K Mean Black

• Name: B.Tech 1st Year
• Branch: B.Tech Printing Technology 1st Sem
• Published: Dec. 18, 2025

Applied Physics

UNIT–I: Harmonic Motion

 

1. Simple Harmonic Motion (SHM)

Definition

Simple Harmonic Motion is a type of periodic motion in which the restoring force acting on a particle is directly proportional to its displacement from the mean position and is always directed towards it.

F=-kx
 

2. Mechanical Simple Harmonic Oscillator

Example: Mass–spring system

• Mass = m
• Spring constant = k

Equation of Motion

m d²x/dt²+kx=0
 

Angular Frequency

ω=√km
 

Displacement

x=Asin⁡(ωt+ϕ)
 

where

• A= amplitude
• ϕ= phase constant

3. Electrical Simple Harmonic Oscillator

Example: LC Circuit

• Inductance = L
• Capacitance = C

Equation

L d²q/dt²+q/C=0
 

Angular Frequency

ω=1/√LC
 

Electrical SHM is analogous to mechanical SHM:

• Mass m↔Inductance L
• Spring constant k↔1/C

4. Complex Number Notation & Phasor Representation

Complex Form of SHM

x=Re{Ae^iωt}
 

This simplifies calculations involving:

• Phase differences
• Superposition of oscillations

Phasor Representation

• SHM is represented as a rotating vector (phasor) in the complex plane
• Projection on real axis gives instantaneous displacement

5. Damped Harmonic Oscillator

Damping occurs when resistive forces (friction, air resistance) are present.

Equation of Motion

m d²x/dt²+b dx/dt+kx=0
 

where bis the damping constant.

6. Types of Damping

(a) Light Damping (Underdamped)

• Oscillations continue with decreasing amplitude
• Most common in physical systems

x=Ae^-βtsin⁡(ω't+ϕ)
 

(b) Critical Damping

• System returns to equilibrium in minimum time
• No oscillations

Used in:

• Shock absorbers
• Measuring instruments

(c) Heavy Damping (Overdamped)

• No oscillations
• Returns slowly to equilibrium

7. Energy Decay in Damped Oscillator

• Energy decreases exponentially with time

E=E₀e^-2βt
 

where β=b/2m

8. Quality Factor (Q-Factor)

Definition

Quality factor measures how sharp the resonance of an oscillator is.

Q=ω₀/2β
 

Significance

• High Q → low energy loss
• Used in tuning circuits and oscillators

9. Forced Harmonic Oscillator

When an external periodic force acts on a system:

F=F₀sin ⁡ωt
 

Equation

m d²x/dt²+b dx/dt+kx= F₀sin ⁡ωt
 

10. Electrical and Mechanical Impedance

Mechanical Impedance

Z_m=F/v
 

Electrical Impedance

Z=√R2+(ωL-1/ωC)2
 

Impedance represents opposition to motion/current.

11. Steady-State Motion of Forced Damped Oscillator

• After transient effects die out
• Oscillator vibrates with same frequency as applied force
• Amplitude depends on:
  • Driving frequency
  • Damping

Resonance

Occurs when amplitude is maximum.

12. Power Absorbed by Oscillator

Average Power

P=1/2F₀v₀cos⁡ϕ
 

• Maximum power absorbed at resonant frequency
• Phase difference between force and velocity is zero

13. Mechanical–Electrical Analogy

Mechanical System	Electrical System
Mass m	Inductance L
Spring constant k	1/C
Damping b	Resistance R

 

 

UNIT–II: Waves and Dispersion

 

1. Wave Motion

Definition

A wave is a disturbance that travels through a medium, transferring energy without transporting matter.

Types of Waves

• Transverse waves – particle vibration ⟂ direction of propagation
• Longitudinal waves – particle vibration ∥ direction of propagation

2. Transverse Wave on a Stretched String

Description

• String under tension T
• Linear mass density μ

Wave Velocity

v=√T/μ
 

Wave Equation

∂²y/ ∂t²=v² ∂²y/∂x²
 

where

• y= transverse displacement

3. Harmonic Waves

A harmonic wave is a sinusoidal wave.

General Equation

y(x,t)=Asin⁡(ωt-kx+ϕ)
 

where

• A= amplitude
• ω=2πf= angular frequency
• k=2π/λ= wave number

4. Waves at a Boundary

When a wave reaches a boundary between two media:

Possible Effects

• Reflection
• Transmission
• Phase change

Fixed End

• Wave reflects with phase reversal (180°)

Free End

• Wave reflects without phase change

5. Impedance and Impedance Matching

Mechanical Wave Impedance

Z=T/v
 

Impedance Matching

• When impedances of two media are equal
• Maximum energy transfer
• Minimum reflection

Important in:

• Transmission lines
• Acoustics
• Optical fibers

6. Standing Waves

Standing waves are formed by superposition of two identical waves traveling in opposite directions.

Characteristics

• Nodes – points of zero displacement
• Antinodes – points of maximum displacement

Equation

y=2Asin ⁡kxcos ⁡ωt
 

7. Eigen Frequencies (Natural Frequencies)

For a stretched string of length L:

Fundamental Frequency

f1=v/2L
 

Higher Harmonics

fn=nv/2L,n=1,2,3,...
 

These frequencies are called eigen frequencies.

8. Longitudinal Waves

Definition

Particles oscillate parallel to the direction of wave propagation.

Examples

• Sound waves in air
• Waves in springs

9. Equation of Longitudinal Waves

Wave Equation

∂²ξ/∂t²=v² ∂²ξ/ ∂x2
 

where

• ξ= longitudinal displacement

Velocity of Longitudinal Wave

v=√E/ρ
 

E= elastic modulus
ρ= density

10. Acoustic Waves

Sound waves are mechanical longitudinal waves.

Speed of Sound

• In air:

v=√γP/ρ
 

Characteristics

• Frequency → pitch
• Amplitude → loudness
• Waveform → quality (timbre)

11. Standing Sound Waves

Formed due to reflection of sound waves.

(a) Closed Pipe (One End Closed)

• Node at closed end
• Antinode at open end

Frequencies

fn=(2n-1)v/4L,n=1,2,3...
 

(b) Open Pipe (Both Ends Open)

• Antinode at both ends

Frequencies

fn=nv/2L,n=1,2,3...
 

12. Applications

• Musical instruments
• Communication systems
• Acoustics engineering
• Noise control

 

13. Fermat’s Principle of Stationary Time

Statement

Light travels between two points along the path for which the optical path length (or travel time) is stationary (minimum, maximum, or saddle point).

Optical Path Length (OPL)=∫n ds
 

where

• n= refractive index
• ds= path element

Significance

• Explains reflection and refraction
• Forms the basis of geometrical optics

14. Laws of Reflection from Fermat’s Principle

Laws

1. Angle of incidence = Angle of reflection
2. Incident ray, reflected ray, and normal lie in the same plane

Explanation

From Fermat’s principle, the shortest optical path between two points leads to:

θi=θr
 

15. Laws of Refraction (Snell’s Laws)

Snell’s Law

n₁sin⁡θ₁=n₂sin⁡θ₂
 

Derivation

Obtained using Fermat’s principle by minimizing optical path length across two media.

16. Mirage Effect

Definition

A mirage is an optical illusion caused by continuous variation of refractive index due to temperature gradients in air.

Explanation

• Hot air near ground → lower refractive index
• Light bends gradually and undergoes total internal reflection
• Eye perceives inverted images (e.g., water on a hot road)

Types

• Inferior mirage (deserts, roads)
• Superior mirage (polar regions)

17. Light as an Electromagnetic Wave

Maxwell’s Prediction

Light is a transverse electromagnetic wave consisting of:

• Oscillating electric field E
• Oscillating magnetic field B
• Both perpendicular to direction of propagation

Speed

c=1 / √μ0ε0
 

18. Fresnel Equations

Fresnel equations give the reflection and transmission coefficients at an interface.

For Perpendicular (s-polarized) Light

r_s=n₁cos⁡θ_i - n₂cos⁡θ_{t /} n₁cos⁡θ_i+n₂cos⁡θ_t
 

For Parallel (p-polarized) Light

r_p=n₂cos⁡θ_i - n₁cos⁡θ_t  / n₂cos⁡θ_i+n₁cos⁡θ_t
 

19. Reflectance and Transmittance

Reflectance (R)

R=∣r∣2
 

Fraction of incident power reflected.

Transmittance (T)

T=n₂cos⁡θ_t  / n₁cos⁡θ_i ∣t∣2
 

Fraction of incident power transmitted.

Energy Conservation

R+T=1
 

20. Brewster’s Angle

Definition

Angle of incidence at which reflected light is completely plane polarized.

Condition

θ_B+θ_t=90∘
 

Formula

tan⁡θ_B=n₂n₁
 

Application

• Polarizing sunglasses
• Optical instruments

21. Total Internal Reflection (TIR)

Conditions

1. Light travels from denser to rarer medium
2. Angle of incidence > critical angle

Critical Angle

sin⁡θ_c=n₂ / n₁, n1>n2
 

Applications

• Optical fibers
• Prisms
• Endoscopy

22. Evanescent Wave

Definition

A non-propagating electromagnetic wave formed during total internal reflection.

Characteristics

• Exists in the rarer medium
• Amplitude decays exponentially

E∝e^_-kx
 

Uses

• TIR microscopy
• Optical sensors
• Frustrated total internal reflection

23. Summary Table

Concept	Key Idea
Fermat’s Principle	Light follows stationary time path
Mirage	Refraction + TIR due to temperature gradient
Fresnel Equations	Reflection & transmission coefficients
Brewster’s Angle	Zero reflection for p-polarization
TIR	Complete reflection beyond critical angle
Evanescent Wave	Exponentially decaying field

 

 

UNIT–III: Wave Optics

1. Huygens’ Principle

Statement

1. Every point on a wavefront acts as a source of secondary wavelets.
2. The new wavefront at any later time is the envelope of these secondary wavelets.

Applications

• Explanation of reflection
• Explanation of refraction
• Basis of wave optics

2. Superposition of Waves

Principle

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

Result

• Constructive interference → increased intensity
• Destructive interference → reduced or zero intensity

3. Interference of Light

Interference is redistribution of light intensity due to superposition of coherent waves.

Conditions for Sustained Interference

• Same frequency
• Constant phase difference
• Same polarization

4. Methods of Producing Interference

(a) Wavefront Splitting

• Single wavefront divided into parts
• Example: Young’s Double Slit Experiment

(b) Amplitude Splitting

• Amplitude of a single beam split
• Example: Newton’s Rings, Michelson Interferometer

5. Young’s Double Slit Experiment (YDSE)

Arrangement

• Two narrow slits separated by distance d
• Screen at distance D

Path Difference

Δ=dsin⁡θ≈dx/D
 

Fringe Width

β=λD/d
 

Applications

• Measurement of wavelength
• Verification of wave nature of light

6. Newton’s Rings

Formation

Interference due to reflection from upper and lower surfaces of a thin air film between a lens and a glass plate.

Condition for Dark Rings

2t=nλ
 

Radius of nth Dark Ring

rn=√nλR
 

Applications

• Determination of wavelength
• Measurement of refractive index

7. Michelson Interferometer

Principle

Interference by amplitude splitting using a beam splitter.

Working

• Incident light split into two beams
• Beams travel different paths
• Recombine to produce interference fringes

Applications

• Measurement of wavelength
• Determination of small distances
• Testing optical components

8. Fraunhofer Diffraction

Occurs when source and screen are effectively at infinity (using lenses).

(a) Single Slit Diffraction

Condition for Minima

asin⁡θ=nλ(n=1,2,3...)
 

Intensity Pattern

• Central maximum is brightest and widest
• Side maxima decrease in intensity

(b) Circular Aperture Diffraction

Airy Disc

• Central bright spot surrounded by rings
• Important in optical instruments

9. Rayleigh Criterion for Limit of Resolution

Statement

Two point sources are just resolved when the principal maximum of one coincides with the first minimum of the other.

Limit of Resolution

θ_min=1.22 λ/D
 

where

• D= diameter of aperture

10. Application to Vision

Human Eye

• Pupil diameter ≈ 2–5 mm
• Resolution depends on wavelength and pupil size

Telescopes & Microscopes

• Larger aperture → better resolution

11. Diffraction Grating

Definition

An optical element with large number of equally spaced slits.

Grating Equation

dsin⁡θ=nλ
 

12. Resolving Power of Grating

Definition

Ability to distinguish between two close wavelengths.

Formula

Resolving Power=λ/Δλ=nN
 

where

• n= order of diffraction
• N= number of slits illuminated

13. Summary Table

Phenomenon	Key Formula
Fringe width (YDSE)	β=λD/d
Single slit minima	asin⁡θ=nλ
Rayleigh criterion	θ=1.22λ/D
Grating resolution	nN

 

UNIT–IV: LASERS

1. Laser: Basic Idea

LASER stands for Light Amplification by Stimulated Emission of Radiation.

Essential Requirements of a Laser

1. Active medium
2. Population inversion
3. Optical resonator (cavity)

2. Einstein’s Theory of Matter–Radiation Interaction

Einstein proposed three processes of interaction between atoms and radiation.

(a) Absorption

• Atom in lower energy state E1absorbs a photon
• Moves to higher energy state E2

E2 - E1=hν
 

Einstein Coefficient: B12

(b) Spontaneous Emission

• Excited atom returns to lower state on its own
• Emits photon in random direction and phase

Einstein Coefficient: A21

(c) Stimulated Emission

• Incident photon forces excited atom to emit another photon
• Emitted photon has:
  • Same frequency
  • Same phase
  • Same direction

Einstein Coefficient: B21

3. Relation Between Einstein A and B Coefficients

A₂₁/B₂₁=8πhν³/c³
 

This shows that stimulated emission dominates at lower frequencies, enabling laser action.

4. Population Inversion

Definition

Population inversion is the condition where number of atoms in excited state exceeds those in ground state.

N2>N1
 

Importance

• Necessary for light amplification
• Not possible in thermal equilibrium

5. Amplification of Light

When a photon passes through an inverted medium:

• Stimulated emission dominates
• Number of photons increases
• Light intensity grows exponentially

6. Three-Level Laser

Energy Levels

1. Ground state E1
2. Pumping level E3
3. Metastable level E2

Working

• Atoms pumped from E1→E3
• Rapid decay to metastable state E2
• Laser transition E2→E1

Example

• Ruby laser

7. Types of Lasers

(a) Gas Laser – He–Ne Laser

Active Medium

Mixture of helium and neon gases

Wavelength

• 632.8 nm (red)

Features

• Continuous wave operation
• High coherence
• Long lifetime

Applications

• Holography
• Alignment
• Optical experiments

(b) Solid-State Lasers

(i) Ruby Laser

• Active medium: Cr³⁺ doped Al₂O₃
• Wavelength: 694.3 nm
• Pulsed laser
• Three-level system

(ii) Nd:YAG Laser

• Active medium: Nd³⁺ doped YAG crystal
• Wavelength: 1064 nm
• Four-level laser
• High efficiency

(c) Semiconductor Lasers

Principle

Laser action due to electron–hole recombination at a p–n junction.

Features

• Small size
• High efficiency
• Low operating voltage

Applications

• Optical fiber communication
• Barcode scanners
• CD/DVD players

8. Properties of Laser Beams

(a) Monochromaticity

• Very narrow wavelength spread

(b) Coherence

• Fixed phase relationship
• High temporal and spatial coherence

(c) Directionality

• Highly collimated beam
• Very small divergence

(d) High Brightness

• Large power concentrated in small area

9. Applications of Lasers

(a) Science

• Spectroscopy
• Holography
• Interferometry

(b) Engineering

• Cutting and welding
• Material processing
• Range finding (LIDAR)

(c) Medicine

• Eye surgery (LASIK)
• Cancer treatment
• Dermatology

10. Comparison of Different Lasers

Laser Type	Medium	Nature	Wavelength
He–Ne	Gas	Continuous	632.8 nm
Ruby	Solid	Pulsed	694.3 nm
Nd:YAG	Solid	Continuous/Pulsed	1064 nm
Semiconductor	p–n junction	Continuous	IR/Visible