PHYSICS-II (PHY C132) ELECTRICITY & MAGNETISM

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Presentation transcript:

PHYSICS-II (PHY C132) ELECTRICITY & MAGNETISM Introduction to Electrodynamics: by David J. Griffiths (3rd Ed.) Dr. Champak B. Das ( BITS, Pilani)

Dr. Champak B. Das ( BITS, Pilani) VECTOR ANALYSIS Differential Calculus Integral Calculus Revisited Curvilinear Coordinates The Dirac Delta Function Theory of Vector Fields Dr. Champak B. Das ( BITS, Pilani)

Differential Calculus Derivative of any function f(x,y,z): Dr. Champak B. Das ( BITS, Pilani)

Gradient of function f  f is a VECTOR Change in a scalar function f corresponding to a change in position : Gradient of function f  f is a VECTOR Dr. Champak B. Das ( BITS, Pilani)

Dr. Champak B. Das ( BITS, Pilani) Geometrical interpretation of Gradient Z P Q dl Y change in f : X =0 => f  dl Dr. Champak B. Das ( BITS, Pilani)

Dr. Champak B. Das ( BITS, Pilani) Z Q dl P Y X Dr. Champak B. Das ( BITS, Pilani)

The rate of change of f is max. for The max. value of rate of change of f is f increases in the direction of Grad f is in the direction of the normal to the surface of constant f Dr. Champak B. Das ( BITS, Pilani)

Dr. Champak B. Das ( BITS, Pilani) Gradient of a function slope of the function along the direction of maximum rate of change of the function Dr. Champak B. Das ( BITS, Pilani)

If  f = 0 at some point (x0,y0,z0) => df = 0 for small displacements about the point (x0,y0,z0) (x0,y0,z0) is a stationary point of f(x,y,z) Dr. Champak B. Das ( BITS, Pilani)

Dr. Champak B. Das ( BITS, Pilani) Prob. 1.12 The height of a certain hill (in feet) is: h(x,y) = 10(2xy – 3x2 -4y2 -18x + 28y +12) where x is distance (in mile) east and y north of Pilani. (a) Where is the top located ? Ans: 3 miles North & 2 miles West Dr. Champak B. Das ( BITS, Pilani)

Dr. Champak B. Das ( BITS, Pilani) Prob. 1.12 (contd.) h(x,y) = 10(2xy – 3x2 -4y2 -18x – 28y +12) (b) How high is the hill ? Ans: 720 ft (c) How steep is the slope at 1 mile north and 1 mile east of Pilani? In what direction the slope is steepest, at that point ? Ans: 311 ft/mile, direction is Northwest Dr. Champak B. Das ( BITS, Pilani)

Dr. Champak B. Das ( BITS, Pilani) Prob. 1.13 Let rs is the separation vector from (x,y,z) to (x,y,z) . Dr. Champak B. Das ( BITS, Pilani)

Dr. Champak B. Das ( BITS, Pilani) The Operator   is NOT a VECTOR, but a VECTOR OPERATOR Satisfies: Vector rules Partial differentiation rules Dr. Champak B. Das ( BITS, Pilani)

Dr. Champak B. Das ( BITS, Pilani)  can act: On a scalar function f : f GRADIENT On a vector function F as: . F DIVERGENCE On a vector function F as:  × F CURL Dr. Champak B. Das ( BITS, Pilani)

Divergence of a vector Divergence of a vector is a scalar. Dr. Champak B. Das ( BITS, Pilani)

Geometrical interpretation of Divergence .F is a measure of how much the vector F spreads out/in (diverges) from/to the point in question. Dr. Champak B. Das ( BITS, Pilani)

Physical interpretation of Divergence Flow of a compressible fluid: (x,y,z)  density of the fluid at a point (x,y,z) v(x,y,z)  velocity of the fluid at (x,y,z) Z X Y dy dx dz A C D B E F G H Dr. Champak B. Das ( BITS, Pilani)

Dr. Champak B. Das ( BITS, Pilani) Net rate of flow out through all pairs of surfaces (per unit time): Dr. Champak B. Das ( BITS, Pilani)

Dr. Champak B. Das ( BITS, Pilani) Net rate of flow of the fluid per unit volume per unit time: DIVERGENCE Dr. Champak B. Das ( BITS, Pilani)

Dr. Champak B. Das ( BITS, Pilani) Example: Calculate, Dr. Champak B. Das ( BITS, Pilani)

Dr. Champak B. Das ( BITS, Pilani) Prob. 1.16 Sketch the vector function and compute its divergence. Explain the answer ! ! Dr. Champak B. Das ( BITS, Pilani)

Curl Curl of a vector is a vector Dr. Champak B. Das ( BITS, Pilani)

Geometrical interpretation of Curl ×F is a measure of how much the vector F “curls around” the point in question. Dr. Champak B. Das ( BITS, Pilani)

Physical significance of Curl Circulation of a fluid around a loop about a point : Y 3 2 y 4 1 x X Circulation Dr. Champak B. Das ( BITS, Pilani)

Dr. Champak B. Das ( BITS, Pilani) Circulation per unit area z-component of CURL Dr. Champak B. Das ( BITS, Pilani)

Dr. Champak B. Das ( BITS, Pilani) Sum Rules For Gradient: For Divergence: For Curl: Dr. Champak B. Das ( BITS, Pilani)

Rules for multiplying by a constant For Gradient: For Divergence: For Curl: Dr. Champak B. Das ( BITS, Pilani)

Dr. Champak B. Das ( BITS, Pilani) Product Rules For a Scalar from two functions: Gradients: Dr. Champak B. Das ( BITS, Pilani)

Dr. Champak B. Das ( BITS, Pilani) Product Rules For a Vector from two functions: Divergences: Dr. Champak B. Das ( BITS, Pilani)

Dr. Champak B. Das ( BITS, Pilani) Product Rules Curls: Dr. Champak B. Das ( BITS, Pilani)

Prob. 1.21 (a) Prob. 1.21 (b) Ans: What does the expression mean ? Compute: Ans: Dr. Champak B. Das ( BITS, Pilani)

Dr. Champak B. Das ( BITS, Pilani) Quotient Rules Dr. Champak B. Das ( BITS, Pilani)

Dr. Champak B. Das ( BITS, Pilani) Second Derivatives Of a gradient: Divergence : Laplacian Curl : ( Prob. 1.27: Prove it ! ) Dr. Champak B. Das ( BITS, Pilani)

Dr. Champak B. Das ( BITS, Pilani) Second Derivatives Of a divergence: Gradient : Dr. Champak B. Das ( BITS, Pilani)

Dr. Champak B. Das ( BITS, Pilani) Second Derivatives Of a Curl: Divergence : Prob. 1.26: Prove it ! Curl : Dr. Champak B. Das ( BITS, Pilani)

Dr. Champak B. Das ( BITS, Pilani) Integral Calculus Line Integral: Surface Integral: Volume Integral: Dr. Champak B. Das ( BITS, Pilani)

Fundamental theorem for gradient Line integral of gradient of a function is given by the value of the function at the boundaries of the line. Dr. Champak B. Das ( BITS, Pilani)

Dr. Champak B. Das ( BITS, Pilani) Corollary 1: Corollary 2: Dr. Champak B. Das ( BITS, Pilani)

Fundamental theorem for Divergence The integral of divergence of a vector over a volume is equal to the value of the function over the closed surface that bounds the volume. Gauss’ theorem, Green’s theorem Dr. Champak B. Das ( BITS, Pilani)

Fundamental theorem for Curl Integral of a curl of a vector over a surface is equal to the value of the function over the closed boundary that encloses the surface. Stokes’ theorem Dr. Champak B. Das ( BITS, Pilani)

Dr. Champak B. Das ( BITS, Pilani) Corollary 1: Corollary 2: Dr. Champak B. Das ( BITS, Pilani)

Curvilinear coordinates: used to describe systems with symmetry. Spherical Polar coordinates (r, , ) Cylindrical coordinates (s, , z) Dr. Champak B. Das ( BITS, Pilani)

Spherical Polar Coordinates A point is characterized by: r : distance from origin Z  : polar angle P r   : azimuthal angle Y  X Dr. Champak B. Das ( BITS, Pilani)

Cartesian coordinates in terms of spherical coordinates: Z P r  Y  X Dr. Champak B. Das ( BITS, Pilani)

Spherical coordinates in terms of Cartesian coordinates: Z P r  Y  X Dr. Champak B. Das ( BITS, Pilani)

Unit vectors in spherical coordinates Prob. 1.37 : Unit vectors in spherical coordinates Z r  Y  X Dr. Champak B. Das ( BITS, Pilani)

Dr. Champak B. Das ( BITS, Pilani) Line element in spherical coordinates: Volume element in spherical coordinates: Dr. Champak B. Das ( BITS, Pilani)

Dr. Champak B. Das ( BITS, Pilani) Area element in spherical coordinates: on a surface of a sphere (r const.) on a surface lying in xy-plane ( const.) Dr. Champak B. Das ( BITS, Pilani)

Dr. Champak B. Das ( BITS, Pilani) Ranges of r,  and  r : 0    : 0    : 0  2 Dr. Champak B. Das ( BITS, Pilani)

The Operator  in Spherical Polar Coordinates Dr. Champak B. Das ( BITS, Pilani)

Dr. Champak B. Das ( BITS, Pilani) Gradient: Divergence: Dr. Champak B. Das ( BITS, Pilani)

Dr. Champak B. Das ( BITS, Pilani) Curl: Dr. Champak B. Das ( BITS, Pilani)

Dr. Champak B. Das ( BITS, Pilani) Laplacian: Dr. Champak B. Das ( BITS, Pilani)

Cylindrical Coordinates A point is characterized by: s : distance from z-axis Z z : cartesian coordinate s P  : azimuthal angle z Y  X Dr. Champak B. Das ( BITS, Pilani)

Prob. 1.41 : Unit vectors in cylindrical coordinates Z s z Y  X Dr. Champak B. Das ( BITS, Pilani)

Dr. Champak B. Das ( BITS, Pilani) Line element in cylindrical coordinates: Volume element in cylindrical coordinates: Dr. Champak B. Das ( BITS, Pilani)

Dr. Champak B. Das ( BITS, Pilani) Ranges of s,  and z s : 0    : 0  2 z : -    Dr. Champak B. Das ( BITS, Pilani)

The Operator  in Cylindrical Coordinates Dr. Champak B. Das ( BITS, Pilani)

Dr. Champak B. Das ( BITS, Pilani) Gradient: Divergence: Dr. Champak B. Das ( BITS, Pilani)

Dr. Champak B. Das ( BITS, Pilani) Curl: Laplacian: Dr. Champak B. Das ( BITS, Pilani)

Dr. Champak B. Das ( BITS, Pilani) General expressions for the Derivatives in different coordinate systems u, v, w Coordinate System: Line element : Dr. Champak B. Das ( BITS, Pilani)

Dr. Champak B. Das ( BITS, Pilani) System u v w f g h Cartesian x y z 1 Spherical r   r sin Cylindrical s Dr. Champak B. Das ( BITS, Pilani)

Dr. Champak B. Das ( BITS, Pilani) GRADIENT Dr. Champak B. Das ( BITS, Pilani)

Dr. Champak B. Das ( BITS, Pilani) DIVERGENCE Dr. Champak B. Das ( BITS, Pilani)

Dr. Champak B. Das ( BITS, Pilani) CURL : Dr. Champak B. Das ( BITS, Pilani)

Dr. Champak B. Das ( BITS, Pilani) LAPLACIAN Dr. Champak B. Das ( BITS, Pilani)

Recall  Prob. 1.16 Sketch the vector function and compute its Divergence Dr. Champak B. Das ( BITS, Pilani)

Dr. Champak B. Das ( BITS, Pilani) Calculation of Divergence => Divergence theorem => Dr. Champak B. Das ( BITS, Pilani)

And its integral over ANY volume containing the point r = 0 Note: as r  0; v  ∞ And its integral over ANY volume containing the point r = 0 Dr. Champak B. Das ( BITS, Pilani)

Dr. Champak B. Das ( BITS, Pilani) THE DIRAC DELTA FUNCTION Dr. Champak B. Das ( BITS, Pilani)

The Dirac Delta Function An infinitely high, infinitesimally narrow “spike” with area 1  Dirac Delta Function is NOT a Function Dr. Champak B. Das ( BITS, Pilani)

A Generalized Function OR distribution The Defining Characteristic Integral : A Generalized Function OR distribution Dr. Champak B. Das ( BITS, Pilani)

Dr. Champak B. Das ( BITS, Pilani) Delta function is something that is always intended for use under an integral sign. Let D1(x) & D2(x) are two expressions involving Delta functions and f(x) is any ordinary function Dr. Champak B. Das ( BITS, Pilani)

Dr. Champak B. Das ( BITS, Pilani) One can show: ………..for a proof, see Example 1.15 Dr. Champak B. Das ( BITS, Pilani)

Dr. Champak B. Das ( BITS, Pilani) The Dirac Delta Function Shifting the singularity from 0 to a; Dr. Champak B. Das ( BITS, Pilani)

Dr. Champak B. Das ( BITS, Pilani) The Dirac Delta Function & the Defining Characteristic Integral : Dr. Champak B. Das ( BITS, Pilani)

Dr. Champak B. Das ( BITS, Pilani) Prob. 1.43: Dr. Champak B. Das ( BITS, Pilani)

Dr. Champak B. Das ( BITS, Pilani) Prob. 1.45 : Dr. Champak B. Das ( BITS, Pilani)

Dr. Champak B. Das ( BITS, Pilani) THE DIRAC DELTA FUNCTION Dr. Champak B. Das ( BITS, Pilani)

Dr. Champak B. Das ( BITS, Pilani) The Dirac Delta Function Shifting the singularity from 0 to a; Dr. Champak B. Das ( BITS, Pilani)

Dr. Champak B. Das ( BITS, Pilani) The Dirac Delta Function (in three dimension) Dr. Champak B. Das ( BITS, Pilani)

Why such a function ? Describe very short range forces as nuclear force Describe a point particle in terms of a mass density Describe a point charge in terms of a charge density Dr. Champak B. Das ( BITS, Pilani)

Dr. Champak B. Das ( BITS, Pilani) Prob. 1.46: Charge density of a point charge q at r : Charge density of a dipole with -q at 0 and +q at a: Dr. Champak B. Das ( BITS, Pilani)

Dr. Champak B. Das ( BITS, Pilani) Prob. 1.46: (contd.) Charge density of a thin spherical shell of radius R and total charge Q: Dr. Champak B. Das ( BITS, Pilani)

Dr. Champak B. Das ( BITS, Pilani) The Paradox of Divergence of From calculation of Divergence: By using the Divergence theorem: Dr. Champak B. Das ( BITS, Pilani)

Dr. Champak B. Das ( BITS, Pilani) So now we can write: Such that: Dr. Champak B. Das ( BITS, Pilani)

Theory of Vector Fields By specifying appropriate boundary conditions, Helmholtz theorem implies that the field can be uniquely determined from its divergence and curl. Dr. Champak B. Das ( BITS, Pilani)

Dr. Champak B. Das ( BITS, Pilani) Potentials THEOREM 1: ( For Curl-less fields ) Dr. Champak B. Das ( BITS, Pilani)

Conclusions from theorem 1 If curl of a vector field vanishes, (everywhere), then the field can always be written as the gradient of a scalar potential ( not unique ) Dr. Champak B. Das ( BITS, Pilani)

Dr. Champak B. Das ( BITS, Pilani) Potentials THEOREM 2: For Divergence-less fields Dr. Champak B. Das ( BITS, Pilani)

Conclusions from theorem 2 If divergence of a vector field vanishes, (everywhere), then the field can always be written as the curl of a vector potential ( not unique ) Dr. Champak B. Das ( BITS, Pilani)

Dr. Champak B. Das ( BITS, Pilani) Helmholtz theorem: Any vector field F with both source and circulation densities vanishing at infinity may be written as the sum of two parts: one of which is curl-less and the other is divergence-less. (Always) Dr. Champak B. Das ( BITS, Pilani)