Course Name: Physics-II Course Code: 10B11PH211 Course Credits: 4 (3 1 0)

Course Name: Physics-II Course Code: 10B11PH211 Course Credits: 4 (3 1 0) Total number of Lectures: 40 • COURSE DESCRIPTION • Broad Area: Electromagnetism • Fiber Optics • Thermodynamics • Quantum mechanism • Solid state physics

Electromagnetism • Electrostatics: • Coulomb’s law, • Gauss law and its applications, • Treatment of electrostatic problems by solution • of Laplace and Poisson’s equations. • Magnetostatics: • Biot-Savart law, • Faraday’s Law, • Ampere’s law, • Maxwell’s equations in free space and dielectric media. • Propagation of EM waves through boundary- • Reflection • Refraction • Absorption and • Total Internal Reflection.

Books: • Introduction to Electrodynamics • by D.J. Griffith • Principles of Electromagnetics • by Matthew N. O. Sadiku • Electromagnetics • by Edminister (Schuam series) • Engineering Electromagnetic • by W H Hayt & J A Buck

What is Electromagnetics?

Electromagnetism Optics Electricity Magnetism

Choice is based on symmetry of problem To understand the Electromagnetic, we must know basic vector algebra and coordinate systems. So let us start the coordinate systems. COORDINATE SYSTEMS • RECTANGULAR or Cartesian Sheets - RECTANGULAR • CYLINDRICAL Examples: Wires/Cables - CYLINDRICAL • SPHERICAL Spheres - SPHERICAL

Orthogonal Coordinate Systems: 1. Cartesian Coordinates z P(x,y,z) Or y Rectangular Coordinates x P (x, y, z) z z P(r, , z) 2. Cylindrical Coordinates P (r, , z) y r x Φ z 3. Spherical Coordinates P(r, θ,) θ r P (r, θ, ) y x Φ

z z Cartesian Coordinates P(x, y, z) P(x,y,z) P(r, θ, Φ) θ r y x y x Φ Cylindrical Coordinates P(r, Φ, z) Spherical Coordinates P(r, θ, Φ) z z P(r, Φ, z) y r x Φ

Cartesian Coordinates Differential quantities:

dy dx y 6 2 3 7 x AREA INTEGRALS • integration over 2 “delta” distances Example: AREA = = 16 Note that: z = constant

Cylindrical coordinate system (r,φ,z) Z Z Y r φ X

Cylindrical Coordinates: Visualization of Volume element Differential quantities: Limits of integration of r, φ,z are 0<r<∞ , o<φ <2π ,0<z <∞

Spherical coordinate system (r,,φ) Radius=r 0<r<∞ • -Zenith angle • 0<θ < ( starts from +Z reaches up to –Z) ,  -Azimuthal Angle 0<φ <2 (starts from +X direction and lies in x-y plane only)

Spherical coordinate system (r,,φ) P(r, θ, φ) Z r sin θ dr P r cosθ r dθ θ r dθ Y dφ φ r sinθ dφ r sinθ X

Spherical Coordinates Differential quantities: Length Element: Area Element: Volume Element:

Spherical coordinate system (r,,φ) P(r, θ, φ) Z r sin θ dr P r cos θ r dθ θ r dθ Y dφ φ r sinθ dφ r sinθ X Try Yourself: Surface area of the sphere= 4πR2 .

Points to remember

Determine a) Areas S1, S2 and S3. b) Volume covered by these surfaces.

Assignment 1: Basics of fields, Gradient, Divergence and Curl.

Lecture – 2 Coulomb’s law, Electric Flux & Gauss’s law

Vector Analysis n B )  A

Scalar and Vector Fields • A scalar field is a function that gives us a single value of some variable for every point in space. voltage, current, energy, temperature • A vector is a quantity which has both a magnitude and a direction in space. velocity, momentum, acceleration and force

Gradient, Divergence and Curl The Del Operator  • Gradient of a scalar function is a vector quantity. • Divergenceof a vector is a scalar quantity. • Curl of a vector is a vector quantity.

Operator in Cartesian Coordinate System gradT: points the direction of maximum increase of the function T. Divergence: Curl: where

Operator in Cylindrical Coordinate System Gradient: Divergence: Curl:

Operator in Spherical Coordinate System Gradient : Divergence: Curl:

Fundamental theorem for divergence and curl • Gauss divergence theorem: • Stokes curl theorem Conversion of volume integral to surface integral and vice verse. Open S Closed L Conversion of surface integral to line integral and vice verse.

Coulomb’s Law Like charges repel, unlike charges attract The electric force acting on a point charge q1 as a result of the presence of a second point charge q2 is given by Coulomb‘sLaw: where e0 = permittivity of space

Electric Flux The number of electric field lines through a surface A  E  E=EA, • Conclusion: • The total flux depends on • strength of the field, • the size of the surface area it passes through, • and on how the area is oriented with respect to the field.

Gauss's Law • The total of the electric flux out of a closed surface is equal to the charge enclosed divided by the permittivity (eo.). • eo = the permittivity of free space 8.854x10-12 C2/(N m2) da E +q Integral Form Differential Form where

-ve flux +ve Flux

The defining conditions of a Guassian surface: • The surface is closed. • At each point of the surface E is either normal or tangential to the surface. • E is sectionally constant over that part of the surface where E is normal.

Electric lines of flux and Derivation of Gauss’ Law using Coulombs law • Consider a sphere drawn around a positive point charge. Evaluate the net flux through the closed surface. Net Flux = For a Point charge dA Gauss’ Law

Differential form of Gauss Law: Proof: Gauss Law Gauss divergence theorem: or Note: Gauss law is also known as Maxwell’s first equation.

+q C A da E da D B Where dΩ is solid angle Asmnt 2: Proof of the Gauss’s law for the charge inside da E +q and outside the Gaussian surface

Applications of Gauss law(Spherical distribution systems) • Conducting Sphere of charge ‘q’ and radius ‘R’: • E at an external point: Eo • E at the surface: Es • E at an internal point: Ei • Nonconducting Sphere • E at an external point: Eo • E at the surface: Es • E at an internal point: Ei

(Spherical systems: Conducting Sphere) Gaussian surface • Conducting Sphere of charge ‘q’ and radius ‘R’: • E at an external point: Eo r>R • E at the surface: Es r=R • E at an internal point: Ei r<R Case-I: E at an external point; Net electric fux through ‘P’: R P r S1 The Electric field strength at any point outside a spherical charge distribution is the same as through the whole charge were concentrated at the centre.

Gaussian surface r=R Gaussian surface R r (Spherical systems: Conducting Sphere) Case-II: E at the Surface; Case-III: E at an internal point;

R P r E Es Eo Ei=0 r=0 r=R r (Spherical systems: Conducting Sphere)

(Spherical systems: Nonconducting Sphere) Nonconducting sphere (Volume charge density) • E at an external point: Eo • E at the surface: Es • E at an internal point: Ei

Gaussian surface R P r S1 (Spherical systems: Nonconducting Sphere) • Nonconducting Sphere of charge ‘q’ and radius ‘R’: • E at an external point: Eo r>R • E at the surface: Es r=R • E at an internal point: Ei r<R Case-I: E at an external point; Net electric flux through ‘P’:

Course Name: Physics-II Course Code: 10B11PH211 Course Credits: 4 (3 1 0)