Electrodynamics and Microwaves 3. Gradient, Curl and Divergence

Transcription

1 1 Module 3 Gradient, Divergence and Curl 1. Introduction 2. The operators & 2 3. Gradient 4. Divergence 5. Curl 6. Mathematical expressions for gradient, divergence and curl in different coordinate systems. 7. Inter conversion of operator & 2 between coordinate systems 8. Mathematical Theorems and Differential Equations involving & 2 9.Vector identities involving & Summary Learning outcome: After completing this module, you will be able to 1. Understand the concepts of gradient, divergence and curl 2. Know about the Del operator in different coordinate systems 3. Study the mathematical tools computation of divergence, curl and gradient 4. Know about the vector identities useful in theoretical physics.

2 2 1. Introduction The spatial variation of the vector and scalar physical quantities is an important aspect in study of electrodynamics. The term spatial variation has a reference to the position of the system. In most of the real life applications, we are mainly interested in knowing how a physical quantity undergoes a variation from point to point in space at a given time instant. What is the net result of such variation on the system? Can any one of such results be used for real life application? We observe changes of various types of physical quantities. For example, while climbing up the hill, we might get tired due to steepness of the path. Hence, we try to search for an easier path having less steepness. The barometric pressure also goes on decreasing with increase in height of the place from the surface of the earth. A wheel free to rotate about its axle rotates fast only for a particular position in flow of water through a river or a leveled cylindrical pipe. The variation in the magnitude of a scalar physical quantity may be different in different directions giving rise to a vector quantity. Or a vector quantity may vary differently in different directions and may give rise to a new type of physical quantity which may be a scalar or a vector. Spatial variations of physical quantities can be studied with the help of an operator known as del operator through the concepts of gradient, divergence and curl. Many of the famous differential equations in theoretical physics involve these concepts and Del operator. Hence adequate knowledge about the operator DEL and the related mathematical aspects will definitely help everyone in the study of electrodynamics. 2The operators & 2 Many of the physical quantities both vectors and scalars- change with position coordinates. The vector operator DEL also known as Nabla is used to account for the directional variation of physical quantities.

3 3 Another Scalar operator Del square ( 2) has also many applications. Expression For Gradient Operator ( ) 1. Rectangular Co-ordinate System i + j + k x y 2. Cylindrical Co-ordinate System ρ k ρ ρ 3. Spherical Co-ordinate System r r + θ 1 r θ + 1 rsinθ Expression for Laplacian Operator ( 2 ) 1. Rectangular Co-ordinate System 2 = 2 x y Cylindrical Co-ordinate System 2 = 1 ρ ρ (ρ ρ ) ρ Spherical Co-ordinate System 2 = 1 r 2 r r 2 r + 1 r 2 sin θ θ sinθ θ + 1 r 2 sin 2 θ Gradient In science we have to mainly account for variation of physical quantities with respect to time and space. The choice of any action may depend upon the way in which the physical quantity involved in the process changes with distance. When we are climbing the hill we choose a curved path instead of a steep one. As we go away from the earth, the air (atmospheric) pressure decreases. While heating a metal rod the temperature is different at different points on the rod. The velocity of the water flowing through a river changes with distance of the layer from the bottom. Thus we can have many situations where some

4 4 physical quantity is changing with distance. This fact is described in terms of a mathematical quantity gradient and is denoted by. Let = A. If A is electrical or magnetic field, then is called as electric potential or magnetic potential respectively. Note that the gradient of a scalar quantity will have directional dependence and hence it is a vector quantity. It can be computed using appropriate mathematical expressions for the operator. 4 Divergence We know that the electric field produced by a charge distribution can be described in terms of the lines of force. Consider the electric field produced by electric charge Q. the electric field at point P(r ) can be measured in terms of the electric lines of force crossing normally through unit area held perpendicular to the position vector of the point under consideration. Consider two charges + Q1 and Q2 separated by a distance d between them. The lines of force due to + Q1 charge will be going away from Q1 while those due to Q2 will be coming toward Q2. If we consider a small volume v then and if n1 number of electric lines enters the volume and n 2 lines of force leave the same volume then the net change in the lines of force through the volume will be n 1 -n 2. Case 1: If n1> n2 we say that divergence of electric field is negative. Case 2: if n1<n2 we say that the divergence of electric field is positive. Case 3: if n1= n2 we say that the divergence of electric field is ero. Similarly we can think of divergence o f magnetic field, thermal field etc. For measurement, divergence of a vector is defined as the net change in the physical quantity represented by the vector per unit volume. It is denoted by. A. It be noted that the divergence of a quantity is a scalar quantity.

5 5 5. Curl 5.1 Concept of curl In everyday life we come across the phenomena wherein the application of force can cause rotation of the system and the effect depends upon the magnitude, direction and the point of application of the force to the system. As an example,consider a wheel which can be set in rotation by the water flowing in a river if proper inclination of axis of the wheel with respect to the direction of water flow is maintained. If an electric field is different at different points of a copper plate then it can cause circulation of the free electric charge in the copper plate. In all such examples, a f rotation/ circulation of is the main process involved. To explain this fact the mathematical concept f curl of vector is developed. Curl of a vector is ero then implies no rotation. it is symbolically denoted by A let A = B. If Bis force field then Ais called as the vector potential giving rise to field B. In fact A is called magnetic vector potential corresponding to magnetic induction B? In general, force fields are set to be derivable from the corresponding potential. It is a vector quantity and can be computed using appropriate mathematical formulae in the different co ordinate. 5.2 Conservative Field When we talk about force, it be remembered that the total force F is many a times expressed as a sum of conservative, non conservative and some other type of forces if any. Let us discuss about the conservative force. There are different ways in which a conservative force field can be described. i) If curl of a vector A is ero, then the vector A is said to be conservative. Any force field varying inversely with square of the distance is conservative. Electrostatic field is conservative. ii) If the work done in displacing a charge in the force field A is independent of the path chosen, then the A is said to be conservative.

6 6 Let us assume that a charge is displaced from point A to point D by two independent paths namely ABCD and APD respectively. Let W1 = work done for path ABCD W2 = work done for path APD If W1 = W2 then the field is a Conservative force field. iii) If the total work done in displacing a charge in the field A in closed path is ero, then the vector field A is said to be conservative. 6. Mathematical expressions for gradient, divergence and curl in different coordinate systems. 6.1 Rectangular Co-ordinate system Grad = = i + j + k x y Div Ā = Ā = A x x + A y y + A Curl Ā = Ā = i j k x y A x A y A

7 7 6.2 Cylindrical Co-Ordinate System Grad = ρ k ρ ρ Div Ā = A ρ + A p + 1 A + A ρ ρ ρ Curl Ā = 1 ρ ρ ρ ρ A ρ ρa A 6.3 Spherical Co-Ordinate System Grad = r + θ r r θ rsin θ Div Ā = 2 r A r + A r r + 1 r A θ + cot θ A θ r θ + 1 A rsin θ Curl Ā = 1 r 2 sin θ r rθ rsinθ r θ A r ra θ rsinθa 7. Inter conversions of operator & 2 between coordinate systems For conversion of from one coordinate system to the other, a step-wise procedure can be developed.. As an example, note the procedural steps for conversion of del operator from Cartesian to Cylindrical System. Step 1:- Note the expression for del in Cartesian coordinate system

8 8 Step 2 :- Note the relations between coordinates in cylindrical and Cartesian coordinate systems. Step 3:- Note the relation between unit vectors in the two coordinate systems Step 4:-Find the expressions for i x using the given expression. Step 5:-Find the expressions for j y using the given expression. Step 6:- Find the expressions for k using the given expression. Step 7:-Substitute the values of step 4,5 and 6 in the expression for in Cartesian coordinate system and simplify by rearranging proper terms. Step 8:-Use relation between unit vectors in two coordinate systems to obtain the expression for Del in cylindrical coordinate system. It be noted that, for conversion of from one coordinate system to the other it is essential to know how the unit vectors in different coordinate system are related with each other. Such relations can be obtained using the concept of resolution of a vector along a specified direction and addition of vectors. 8. Theorems and equations involving & Mathematical Theorems: - There are many mathematical theorems which are of great help in deduction of many useful results in electrodynamics. Some of such theorems are as listed below. x x x ρ ρ x y y y ρ ρ y ρ ρ

9 9 i) Fundamental theorem of gradient This theorem can be used to prove that Electric field is the negative gradient of electrostatic potential. ii) Stokes Theorem: Here C is the boundary of the surface S. b a b a dl c F.dr F ndσ s iii) Gauss Divergence Theorem FdV Fdσˆ V s Gauss Divergence Theorem can be applied to derive coulombs law in electrostatics. 8.2 Differential Equations:- Differential equations form the basis of mathematical description of many physical systems and phenomenon. Some of such important equations are as mentioned below. Equation of continuity: This is the Fundamental Equation in Physics and must be satisfied in every physically acceptable situation. Interpretation of Current density J and charge density ρ depends upon the system. For example, in quantum mechanics the current density j corresponds to the number of particles crossing the potential barrier.

10 10 Poisson s Equation Relation between charge density ρ and potential at a point is given by Poisson s equation which is of prime importance in force field theory. Laplace Equation: Poisson Equation with ρ = 0 reduces to laplace equation 2. It is useful for solving potential problems in charge free region. The standard solutions in different coord inate systems exist. Maxwell s Equations: Maxwell s equations involve curl and divergence of EM vector fields E, H, B and D and are known as source density and circulation density equations. Using Stokes Theorem and Gauss divergence Theorem, these equations can be converted from differential form to Integral form. Lorent and Coulomb gauge conditions in electrodynamics involve Del operator. Diffusion Equation and Schrodinger Equation are the examples which are of prime importance in the field of semiconductors and nanophysics. Visualiation of mathematical equations for real life application is one of the important aspect of studying science. As an example, consider Maxwell s third equation. E = B t This equation can be visualied as a physical system which can be developed for generation of Electric current through a coil rotating in a magnetic field. The magnetic induction B corresponds to the magnetic lines of force associated with the area of the conductor. If the conductor is taken in the form of a coil then the number of magnetic lines of force passing through the coil can be changed with respect to time by rotating it in the magnetic field of bar magnets as shown. As seen from the L.H.S. of the

11 11 equation, there is a generation of space varying electric field in the coil. Due to this field, the free electrons in the coil move giving rise to electric current. 9. Vector identities involving & 2 Some of the useful and important identities are enlisted below I. = 0 i.e. Curl of Gradient of is 0. II. A = A 2 A III. A B C = B A C C (A B ) IV. (A B ) C = B A C A (C B ) V. A B C + B A C + C (A B ) = 0 VI. (A B ) C D = A C B D B C A D VII.. A + B =.A +.B VIII. IX. A + B = A + B A + B = A + B X. V = V + V XI. Ā = Ā + V Ā XII. A B = A B + B A + A B + ( B)A

12 12 XIII. A B = B A A B XIV. ØA = Ø A + Ø( A) XV. = 0 XVI. Ø V = V + 2 V Various vector identities and the Mathematical Theorems described above are very much useful in many ways while studying Electrodynamics. Use of some of such important identities and theorems are stated below. a) Curl(grad Φ ) = 0 This identity is used to establish the relation between electrostatic field E and the corresponding electric potential V. It is known that E is conservative, hence we have, curl E = 0 From vector identity under consideration, Curl (grad Φ) = 0 By comparison we find E= ± grad Φ Mathematically, both signs vi + and are allowed for grad Φ.However, knowing the physical reality that the electrostatic potential decreases with distance of the field point from the source, we accept only sign. Thus, we have, = Φ. b) Div(curl A)= 0 OR. ( A )= 0 From this identity we get the relation between Magnetic induction B and Magnetic vector potential A as follows. From Maxwell s equation we have.b = 0. From vector identity.( A ) = 0, by comparison of the two equations we obtain A = B where A is known as magnetic vector potential. c) Gauss divergence theorem is used to derive Coulombs law in electrostatics. d) Stokes theorem is used in the derivation of Poynting vector Theorem.

13 13 e) The identity A =.A. A is used in deriving the wave equations for E and H. 10. Summary The spatial variation of physical quantities can be studied using the concept of gradient, divergence and curl.the vector operator Del is involved in the mathematical description of these concepts. Divergence represents the net change in the quantity per unit volume and is scalar while the curl and gradient accounting for directional change of the quantity are vectors. Many differential equations involve the operators del and del square. The equation of continuity and Poisson equation are the basic equations in ED. The realiation of mathematical equations can lead to several useful applications in everyday life.