JEE Main Physics — Waves, SHM & Optics Formula Sheet

This comprehensive formula sheet is meticulously crafted for JEE Main aspirants, providing a deep dive into the essential concepts, formulas, and problem-solving techniques for Waves, Simple Harmonic Motion (SHM), and Optics. Mastering these topics is crucial for securing a high rank, as they consistently contribute a significant portion of marks in the JEE Main Physics section.

JEE Main Physics: Waves, SHM & Optics Formula Sheet

Exam Weightage and Importance

The chapters on Waves (including SHM and Wave Optics) and Geometrical Optics collectively form a high-weightage section in JEE Main Physics. Typically, one to two questions appear from SHM and Waves, and another one to two questions from Geometrical and Wave Optics. This translates to 8-16 marks, which can be a significant differentiator in competitive exams. A strong grasp of the underlying principles and quick recall of formulas are paramount for success in these areas.

I. Simple Harmonic Motion (SHM)

Simple Harmonic Motion is a special type of periodic motion where the restoring force is directly proportional to the displacement from the mean position and acts opposite to the displacement.

1. Defining Equations and Characteristics

• Restoring Force: $F = -kx$
  • $k$: force constant (N/m)
  • $x$: displacement from mean position (m)
• Differential Equation of SHM: $$ \frac{d^2x}{dt^2} + \omega^2 x = 0 $$
  • $\omega = \sqrt{\frac{k}{m}}$: angular frequency (rad/s)
  • $m$: mass of the oscillating body (kg)
• Displacement Equation: $$ x(t) = A \sin(\omega t + \phi) \quad \text{or} \quad x(t) = A \cos(\omega t + \phi') $$
  • $A$: amplitude (maximum displacement) (m)
  • $\phi$: initial phase angle (rad)
  • $\phi' = \phi + \pi/2$
• Velocity Equation: $$ v(t) = \frac{dx}{dt} = A\omega \cos(\omega t + \phi) = \pm \omega \sqrt{A^2 - x^2} $$
  • Maximum velocity: $v_{max} = A\omega$ (occurs at mean position $x=0$)
• Acceleration Equation: $$ a(t) = \frac{dv}{dt} = -A\omega^2 \sin(\omega t + \phi) = -\omega^2 x $$
  • Maximum acceleration: $a_{max} = A\omega^2$ (occurs at extreme positions $x=\pm A$)
• Time Period (T): Time taken for one complete oscillation. $$ T = \frac{2\pi}{\omega} = 2\pi \sqrt{\frac{m}{k}} $$
• Frequency (f): Number of oscillations per unit time. $$ f = \frac{1}{T} = \frac{\omega}{2\pi} $$

2. Energy in SHM

• Kinetic Energy (KE): $$ KE = \frac{1}{2} mv^2 = \frac{1}{2} m \omega^2 (A^2 - x^2) = \frac{1}{2} k (A^2 - x^2) $$
• Potential Energy (PE): $$ PE = \frac{1}{2} kx^2 = \frac{1}{2} m \omega^2 x^2 $$
• Total Mechanical Energy (E): $$ E = KE + PE = \frac{1}{2} kA^2 = \frac{1}{2} m \omega^2 A^2 = \text{constant} $$
  • At mean position ($x=0$): $KE = \frac{1}{2} kA^2$, $PE = 0$
  • At extreme positions ($x=\pm A$): $KE = 0$, $PE = \frac{1}{2} kA^2$

3. Different Systems Executing SHM

4. Damped and Forced Oscillations

• Damped Oscillations: Oscillations where amplitude decreases with time due to dissipative forces (e.g., air resistance).
  • Equation: $m\frac{d^2x}{dt^2} + b\frac{dx}{dt} + kx = 0$
  • Term b dx/dt is the damping force.
  • Amplitude decay: $A(t) = A_0 e^{-bt/2m}$
  • Angular frequency: $\omega' = \sqrt{\omega_0^2 - (b/2m)^2}$
• Forced Oscillations and Resonance: When an oscillating system is subjected to an external periodic force.
  • Driving frequency: $\omega_d$
  • Amplitude of forced oscillation is maximum when $\omega_d$ is close to the natural frequency of the oscillator ($\omega_0$). This phenomenon is called resonance.

II. Waves

Waves are disturbances that propagate through a medium or space, transferring energy without significant transport of matter.

1. General Wave Equation

• One-dimensional wave: $$ y(x, t) = A \sin(kx - \omega t + \phi) \quad \text{or} \quad y(x, t) = A \sin(\omega t - kx + \phi') $$
  • $A$: amplitude
  • $k = \frac{2\pi}{\lambda}$: angular wave number (rad/m)
  • $\omega = \frac{2\pi}{T}$: angular frequency (rad/s)
  • $\lambda$: wavelength (m)
  • $T$: time period (s)
  • $\phi$: initial phase
• Wave Speed (v): $$ v = f\lambda = \frac{\omega}{k} $$
• Relation between $v$, $\lambda$, $f$:
  • If wave travels in positive x-direction: $y(x,t) = f(x - vt)$
  • If wave travels in negative x-direction: $y(x,t) = f(x + vt)$

2. Types of Waves

• Transverse Waves: Particles of the medium oscillate perpendicular to the direction of wave propagation. (e.g., light waves, waves on a string).
• Longitudinal Waves: Particles of the medium oscillate parallel to the direction of wave propagation. (e.g., sound waves).
• Mechanical Waves: Require a medium for propagation (e.g., sound waves, water waves).
• Electromagnetic Waves: Do not require a medium and can travel through vacuum (e.g., light, radio waves).

3. Speed of Mechanical Waves

• Speed of Transverse Wave on a Stretched String: $$ v = \sqrt{\frac{T}{\mu}} $$
  • $T$: tension in the string (N)
  • $\mu$: linear mass density (mass per unit length) (kg/m)
• Speed of Longitudinal Wave (Sound) in a Fluid: $$ v = \sqrt{\frac{B}{\rho}} $$
  • $B$: bulk modulus of the fluid (Pa)
  • $\rho$: density of the fluid (kg/m$^3$)
• Speed of Sound in Solids (Longitudinal): $$ v = \sqrt{\frac{Y}{\rho}} $$
  • $Y$: Young's modulus (Pa)
• Speed of Sound in Gases (Newton-Laplace Formula): $$ v = \sqrt{\frac{\gamma P}{\rho}} $$
  • $\gamma = C_P/C_V$: adiabatic index
  • $P$: pressure (Pa)
  • $\rho$: density of gas (kg/m$^3$)
  • Also $v = \sqrt{\frac{\gamma RT}{M}}$, where $R$ is universal gas constant, $T$ is temperature, $M$ is molar mass.

4. Intensity of Waves

• Intensity (I): Power transmitted per unit area. $I = \frac{P}{A}$. $$ I = \frac{1}{2} \rho v \omega^2 A^2 $$
  • For a spherical wave radiating uniformly in all directions, $I \propto \frac{1}{r^2}$ (where $r$ is distance from source).

5. Superposition of Waves

When two or more waves overlap, the resultant displacement at any point and time is the vector sum of the displacements due to individual waves.

• Interference: Superposition of two waves of same frequency and constant phase difference.
  • Constructive Interference: Path difference $\Delta x = n\lambda$ or Phase difference $\Delta\phi = 2n\pi$. Resultant amplitude $A_R = A_1+A_2$. Resultant intensity $I_R = (\sqrt{I_1} + \sqrt{I_2})^2$.
  • Destructive Interference: Path difference $\Delta x = (n + \frac{1}{2})\lambda$ or Phase difference $\Delta\phi = (2n+1)\pi$. Resultant amplitude $A_R = |A_1-A_2|$. Resultant intensity $I_R = (\sqrt{I_1} - \sqrt{I_2})^2$.
  • If $A_1=A_2=A_0$, then $I_{max} = (2A_0)^2 \propto 4I_0$ and $I_{min} = 0$.
  • Resultant amplitude: $A_R^2 = A_1^2 + A_2^2 + 2A_1A_2 \cos \Delta\phi$
  • Resultant intensity: $I_R = I_1 + I_2 + 2\sqrt{I_1I_2} \cos \Delta\phi$

6. Standing Waves (Stationary Waves)

Formed by the superposition of two identical waves traveling in opposite directions.

• Equation: $y(x, t) = (2A \sin kx) \cos \omega t$ (for two waves $y_1 = A \sin(kx-\omega t)$ and $y_2 = A \sin(kx+\omega t)$)
• Nodes: Points of zero displacement ($x = n\frac{\lambda}{2}$, where $n=0,1,2,...$).
• Antinodes: Points of maximum displacement ($x = (n + \frac{1}{2})\frac{\lambda}{2}$, where $n=0,1,2,...$).
• Distance between two consecutive nodes or antinodes is $\frac{\lambda}{2}$.
• Distance between a node and an adjacent antinode is $\frac{\lambda}{4}$.

7. Standing Waves in Strings

• Fixed at both ends:
  • Possible wavelengths: $\lambda_n = \frac{2L}{n}$, where $n = 1, 2, 3, ...$
  • Possible frequencies: $f_n = \frac{nv}{2L} = n f_1$
  • $f_1 = \frac{v}{2L} = \frac{1}{2L}\sqrt{\frac{T}{\mu}}$ (fundamental frequency or first harmonic)
  • All harmonics (multiples of fundamental) are present.

8. Standing Waves in Organ Pipes (Air Columns)

• Open at both ends:
  • Possible wavelengths: $\lambda_n = \frac{2L}{n}$, where $n = 1, 2, 3, ...$
  • Possible frequencies: $f_n = \frac{nv}{2L} = n f_1$
  • $f_1 = \frac{v}{2L}$ (fundamental frequency or first harmonic)
  • All harmonics are present.
• Closed at one end:
  • Possible wavelengths: $\lambda_n = \frac{4L}{n}$, where $n = 1, 3, 5, ...$ (only odd integers)
  • Possible frequencies: $f_n = \frac{nv}{4L} = n f_1$
  • $f_1 = \frac{v}{4L}$ (fundamental frequency or first harmonic)
  • Only odd harmonics are present.

9. Beats

When two sound waves of slightly different frequencies ($f_1$ and $f_2$) traverse the same medium simultaneously, intensity alternately rises and falls.

• Beat frequency ($f_{beat}$): Number of beats heard per second. $$ f_{beat} = |f_1 - f_2| $$

10. Doppler Effect in Sound

Apparent change in frequency of sound due to relative motion between source and observer.

• Formula: $$ f' = f \left( \frac{v_{sound} \pm v_{observer}}{v_{sound} \mp v_{source}} \right) $$
  • $v_{sound}$: speed of sound in the medium.
  • $v_{observer}$: speed of observer.
  • $v_{source}$: speed of source.
  • Convention:
    • Numerator: Use '+' if observer moves towards source, '-' if observer moves away.
    • Denominator: Use '-' if source moves towards observer, '+' if source moves away.
  • Important: All velocities are relative to the medium.
  • If medium itself is moving (wind), replace $v_{sound}$ with $v_{sound} \pm v_{wind}$ (add if wind direction is same as sound, subtract if opposite).

III. Geometrical Optics

Geometrical Optics deals with light propagation in terms of rays and is based on the laws of reflection and refraction.

1. Reflection of Light

• Laws of Reflection:
  1. The incident ray, the reflected ray, and the normal to the surface at the point of incidence all lie in the same plane.
  2. The angle of incidence ($i$) is equal to the angle of reflection ($r$). ($i=r$)
• Plane Mirrors:
  • Image formed is virtual, erect, and laterally inverted.
  • Image distance = Object distance ($|v|=|u|$).
  • Magnification $m = 1$.
  • If object moves with $v_o$, image moves with $v_i = -v_o$.
  • If mirror rotates by angle $\theta$, reflected ray rotates by $2\theta$.
• Spherical Mirrors (Concave and Convex):
  • Mirror Formula: $$ \frac{1}{f} = \frac{1}{v} + \frac{1}{u} $$
    • $f$: focal length, $v$: image distance, $u$: object distance
    • Sign Convention (Cartesian): Origin at pole, light from left. Distances to the right of pole are positive, to the left are negative. Heights above principal axis are positive, below are negative.
    • Concave mirror: $f < 0$. Convex mirror: $f > 0$.
  • Relation between $f$ and $R$ (radius of curvature): $f = R/2$
  • Magnification (m): $$ m = \frac{\text{Height of image}}{\text{Height of object}} = -\frac{v}{u} = \frac{f-v}{f} = \frac{f}{f-u} $$
  • Nature of Images:
    • Real and Inverted if $m < 0$.
    • Virtual and Erect if $m > 0$.
    • Magnified if $|m| > 1$.
    • Diminished if $|m| < 1$.

2. Refraction of Light

• Laws of Refraction (Snell's Law):
  1. The incident ray, the refracted ray, and the normal to the interface at the point of incidence all lie in the same plane.
  2. $\frac{\sin i}{\sin r} = \frac{n_2}{n_1} = \frac{v_1}{v_2} = \frac{\lambda_1}{\lambda_2}$ (constant for a given pair of media and given color of light)
     • $n_1, n_2$: refractive indices of media 1 and 2
     • $v_1, v_2$: speed of light in media 1 and 2
     • $\lambda_1, \lambda_2$: wavelength of light in media 1 and 2
     • Absolute Refractive Index: $n = \frac{c}{v_{medium}}$ ($c$: speed of light in vacuum)
• Apparent Depth: When viewed from a denser medium (e.g., water) into a rarer medium (e.g., air), objects appear closer. $$ n_{rarer} (\text{real depth}) = n_{denser} (\text{apparent depth}) \quad \implies \quad d_{apparent} = d_{real} \frac{n_2}{n_1} $$
  • Where $n_1$ is the refractive index of the medium where the object is, and $n_2$ is the refractive index of the medium from which it is observed.
• Total Internal Reflection (TIR): When light travels from a denser medium to a rarer medium and the angle of incidence exceeds the critical angle.
  • Critical Angle ($\theta_c$): $$ \sin \theta_c = \frac{n_{rarer}}{n_{denser}} $$
  • Conditions for TIR: 1. Light travels from denser to rarer medium. 2. Angle of incidence $i > \theta_c$.

3. Refraction at Spherical Surfaces

• Lens Maker's Formula (for thin lenses): $$ \frac{1}{f} = (n-1) \left( \frac{1}{R_1} - \frac{1}{R_2} \right) $$
  • $n$: refractive index of lens material relative to surrounding medium ($n_{lens}/n_{medium}$)
  • $R_1, R_2$: radii of curvature of the two lens surfaces.
  • Sign Convention: $R$ is positive if surface is convex towards the object, negative if concave.
• Lens Formula: $$ \frac{1}{v} - \frac{1}{u} = \frac{1}{f} $$
  • Sign Convention (Cartesian): Similar to mirrors. $f$ is positive for convex lens, negative for concave lens.
• Magnification (m): $$ m = \frac{\text{Height of image}}{\text{Height of object}} = \frac{v}{u} = \frac{f-v}{-f} = \frac{f}{f+u} $$
• Power of a Lens (P): $P = \frac{1}{f}$ (in dioptres (D) if $f$ is in metres).
• Combination of Thin Lenses in Contact:
  • Equivalent focal length: $\frac{1}{f_{eq}} = \frac{1}{f_1} + \frac{1}{f_2} + \frac{1}{f_3} + \dots$
  • Equivalent power: $P_{eq} = P_1 + P_2 + P_3 + \dots$
  • Magnification: $m_{eq} = m_1 \times m_2 \times m_3 \times \dots$

4. Prism

• Angle of deviation ($\delta$): Angle between incident and emergent rays. $$ \delta = i+e-A $$
  • $A$: angle of prism, $i$: angle of incidence, $e$: angle of emergence
• Relation between angles: $A = r_1+r_2$ ($r_1, r_2$: angles of refraction at first and second surfaces)
• Minimum Deviation ($\delta_{min}$): Occurs when the ray within the prism is parallel to the base ($i=e$ and $r_1=r_2=A/2$). $$ n = \frac{\sin((A+\delta_{min})/2)}{\sin(A/2)} $$
• Dispersion: Splitting of white light into its constituent colors. Caused by varying refractive index for different wavelengths ($n_{red} < n_{violet}$).
  • Angular Dispersion: $\theta = \delta_v - \delta_r = (n_v - n_r)A$
  • Dispersive Power ($\omega$): $\omega = \frac{n_v - n_r}{n_{yellow}-1} = \frac{\delta_v-\delta_r}{\delta_{mean}}$

5. Optical Instruments

• Human Eye:
  • Near Point: 25 cm (normal eye)
  • Far Point: Infinity (normal eye)
• Simple Microscope (Magnifying Glass): Converts a close object into a virtual, erect, and magnified image.
  • Angular Magnification ($M$):
    • Image at near point (25 cm): $M = 1 + \frac{D}{f}$ (D=25 cm)
    • Image at infinity: $M = \frac{D}{f}$
• Compound Microscope: Two lenses: objective (short focal length) and eyepiece (longer focal length).
  • Angular Magnification ($M$):
    • Final image at D: $M = \frac{v_0}{u_0} (1+\frac{D}{f_e}) = \frac{L}{f_0}(1+\frac{D}{f_e})$ (approx. for $u_0 \approx f_0$)
    • Final image at infinity: $M = \frac{v_0}{u_0} (\frac{D}{f_e}) = \frac{L}{f_0}(\frac{D}{f_e})$
    • $L$: length of microscope tube, $f_0$: objective focal length, $f_e$: eyepiece focal length.
• Astronomical Telescope: For distant objects. Objective has long focal length, eyepiece short focal length.
  • Angular Magnification ($M$):
    • Final image at infinity (normal adjustment): $M = -\frac{f_0}{f_e}$
    • Final image at D: $M = -\frac{f_0}{f_e} (1+\frac{f_e}{D})$
  • Length of telescope in normal adjustment: $L = f_0 + f_e$.
• Reflecting Telescope (Cassegrain type): Uses mirrors for objective, eliminating chromatic aberration.

IV. Wave Optics

Wave Optics explains phenomena like interference, diffraction, and polarization, which cannot be explained by ray optics.

1. Huygens' Principle

• Every point on a wavefront is a source of secondary wavelets, which spread out in all directions with the speed of light in that medium.
• The new wavefront is the envelope of these secondary wavelets.

2. Interference of Light (Young's Double Slit Experiment - YDSE)

• Two coherent sources (from a single source split into two) creating interference pattern.
• Path difference ($\Delta x$): $d \sin\theta \approx d \frac{y}{D}$
  • $d$: slit separation
  • $D$: distance from slits to screen
  • $y$: distance of point from central maximum
• Condition for Constructive Interference (Bright Fringes): $$ \Delta x = n\lambda \quad \implies \quad y_n = \frac{n\lambda D}{d} $$
  • $n = 0$ for central maximum.
• Condition for Destructive Interference (Dark Fringes): $$ \Delta x = (n + \frac{1}{2})\lambda \quad \implies \quad y_n = \frac{(2n+1)\lambda D}{2d} $$
• Fringe Width ($\beta$): Distance between two consecutive bright or dark fringes. $$ \beta = \frac{\lambda D}{d} $$
• Angular Fringe Width ($\omega$): Angle subtended by a fringe at the slits. $$ \omega = \frac{\beta}{D} = \frac{\lambda}{d} $$
• Shift of Fringes by Transparent Sheet: $$ \Delta y = \frac{D}{d} (n-1)t $$
  • $n$: refractive index of the sheet material
  • $t$: thickness of the sheet.
  • Shift is towards the side where the sheet is placed.

3. Diffraction of Light (Single Slit)

Bending of light around obstacles or apertures.

• Condition for minima: $a \sin\theta_n = n\lambda$ for $n = \pm 1, \pm 2, ...$
  • $a$: width of the slit
• Condition for maxima: $a \sin\theta_n = (n + \frac{1}{2})\lambda$ for $n = \pm 1, \pm 2, ...$ (approximately)
• Central Maximum: Extends from $\theta = -\frac{\lambda}{a}$ to $\theta = \frac{\lambda}{a}$.
  • Angular width of central maximum: $2\theta_1 = 2\frac{\lambda}{a}$
  • Linear width of central maximum: $W = 2\frac{\lambda D}{a}$
• Intensity pattern: Diminishes rapidly away from the central maximum. Central maximum intensity is about 20 times that of the first secondary maximum.

4. Resolving Power

• Rayleigh's Criterion: Two closely spaced objects are just resolved when the central maximum of the diffraction pattern of one coincides with the first minimum of the diffraction pattern of the other.
• Resolving Power of a Microscope: $RP = \frac{2n \sin\theta}{\lambda}$
  • $n \sin\theta$: numerical aperture.
• Resolving Power of a Telescope: $RP = \frac{1}{\Delta \theta_{min}} = \frac{D}{1.22\lambda}$
  • $D$: diameter of the objective lens
  • $\Delta \theta_{min}$: minimum resolvable angular separation.

5. Polarization of Light

Light waves are transverse, meaning electric and magnetic fields oscillate perpendicular to the direction of propagation. Unpolarized light has oscillations in all planes perpendicular to propagation. Polarized light has oscillations restricted to a single plane.

• Methods of Polarization: Reflection, refraction, scattering, double refraction, selective absorption.
• Brewster's Law (Polarization by Reflection):
  • When light is incident at an angle $i_p$ (Brewster's angle) such that the reflected ray is perpendicular to the refracted ray, the reflected light is completely plane-polarized. $$ \tan i_p = n $$
  • $n$: refractive index of the medium.
• Malus's Law: If an unpolarized light of intensity $I_0$ passes through a polarizer, its intensity becomes $I_0/2$. When this plane-polarized light passes through an analyser, the transmitted intensity $I$ is given by: $$ I = I_0 \cos^2 \theta $$
  • $\theta$: angle between the transmission axes of the polarizer and analyser.

V. Important Tables and Values

1. Common Physical Constants (Approximate)

2. Sign Conventions Summary (Cartesian)

VI. Worked Examples

Example 1: SHM Energy

A particle executes SHM with an amplitude A. At what displacement from the mean position is its kinetic energy equal to its potential energy?

Solution: Let the displacement from the mean position be $x$. Kinetic Energy (KE) $= \frac{1}{2} k(A^2 - x^2)$ Potential Energy (PE) $= \frac{1}{2} kx^2$ According to the problem, KE = PE. $\frac{1}{2} k(A^2 - x^2) = \frac{1}{2} kx^2$ $A^2 - x^2 = x^2$ $A^2 = 2x^2$ $x^2 = \frac{A^2}{2}$ $x = \pm \frac{A}{\sqrt{2}}$

Answer: The kinetic energy is equal to its potential energy at a displacement of $\pm \frac{A}{\sqrt{2}}$ from the mean position.

Example 2: Waves on a String

A string of length 1 m and mass 5 g is stretched with a tension of 10 N. What is the speed of transverse waves on this string? Also, calculate the fundamental frequency.

Solution:

1. Calculate linear mass density ($\mu$): Mass $m = 5 , \text{g} = 5 \times 10^{-3} , \text{kg}$ Length $L = 1 , \text{m}$ $\mu = \frac{m}{L} = \frac{5 \times 10^{-3} , \text{kg}}{1 , \text{m}} = 5 \times 10^{-3} , \text{kg/m}$

2. Calculate wave speed (v): Tension $T = 10 , \text{N}$ $v = \sqrt{\frac{T}{\mu}} = \sqrt{\frac{10 , \text{N}}{5 \times 10^{-3} , \text{kg/m}}} = \sqrt{\frac{10^4}{5}} = \sqrt{2000} , \text{m/s}$ $v = 10\sqrt{20} = 20\sqrt{5} \approx 20 \times 2.236 = 44.72 , \text{m/s}$

3. Calculate fundamental frequency ($f_1$): For a string fixed at both ends, the fundamental frequency is $f_1 = \frac{v}{2L}$. $f_1 = \frac{20\sqrt{5} , \text{m/s}}{2 \times 1 , \text{m}} = 10\sqrt{5} \approx 22.36 , \text{Hz}$

Answer: The speed of transverse waves is approximately $44.72 , \text{m/s}$, and the fundamental frequency is approximately $22.36 , \text{Hz}$.

Example 3: Doppler Effect

A train blows a whistle of frequency 500 Hz while approaching a stationary observer. The speed of sound in air is 340 m/s. If the observer hears a frequency of 550 Hz, what is the speed of the train?

Solution: Given: Original frequency $f = 500 , \text{Hz}$ Apparent frequency $f' = 550 , \text{Hz}$ Speed of sound $v_{sound} = 340 , \text{m/s}$ Observer is stationary, so $v_{observer} = 0$. The train (source) is approaching the observer, so we use '-' in the denominator for $v_{source}$.

Doppler effect formula: $f' = f \left( \frac{v_{sound} + v_{observer}}{v_{sound} - v_{source}} \right)$ $550 = 500 \left( \frac{340 + 0}{340 - v_{source}} \right)$ $550 = 500 \left( \frac{340}{340 - v_{source}} \right)$ $\frac{550}{500} = \frac{340}{340 - v_{source}}$ $1.1 = \frac{340}{340 - v_{source}}$ $1.1 (340 - v_{source}) = 340$ $374 - 1.1 v_{source} = 340$ $1.1 v_{source} = 374 - 340$ $1.1 v_{source} = 34$ $v_{source} = \frac{34}{1.1} \approx 30.91 , \text{m/s}$

Answer: The speed of the train is approximately $30.91 , \text{m/s}$.

Example 4: Mirror Formula

An object is placed 15 cm in front of a concave mirror of focal length 10 cm. Find the position and nature of the image.

Solution: Given: Object distance $u = -15 , \text{cm}$ (using Cartesian sign convention) Focal length of concave mirror $f = -10 , \text{cm}$

Mirror formula: $\frac{1}{f} = \frac{1}{v} + \frac{1}{u}$ $\frac{1}{-10} = \frac{1}{v} + \frac{1}{-15}$ $-\frac{1}{10} = \frac{1}{v} - \frac{1}{15}$ $\frac{1}{v} = \frac{1}{15} - \frac{1}{10}$ $\frac{1}{v} = \frac{2 - 3}{30} = -\frac{1}{30}$ $v = -30 , \text{cm}$

Magnification $m = -\frac{v}{u} = -\frac{(-30 , \text{cm})}{(-15 , \text{cm})} = -2$

Analysis:

• Since $v = -30 , \text{cm}$, the image is formed 30 cm in front of the mirror (on the same side as the object). This indicates a real image.
• Since $m = -2$, the image is inverted (negative sign) and magnified (magnitude > 1).

Answer: The image is formed 30 cm in front of the mirror, is real, inverted, and magnified.

Example 5: Lens Maker's Formula

A biconvex lens has radii of curvature 20 cm and 30 cm. If the refractive index of the lens material is 1.5, calculate its focal length.

Solution: Given: For a biconvex lens, both surfaces are convex. Assuming light comes from the left. First surface (convex): $R_1 = +20 , \text{cm}$ Second surface (convex, but light strikes from inside the lens, so its centre of curvature is to the left): $R_2 = -30 , \text{cm}$ Refractive index $n = 1.5$

Lens Maker's Formula: $\frac{1}{f} = (n-1) \left( \frac{1}{R_1} - \frac{1}{R_2} \right)$ $\frac{1}{f} = (1.5 - 1) \left( \frac{1}{+20} - \frac{1}{-30} \right)$ $\frac{1}{f} = (0.5) \left( \frac{1}{20} + \frac{1}{30} \right)$ $\frac{1}{f} = 0.5 \left( \frac{3+2}{60} \right)$ $\frac{1}{f} = 0.5 \left( \frac{5}{60} \right) = 0.5 \times \frac{1}{12} = \frac{1}{2} \times \frac{1}{12} = \frac{1}{24}$ $f = 24 , \text{cm}$

Answer: The focal length of the biconvex lens is 24 cm.

Example 6: YDSE Interference

In a Young's Double Slit Experiment, the slits are 0.5 mm apart and the screen is 2 m away. If the wavelength of light used is 500 nm, calculate the fringe width. Also, find the distance of the 3rd dark fringe from the central maximum.

Solution: Given: Slit separation $d = 0.5 , \text{mm} = 0.5 \times 10^{-3} , \text{m}$ Distance to screen $D = 2 , \text{m}$ Wavelength $\lambda = 500 , \text{nm} = 500 \times 10^{-9} , \text{m} = 5 \times 10^{-7} , \text{m}$

1. Calculate Fringe Width ($\beta$): $\beta = \frac{\lambda D}{d}$ $\beta = \frac{(5 \times 10^{-7} , \text{m}) \times (2 , \text{m})}{0.5 \times 10^{-3} , \text{m}}$ $\beta = \frac{10 \times 10^{-7}}{0.5 \times 10^{-3}} = \frac{10 \times 10^{-4}}{0.5} = 20 \times 10^{-4} , \text{m} = 2 , \text{mm}$

2. Distance of 3rd Dark Fringe ($y_3$): For a dark fringe, the formula is $y_n = \frac{(2n+1)\lambda D}{2d}$. For the 3rd dark fringe, $n=2$. (Note: $n=0$ for 1st dark, $n=1$ for 2nd dark, $n=2$ for 3rd dark) $y_3 = \frac{(2 \times 2 + 1)\lambda D}{2d} = \frac{5\lambda D}{2d}$ We know $\beta = \frac{\lambda D}{d}$, so $\frac{\lambda D}{d} = 2 , \text{mm}$. $y_3 = \frac{5}{2} \times \left( \frac{\lambda D}{d} \right) = \frac{5}{2} \times \beta$ $y_3 = \frac{5}{2} \times 2 , \text{mm} = 5 , \text{mm}$

Answer: The fringe width is 2 mm, and the distance of the 3rd dark fringe from the central maximum is 5 mm.

VII. Common Mistakes to Avoid

1. Sign Conventions: Incorrect application of Cartesian sign conventions for mirrors and lenses is the most frequent error. Be consistent and draw ray diagrams to visualize.
2. Units: Ensure all quantities are in consistent units (S.I. units are best) before calculation.
3. SHM Mean Position: For SHM, displacement, velocity, and acceleration are all measured with respect to the mean position, not the origin or equilibrium position of the support.
4. Differentiating SHM and Waves: Understand that SHM describes the motion of a single particle, while a wave describes the propagation of a disturbance.
5. Doppler Effect Formula: Remember the sign convention for observer/source velocity relative to the medium. Always consider relative motion carefully.
6. Standing Wave Harmonics: For open pipes and strings, all harmonics are present ($f, 2f, 3f...$). For closed pipes, only odd harmonics are present ($f, 3f, 5f...$).
7. Interference vs. Diffraction: While both involve superposition, interference is due to two (or few) coherent sources, diffraction is due to multiple coherent sources from parts of the same wavefront.
8. TIR Conditions: Total internal reflection only occurs when light travels from a denser to a rarer medium and the angle of incidence exceeds the critical angle.
9. Lens Magnification: Correctly identify whether the image is real/virtual and erect/inverted based on the sign of magnification and final image position. For lenses $m=v/u$, for mirrors $m=-v/u$.

VIII. Memory Tricks and Mnemonics

• VIBGYOR (Colors of visible light): For increasing wavelength, decreasing frequency/energy.
• ROY G BIV: Same as VIBGYOR, just reversed for remembering order (Red, Orange, Yellow, Green, Blue, Indigo, Violet).
• Doppler Effect Sign Convention: "Towards is less in source (denominator), towards is more in observer (numerator)."
  • Source moving toward observer: $v_s$ (denominator) is negative ($v-v_s$).
  • Source moving away from observer: $v_s$ (denominator) is positive ($v+v_s$).
  • Observer moving toward source: $v_o$ (numerator) is positive ($v+v_o$).
  • Observer moving away from source: $v_o$ (numerator) is negative ($v-v_o$).
• Mirror/Lens Sign Convention (for $u, v, f$): Real is usually negative for distance (like real expenses are negative), virtual is positive (like a bonus). This is tricky as sign conventions vary. Best practice is to stick to one (Cartesian is standard) and practice.
• SHM - extreme positions: KE is 0, PE is max. Acceleration is max, velocity is 0.
• SHM - mean position: KE is max, PE is 0. Acceleration is 0, velocity is max.
• Dispersion: Violet bends the most, Red bends the least ($n_v > n_r$).

IX. Previous Year Question Analysis (Typical Patterns)

• SHM:
  • Direct application of $T = 2\pi\sqrt{m/k}$ or $T = 2\pi\sqrt{L/g}$.
  • Energy conservation in SHM (KE = PE at certain position).
  • Finding position, velocity, acceleration from displacement equation.
  • Comparison of SHM properties for different systems (e.g., spring cut into parts).
• Waves:
  • Speed of wave on a string or in gas.
  • Standing waves in strings and pipes (harmonics, overtones).
  • Beat frequency.
  • Doppler effect (most common in sound waves, sometimes light).
  • Wave equation interpretation.
• Geometrical Optics:
  • Mirror and Lens formula-based numericals (single lens/mirror, sometimes combination).
  • Refraction at plane/spherical surfaces (including apparent depth).
  • Total Internal Reflection applications.
  • Prism formula for minimum deviation, sometimes dispersion angle.
  • Optical instruments: Magnifying power, length of microscope/telescope.
• Wave Optics:
  • Young's Double Slit Experiment (YDSE) - fringe width, position of bright/dark fringes, shift due to thin film.
  • Diffraction by a single slit - width of central maximum, intensity distribution (qualitative).
  • Resolving power for telescope/microscope.
  • Brewster's angle, Malus's Law in polarization.

X. Revision Checklist

SHM

• $F=-kx$, $a = -\omega^2 x$
• $x(t) = A\sin(\omega t+\phi)$
• $v(t) = A\omega\cos(\omega t+\phi) = \pm \omega\sqrt{A^2-x^2}$
• $a(t) = -A\omega^2\sin(\omega t+\phi)$
• $T = 2\pi\sqrt{m/k}$, $\omega = \sqrt{k/m}$
• $T = 2\pi\sqrt{L/g}$ (simple pendulum)
• $KE = \frac{1}{2}k(A^2-x^2)$, $PE = \frac{1}{2}kx^2$, $E = \frac{1}{2}kA^2$
• Damped oscillations concepts (amplitude decay)
• Forced oscillations & resonance

Waves

• General wave equation $y(x,t) = A\sin(kx \pm \omega t + \phi)$
• $v = f\lambda = \omega/k$
• Speed on string: $v=\sqrt{T/\mu}$
• Speed of sound: $v=\sqrt{B/\rho}$ (fluid), $v=\sqrt{Y/\rho}$ (solid), $v=\sqrt{\gamma P/\rho}$ (gas)
• Intensity $I \propto A^2$, $I = \frac{1}{2}\rho v \omega^2 A^2$
• Superposition principle
• Interference conditions (path difference, phase difference)
• Standing waves: nodes, antinodes
• Standing waves in strings: $f_n = n\frac{v}{2L}$
• Standing waves in open pipes: $f_n = n\frac{v}{2L}$
• Standing waves in closed pipes: $f_n = n\frac{v}{4L}$ (odd harmonics only)
• Beat frequency: $f_{beat} = |f_1-f_2|$
• Doppler effect for sound: $f' = f \left( \frac{v_s \pm v_o}{v_s \mp v_s} \right)$

Geometrical Optics

• Laws of reflection & refraction (Snell's Law: $n_1\sin i = n_2\sin r$)
• Mirror formula: $\frac{1}{f} = \frac{1}{v} + \frac{1}{u}$
• Lens formula: $\frac{1}{f} = \frac{1}{v} - \frac{1}{u}$
• Magnification: $m = -v/u$ (mirror), $m = v/u$ (lens)
• Lens Maker's Formula: $\frac{1}{f} = (n-1) \left( \frac{1}{R_1} - \frac{1}{R_2} \right)$
• Critical angle: $\sin\theta_c = n_{rarer}/n_{denser}$ (TIR conditions)
• Apparent depth formula
• Prism minimum deviation: $n = \frac{\sin((A+\delta_{min})/2)}{\sin(A/2)}$
• Dispersion & Dispersive power
• Simple/Compound Microscope, Astronomical Telescope (magnification, length)

Wave Optics

• Huygens' Principle
• YDSE: Path difference $\Delta x = d\sin\theta$, Fringe width $\beta = \frac{\lambda D}{d}$
• Bright fringe: $\Delta x = n\lambda$, Dark fringe: $\Delta x = (n+1/2)\lambda$
• Fringe shift due to film: $\Delta y = \frac{D}{d}(n-1)t$
• Single slit diffraction: minima $a\sin\theta = n\lambda$, central max width $2\lambda D/a$
• Resolving power of telescope/microscope.
• Polarization by reflection (Brewster's Law: $\tan i_p = n$)
• Malus's Law: $I = I_0 \cos^2\theta$

XI. Last-Day Strategy for Waves, SHM & Optics

1. Rapid Formula Recall: Go through this formula sheet and mentally (or quickly on paper) recall each formula and its purpose. Don't solve problems, just recall.
2. Sign Convention Check: Revisit the sign conventions for mirrors and lenses to ensure they are crystal clear. This is where many critical marks are lost.
3. Key Terms & Definitions: Quickly review what terms like "node," "antinode," "critical angle," "polarization," "coherence," "phase difference," etc., mean.
4. Diagram Interpretation: Glance at typical diagrams for YDSE, single slit, prism path, and lens/mirror ray tracing. Understand how they relate to the formulas.
5. Unit Consistency: Make a mental note to check units carefully during the exam. Most errors often stem from not converting units to SI.
6. Avoid New Problems: Do not try to solve any new, complex problems on the last day. This can lead to frustration if you get stuck. Only review questions you've previously solved and understood.
7. Focus on High-Yield: Prioritize areas like YDSE, mirror/lens formulas, Doppler effect, and SHM energy concepts, as they frequently appear.
8. Stay Calm: Confidence in your preparation and a calm mind are your greatest assets. Trust your hard work.

By diligently reviewing this comprehensive formula sheet and following the suggested strategies, you will be well-equipped to tackle Waves, SHM, and Optics questions in JEE Main with confidence. Good luck!