = order, \(N\) = total number of slits illuminated.
Polarisation
\[ I = I_0 \cos^2\theta \]
Brewster's Law:
\[
\tan i_B = n
\]
where \(i_B\) = polarising angle.
Quick tips & common mistakes
- Always mention the sign convention before solving lens/mirror numericals.
- Fringe width in YDSE increases only when \(\lambda\) or \(D\) increases — not brightness.
- In diffraction, decreasing slit width increases spreading; the opposite of geometric intuition.
- Resolving power ∝ order (grating). High order = better resolution.
- Polarization only affects transverse waves—sound cannot be polarised.
Practice & PYQs
FAQ (Optics)
Which formulas are most important for JEE?
Mirror/lens formula, lens maker, YDSE fringe width, diffraction minima, resolving power, Brewster’s law.Is sign convention necessary?
Yes. Most numerical mistakes arise from inconsistent sign conventions.How to revise Optics quickly?
Memorize core formulas + practice 25–40 PYQs on Ray + Wave optics.Quick tips & common mistakes
- Always state the sign convention you use (Cartesian / Real-is-positive etc.).
- For lenses in contact, add powers instead of combining focal lengths directly.
- Interference requires coherent sources; check path difference carefully (watch ± signs).
- Diffraction minima locations are not where intensity is zero for finite-width slits — approximations depend on Fraunhofer conditions.
Practice & PYQs
Solve topic-wise PYQs and sample problems to apply formulas. Recommended StudyBeacon links:
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