EE6302 ELCTROMAGNETIC THEORY UNIT I ELECTROSTATICS I

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1 13 EE630 ELCTROMAGNETIC THEORY UNIT I ELECTROSTATICS I 1. Define Scalar and Vector Scalar: Scalar is defined as a quantity that is characterized only by magnitude. Vector: Vector is defined as a quantity that is characterized both by magnitude and direction.. Define Scalar Multiplication of two vectors Scalar multiplication of two vectors is defined as a scalar quantity whose magnitude is the product of the magnitudes of the vectors multiplied by the cosine of the angle between them. It is referred as dot product. A. B = ABcosθ 3. Define Vector Multiplication of two vectors The vector product of two vectors is defined as a vector whose magnitude is the product of the magnitudes of the two vectors and the sine of the angle between them. This multiplication is called Cross Product. A x B = ABsinθ 4. Show that the two vectors A = 6a x + a y -5a z and B = 3(a x + a y -a z ) are perpendicular to each other. For perpendicular dot, product of two vectors should be zero. A. B 0 A =6a x + a y -5a z B =3(a x + a y -a z ) A. B = 6 x 5 x 5 5 x 4 = 0 Vector A and B are perpendicular to each other.

2 5. Show that the two vectors A =4a x - a y +a z and B =-6a x +3 a y -3a z are parallel to each other. For parallel A XB0 A x B = A =4a x - a y +a z B =-6a x +3 a y -3a z a a a x 4 6 y 3 z 3 = a x (6-6) - a y (-6+6) +a z (1-1) = 0 Vectors A and B are parallel to each other. 6. Define Gradient of any scalar function The gradient of any scalar function is defined as the maximum space rate of change of that function. If the scalar V represents electric potential, V represents potential gradient. V V V V= a x + a y + x y z This operation is called the gradient. 7. Define Divergence of a vector The divergence of a vector A at any point is defined as the limit of its surface integrated per unit volume, as the volume enclosed by the surface shrinks to zero. a z 1.A =LtV 0 v S A. n ds. A.A = X + A Y A + Z x y z This operation is called divergence. Divergence of a vector is a scalar quantity. 8. Define Curl of a vector The curl of a vector A at any point is defined as the limit of its cross product with normal over a closed surface per unit volume as the volume shrinks to zero. 1 x A =LtV 0 v S n x Ads. 14

3 9. Show that the vector H = 3y 4 za x + 4x 3 z a y + x 3 y a z is solenoidal. For solenoidal H. 0.H= = x a x + y a y + z (3y 4 z)+ ( 4x 3 z )+ ( x 3 y ) x y z = = 0 Hence H is solenoidal. az. (3y 4 za x + 4x 3 z a y + x 3 y a z ) 10. Find the dot products of the vectors A and B if A = ax 3ay 4az and B ax ay A.B a z =A x B x +A y B y +A z B z = (-1)-3() +4() = Given A= 4ay 8az and B ay 6az find A. B A.B = AxBx +AyBy +AzBz = 4(-) + 8(6) = Write the expression for conversion of Cartesian to Cylindrical system. The Cartesian co-ordinates (x, y, z) can be converted into cylindrical co-ordinates Φ, z). Given Transform x r = x y Φ = tan -1 (y/x) z z = z y (r,

4 16 co- 13. Write the expression for conversion of Cylindrical to Cartesian system. The Cylindrical co-ordinates (r, Φ, z) can be converted into Cartesian ordinates (x, y, z). Given Transform r x = r cosθ Φ y = r sinθ z z = z 14. Write the expression for conversion of Cartesian to Spherical system. The Cartesian co-ordinates ( x, y, z ) can be converted into spherical co-ordinates θ, Φ). Given Transform x r = y θ =cos -1 ( z Φ = tan -1 (y/x) 15. Write the expression for conversion of spherical to Cartesian system. The spherical co-ordinates (r, θ, Φ) can be converted into Cartesian co-ordinates y, z). Given Transform r x = rsinθ.cosφ θ y = r sinθ.sin Φ Φ z = rcosθ x y x z z y z ) (r, (x, 16. Transform the Cartesian co-ordinates x =, y = 1 and z = 3 into Spherical co-ordinates. Given Transform x = r = y = 1 θ =cos -1 ( z = 3 x y z = x z y z = 3.74 ) = cos-1( Φ = tan -1 (y/x) = tan -1 (1/) = 6.56 The spherical co-ordinates = (3.74, 36.7, 6.56). 3 ) =

5 17. Write the Cartesian co-ordinates of a point whose Cylindrical co-ordinates are r = 1, Φ =45, z=. Given Transform r = 1 x = r cosθ = 1.cos45 = Φ =45 y = r sinθ = 1.sin45 = z = z = z = The Cartesian co-ordinates = (0.707, 0.707, ) 18. State Divergence theorem. The volume integral of the divergence of a vector field over a volume is equal to the surface integral of the normal component of this vector over the surface bounding the volume. v. AdV = S A. ds 19. State Stoke s Theorem. The line integral of a vector around a closed path is equal to the surface integral of the normal component of its equal, to the integral of the normal component of its curl over any closed surface. H. dl = xhds S 0. Express the value of differential volume in rectangular and cylindrical co-ordinates systems. For rectangular co-ordinate dv = dxdydz For cyclindrical co-ordinate dv = rdrdθdz. 1. Write the expression for differential length in cylindrical and spherical coordinates. For cylindrical co-ordinates dl = ( dr) ( rd) ( dz For spherical co-ordinates ) 17

6 18 dl = ( dr) ( rd ) ( rsin d). Define Unit Vector A unit vector in a given direction is defined as a direction in that direction divided by its magnitude. (or) A unit vector is having unit magnitude and directed along the co-ordinate axes. a r = r r 3. Find the distance from A (1,, 3) to B (,0,-1) in rectangular co-ordinates. r = = ( x x ) ( y y ) ( z z 1 1 1) ( 1 ) (0 ) ( 1 3 = = 1 1 ) 4. What is the divergence of curl of a vector?. X H 0 5. Fill up the blanks x xa xxa. A A 6. What is the physical significance of divergence of D? The divergence of the vector flux density D is the outflow from a small closed surface per unit volume as the volume shrinks to zero..d = LT v0 Dds. v 7. Express the divergence of a vector in the three systems of orthogonal co-ordination. For rectangular co-ordinate system

7 19.B = Bx By Bz x y z For cylindrical co-ordinate system 1.B = ( rbr) B Bz 1 r r r z For spherical co-ordinate system.b = r 1 ( r sin Br) 1( rsinb ) rb ( ) sin r r 8. Show that the two vectors A = 6a x + a y - 5a z and B = 3( ax -a y + a z ) are perpendicular to each other. A. B = (6x3) + (1x-3) + (-5x-3) = = 0 UNIT II ELECTROSTATICS II 1. Define Electric Flux and Electric Flux Density Electric Flux: Electric flux is defined as the lines of electric force. It is denoted by. = Q (charge) Coulomb. Electric Flux Density: Electric flux density or displacement density is defined as the electric flux per unit area. D = A Q. State the point form of Gauss s law. The divergence of electric flux density is equal to the volume charge density.

8 0 D. ρ v. 3. State Coulomb s law. Coulomb s law states that the force between two very small charged objects separated by a large distance compared to their size is proportional to the charge on each object and inversely proportional to the square of the distance between them. F Q 1 Q 1 F r Q1Q F r Q1Q F = a 1 4r Newton. 4. Write the applications of Gauss s law in electrostatics. Gauss s law is applied to determine the electric field intensity from a closed surface. ( e.g) Electric field can be determined for shell, two concentric shells or cylinders, etc. 5. What is meant by point charge? Point charge is one whose maximum dimension is very small in comparison with any other length. 6. What is meant by linear, surface and volume charge densities? Linear Charge density: It is the charge per unit length ( Col / m) at a point on the line of charge. Q ρ l = Lt l 0 ( ) l Surface charge density: It is the charge per surface area ( C/m ) at a point on the surface of the charge. ρ s = Lt s 0 ( Q ) s

9 Volume charge density: It is the charge per volume ( C/m 3 ) at a point on the volume of the charge. ρ s = Lt v 0 ( Q ) v 7. Define Electric Field Intensity or Electric Field Electric field intensity is defined as the electric force per unit positive charge. It is denoted by E. E = Q F Q E = 4r 8. Define Potential and Potential Difference Potential: Potential at any point is defined as the work done in moving a unit positive charge from infinity to that point in an electric field. Q V = Volts. 4r Potential Difference: Potential difference is defined as the work done in moving a unit positive charge from one point in an electric field. Q 1 1 V = ( ) Volts. 4 r A rb 9. What is the relation between intensity of electric field Eand electric flux density D in free space? D = εe c/m Where ε Permittivity of the medium, ε = ε o ε r 10. Write the relationship between potential gradient and electric field. E = E = - V x a x + y a y + z az v 1

10 11. What is the electric field intensity at a distance of 0 cm from a charge of μc in vacuum? Q E = 4r x10 4x8.854x 10 6 = 1 x9x10 = x(0.) = 450 KV/m. 1. Find the electric potential at a point (4, 3) m due to a charge of 10-9 C located at the origin in free space. Q V = 4 O r r = 4 3 = 5m V = = 1.8V 4x8.854x 101x(5) 13. What is the physical significance of divd? D. ρ v The divergence of a vector flux density is electric flux per unit volume leaving a small volume. This is equal to the volume charge density. 14. Write the Poisson s equation and Laplace equation. Poission equation V ρ/ε ρ Volume charge density ε Permittivity of the medium - Laplacian operator.

11 3 V x V + Y V z + = - ρ/ε Laplace equation V 0 V V + x Y V z + =0 15. Represent in unit vector along a vector R=6a x +8a y 6 y R a Unit Vector a R = = x + 8a R = = 6a x + 8a y 100 6a x + 8a y 10 = 0.6a x +0.8a y 16. A uniform line charge, infinite in extent, with ρ l = 0nc/m lines along the z axis. Find E at (6, 8, 3) m. r = = = 109 E = ρ l / πε o r = 0x10 x8.854x x 109 = V/m 17. State the condition for the vector F to irrotational. The vector F is said to be irrotational if XF = Define Dipole and Dipole Moment

12 Dipole or electric dipole is defined as two equal and opposite point charges separated by a very small distance. The product of electric charge and distance ( spacing) is known as dipole moment. It is denoted by m Q is the charge and l is the length. m = Q. l C/m 19. Define Capacitor A capacitor is defined as a device which consists of two conductors separated by a dielectric medium. 0. Define Capacitance The capacitance of two conducting planes is defined as the ratio of magnitude of charge on either of the conductors to the potential difference between conductors. It is given by Q C V The unit of capacitance is coulomb / volt or Farad. 1. Determine the capacitance of a parallel plate capacitor with two metal plates of size 30cm x 30cm separated by 5mm in air medium. Given data, A = 0.3 X0.3 = 0.09m d =5 x 10-3 m. ` ε o = x 10 - C = A εo 4 = 0.09X 8.854X10 5X = 15.9 nf. Write the expression for the value of capacitance for a co-axial cable. or C = b ln a b outer radius.

13 5 a inner radius. 3. Write the expression for the energy density in electro static field. W 1 E v 1 = DE 4. Find the energy stored in a parallel plate capacitor of 0.5m by 1m if it has a separation of cm and a voltage difference of 10V. C = ε o d A x10 x0.5x1 = x10 =.135x10-10 F Energy stored in a capacitor E = 1/ CV = 1/ X.135 X X10 = X 10-8 Joules. 5. Write down the expression for the capacitance between two co-axial cylinders. o C = d ln a d distance between two transmission lines in m a radius of cylinders in m 6. State the boundary conditions at the interface between two perfect dielectrics. a) The tangential component of electric field E is continuous at the surface. That E is the same just outside the surface as it is just inside the surface. E t1 = E t

14 b) The normal component of electric flux density is continuous if there is no surface charge density. Otherwise D is discontinuous by an amount equal to the surface charge density. D n1 = D n 7. A parallel plate capacitor has a charge of 10-3 C on each plate while the potential difference between the plates is 1000V.Calculate the value of capacitance. Given data, Q = 10-3 C V = 1000V To find C, C = V Q 3 10 C = = 1μF State point form of Ohm s law. Point form of Ohm s law states that the field strength within a conductor is proportional to the current density. J E J = σe σ is conductivity of the material. 9. What is meant by conduction current? Conduction current is nothing but the current flowing through the conductor. Conduction current density is given by Jc = σe Amp / m 30. What is meant by displacement current density? Displacement current is nothing but the current flowing through the capacitor. Displacement current density is given by, D J d = Amp / m t 31. Define Polarization in Dielectric Material 6

15 7 Polarization in dielectric material is defined as a dipole moment per unit volume. P= Lt v0 1 n v v P i i1 UNIT III MAGNETOSTAICS 1. Define Magnetic Flux Magnetic flux is defined as the flux passing through any area. Its unit is Weber. a B. da Wb. Define Magnetic Flux Density Magnetic flux density is defined as the magnetic flux density passing per unit area. Its unit is Weber / meter or Tesla. B = A B = μh 3. Define Magnetic Gauss s Law The total magnetic flux passing thorough any closed surface is equal to zero. B. da 0 a 4. State Biot- Savart law. It states that the magnetic flux density at any point due to current element is proportional to the current element and sine of the angle between the elemental length and the line joining and inversely proportional to the square of the distance between them. sin db Idl o 4r 5. Write the force on a current element. The force on a current element Idl is given by df = I x B dl = BI dl sinθ Newton.

16 6. State the Lorentz force equation. The force on a moving particle due to combined electric and magnetic field is given by F = Q E VxB This force is called Lorentz force. 7. State Ampere s circuital law. Ampere s circuital law states that the line integral of magnetic field intensity H about any closed path is exactly equal to the direct current enclosed by the path. H. dl I 8. What is field due to toroid and solenoid? NI a) Toroid H r I b) Solenoid H l 9. Write down the expression for magnetic field at the centre of the circular coil I H a 10. Define Scalar Magnetic Potential It is defined as dead quantity whose negative gradient gives the magnitude intensity if there is no current source present. H V m V m is the magnetic scalar potential. V m H. dl 11. Define Magnetic Vector Potential Magnetic vector potential is defined as that quantity whose curl gives the magnetic flux density B x A A is the magnetic vector potential. 8

17 9 A 4 V J dr r Web / m 1. Distinguish among diamagnetic, paramagnetic and ferromagnetic materials. Diamagnetic : In diamagnetic materials magnetization is opposed to the applied field. It has magnetic field. Paramagnetic: In paramagnetic materials magnetization is in the same direction as the field. It has a weak magnetic field. Ferromagnetic : In Ferromagnetic materials magnetization is in the same direction as the field. It has strong magnetic field. 13. A solenoid with a radius of cm is wound with 0 turns per cm length and carries 10 ma. Find H at the centre if the total length is10 cm. Given data, N=nl = 0 x 10 = 00 turns. l =10 X 10 - m I = 10 x 10-3 A NI H l 00x10x10 = 10x10 = 0AT/m Define Mechanical Moment The tangential force multiplied by the radial distance at which it acts is called torque or mechanical moment on the loop. 15. Define magnetic moment The magnetic moment is defined as the maximum torque on loop per unit magnetic induction (Flux density).

18 30 A is Area. m = IA 16. Write the force on a current element. The force on a current element Idl is given by df = I x Bdl = BI dl sinθ 17. Write the torque on closed circuits. The torque on closed circuit in a magnetic field is T = BIA cosθ T = mbcosθ m is magnetic moment In vector form T = m x B 18. Write the expression for torque on a solenoid. Torque on a solenoid in a magnetic field is T = n. IAB m = nia = nbia = mb 19. Compare electrostatic field with magnetic field. Electrostatic field Magnetic field

19 31 Electric field intensity E ( volts/m ) Electric flux density D=εE c/m Energy stored is 1/CV Charges are at rest Magnetic field intensity H ( Amp/m ) Magnetic flux density B=μH (web / m ) Energy stored is 1/LI Charges are in motion 0. Determine the force per unit length between two long parallel wires separated by 5 cm in air and carrying currents 40A in the same direction. Force per unit length between two long parallel wires is o I Force / length = I 1 D 7 4X 10 X 40X 40 X 5X 10 = =6.4 x 10-3 N/m 1. Define - Magnetic Dipole A small bar magnet with pole strength Q m and length l may be treated as magnetic dipole whose magnetic moment is Q ml.. Define - Magnetization Magnetization is defined as the ratio of magnetic dipole moment to unit volume. Magnticdip ole Qm M = ; M = a A/m Volume A 3. Define Magnetic Susceptibility Magnetic susceptibility is defined as the ratio of magnetization to the magnetic field intensity. It is dimensionless quantity. m M H 4. What is the relation between relative permeability and susceptibility? 1 r m

20 3 r is relative permeability is susceptibility m 5. What are the different types of magnetic materials? The magnetic materials can be classified into three groups according to their behavior. They are diamagnetic, paramagnetic and ferromagnetic materials. 6. Write the magnetic boundary conditions. 1. The tangential component of magnetic field intensity is continuous across the boundary. H t1 = H t. The normal component of magnetic flux density is continuous across the boundary. Bn 1 = B n 7. Define Self Inductance The self induction of a coil is defined as the ratio of total magnetic flux linkage in the circuit to the current through the coil. N L = i is magnetic flux N is number of turns of coil i is the current. 8. Define Mutual Inductance The mutual inductance between two coils is defined as the ratio of induced magnetic flux linkage in one coil to the current through in other coil. N1 M = i N is number of turns in coil 1 is magnetic flux links in coil i 1 is the current through coil 1 1

21 9. Define Co-efficient of Coupling Co-efficient of coupling is defined as the fraction of the total flux produced by one coil linking the second coil (K). K = 1 = is the flux produced by coil 1 1 is flux links coil K 1 M K = L 1 L What will be the effective inductance, if two inductors are connected in (a) series and (b) parallel? (a) For series L = L 1 + L M + sign for aiding (b) For Parallel L = LL 1 M L L M 31. Write the expression for inductance of a solenoid. 1 A L N o l N is number of turns A is area of cross-section l is length of solenoid is free space permeability. o 3. Write the expression for inductance of a toroid. N is number of turns A L N o = N o r R R - sign for opposition

22 34 r is radius of the coil R is radius of toroid is free space permeability. o 33. Write the expression for inductance per unit length of a co-axial transmission line. L = o b ln H/m. a Where a is the radius of inner conductor b is the radius of outer conductor. 34. Distinguish between solenoid and toroid. Solenoid: Solenoid is a cylindrically shaped coil consisting of a large number of closely spaced turns of insulated wire wound usually on a non magnetic frame. Toroid: If a long, slender solenoid is bent into the form of a ring and thereby closed on itself, it becomes toroid. 35. What is the mutual inductance of two inductively tightly coupled coils with self inductance of 5 mh and 100 mh. L 1 = 5 mh. L = 100 mh M = K L L 1 = 5X 100 = 50 mh UNIT IV ELECTRO DYNAMIC FIELDS 1. State Faraday s law of electromagnetic induction. Faraday s law states that electromagnetic force induced in a circuit is proportional to the rate of change of magnetic flux linking the circuit.

23 35 d emf = N dt. Define Magneto Motive Force Magneto motive force (mmf ) is defined as the product of magnetic flux and reluctance of that magnetic circuit. mmf = flux x reluctance mmf = Φ AT 3. Define Reluctance Reluctance is defined as the ratio of mmf of magnetic circuit to the flux through it. mmf flux () It is also written as l A l is the length in m A is the area of cross- section in m μ is permeability μ = μ 0 μ r μ 0 = 4π x 10-7 H / m 4. In a solenoid with an inductance of 5 mh, current is increasing at the rate of 100 A/sec. What is the value of induced emf? Solution: Given L = 5 mh = 5 x 10-3 H di / dt = 10 A / sec Formula Used di emf = L dt = 5x10-3 x 100 = 0.5V

24 36 5. Write the expression for lifting force of an electro magnet. F = B A o B is flux density A is area of air gap between the poles of the magnet o is permeability of free space μ 0 = 4π x 10-7 H / m 6. What is the expression for energy stored in magnetic field? 1 W = LI Joules Where L is the inductance in Henry I is the current in Amp 7. What is energy density in the magnetic field? 1 Energy density w = BH = 1 H B Magnetic flux density in wb / m H Magnetic field intensity in H / m μ is permeability μ = μ 0 μ r μ 0 = 4π x 10-7 H / m 8. Write down the general, integral and point form of Faraday s law. d emf v dt ( General ) B Edl. ds t ( Integral ) B E t ( Point form )

25 9. Distinguish between transformer emf and motional emf. The emf induced in a stationary conductor due to the change in flux linked with it, is called transformer emf or static induced emf. B emf = -. ds eg. Transformer. t The emf induced due to the movement of conductor in a magnetic field is called motional emf or dynamic induced emf. emf =-v Bdl. eg. Generator c 10. State Lenz s law. Lenz s law states that the induced emf in a circuit produces a current which opposes the change in magnetic flux producing it. 11. State Dot rule. If both the currents enter dotted ends of coupled coils or if the both currents enter undotted ends, then the sign on the M will be same as the sign on the L. If one current enters a dotted end and the other an undotted end, the sign on the M will be opposite to the sign on the L. 1. Compare electric circuit with magnetic circuit emf (volts) Electric circuit emf. Current = resis tance l 3. Resistance R = A 4. Conductance G = R 1 mmf ( Amp-turns ) Magnetic flux = Reluctance Permeance P = 1 Magnetic circuit mmf reluc tance l A

26 A region in free space has a magnetic field intensity of B web/m. What is the energy stored per m 3 of space? Energy density = Energy per volume 1B = Joules / m 3, μ is the permeability of the medium; B Magnetic flux density in wb / m 14. Write the Maxwell s equation in integral form. From Ampere s Law D Hdl. J ds t From Faraday s Law B Edl. ds t From Electric Gauss s Law S S D. ds s v dv From Magnetic Gauss s Law s B. ds Write the Maxwell s equation in point form. From Ampere s Law D H J t From Faraday s Law B E t From Electric Gauss s Law D. From Magnetic Gauss s Law

27 39 B Write the fundamental postulate for electromagnetic induction and explain how its leads to Faraday s Law. A changing magnetic flux (Φ) through a closed loop produces an emf or voltage at the terminals as given by d v dt the voltage is the integral of the electric field E around the loop. For uniform magnetic field Φ = B/A B is the magnetic flux density and A is the area of the loop. B v Edl. ds t This is Faraday s law. It states that the line integral of the electric field around a stationary loop equals the surface integral of the time rate of change of the magnetic flux density B integrated over the loop area. 17. What is the significance of displacement current? Write the Maxwell s equation in which it is used. The displacement current i D through a specified surface is obtained by integration of the normal component of J D over the surface. i D = J. ds D S D =. ds t S E i D = ds t This is a current which directly passes through the capacitor. Maxwell s equation XH J C J D E = E (Differential form ) t D Hdl. ( J ) ds (Integral form ) t C S

28 18. Find the total current in a circular conductor of radius 4mm if the current density 10 varies according to J = r Solution: 10 J = r Current I = J. ds 4 4 A/ m. A/ m =. rdrd r 0 r0 4 = 10 r d = = 80π 19. Write Maxwell s equations in point phasor forms. X H JjD ( j ) E XEjBjH D. B Write Maxwell s equations in integral phasor form. dl ( Jj Dds ) ( H. j ) Eds. S dl j Bds S E. jhds. Dds. S S B. ds 0 dv 1. Write Maxwell s equation in integral form. From Ampere s Law S S 40

29 41 D Hdl. ds t From Faraday s Law B Edl. ds t From Electric Gauss s Law s S D. ds 0 From Magnetic Gauss s Law s B. ds 0. Write Maxwell s equation in point form. From Ampere s Law D H t From Faraday s Law B E t From Electric Gauss s Law D. 0 From Magnetic Gauss s Law B. 0 S 3. Explain why B. 0 B. 0 states that there are no magnetic charges. The net magnetic flux emerging through any closed surface is zero. 4. Explain why X E 0. In a region in which there is no time changing magnetic flux, the voltage around the loop would be zero. By Maxwell s equation

30 4 5. Explain why D. 0 B E =0 t In a free space there is no charge enclosed by the medium. The volume charge density is zero. By Maxwell s equation. D 0 6. Compare circuit theory with field theory. v Circuit Theory Field Theory 1. This analysis originated by its own. Evolved from Transmission theory.. Applicable only for portion of RF range. Beyond RF range ( Microwave ) 3. The dependent and independent parameters I, V are directly obtained for the given circuit. Not directly, through E and H. 4. Parameters of medium are not involved. Parameter of medium ( permittivity and permeability) are involved in the analysis. 5. Laplace Transform is employed. Maxwell s equation is employed 6. Z, Y, and H parameters are used. S parameter is used. 7. Low power is involved. Relatively high power is involved. 8. Simple to understand. Needs visualization ability 9. Two dimensional analysis Three dimensional analysis 10.Frequency is used as reference. Wave length is used as reference 11. Lumped components are involved Distributed components are involved.

31 43 UNIT V ELECTROMAGNETIC WAVES 1. What is meant by wave? If a physical phenomenon that occurs at one place at a given time is reproduced at other places at later times, the time delay being proportional to the space separation from the first location, then the group of phenomenon constitutes a wave.. List out the properties of uniform plane wave. The properties of uniform plane wave are as follows 1. At every point in space, the electric field E and Magnetic field H are perpendicular to each other and to the direction of the travel.. The fields vary harmonically with the time and at the same frequency, every in space. 3. Each field has the same direction, magnitude and phase at every point in any plane perpendicular to the direction of wave travel. 3. Write the wave equations for E and H in a non-dissipative ( free space ) medium. E oo H oo E t H t Write the wave equations for E and H in a conducting medium.

32 44 E E E t t H H t H t 5. Define Plane Wave The plane wave is defined as the phase of a wave that is the same for all points on a plane surface Define Uniform Plane Wave Uniform plane wave is defined as the phase of a wave that is the same for all points on a plane surface it is called plane wave. If the amplitude is also constant in a plane wave, it is called uniform plane wave. 7. What are the properties of uniform plane waves? 1. At every point space electric field (E) and magnetic field (H) are perpendicular to each other and to the direction of travel.. The fields vary harmonically with time and at the same frequency, every in space. 3. Each field has the same direction, magnitudes and phase at every point in any plane perpendicular to the direction of wave travel. 8. Define Intrinsic Impedance or Characteristic Impedance Intrinsic impedance is defined as the ratio of electric field to magnetic field. Otherwise it is the ratio of square root of permeability to permittivity of the medium. E H 9. Calculate intrinsic impedance or characteristic impedance of free space. E H o o 4x x10 7 = 1 = 10π = 377 ohms

33 Write the expression for propagation constant. The propagation constant (γ) is a complex number and it is given by j α is attenuation constant β is phase constant j ( j 11. Define Skin Depth or Depth of Penetration Skin depth or depth of penetration (δ) is defined as the depth in which the wave has been attenuated to 1 / e (or approximately 37% of its original value). 1 for good conductor. 1. Define Polarization Polarization is defined as the polarization of a uniform plane wave refers to the time varying nature of the electric field vector at some fixed point in space. 13. What is meant by linear polarization? If x and y component of electric field Ex and Ey are present and are in phase, the resultant electric field has a direction at an angle of tan -1 E Y ( ) and if this angle is E constant with time, the wave is said to be linearly polarized. 14. What is meant by circular polarization? If x and y component of electric field Ex and Ey have equal amplitude and 90 o phase difference, the locus of the resultant electric field E is an circle and the wave is said to be circularly polarized. 15. What is meant by Elliptical polarization? If x and y component of electric field Ex and Ey have different amplitude and 90 o phase difference, the locus of the resultant electric field E is a ellipse and the wave is said to be elliptically polarized. X

34 16. Find the skin depth at a frequency of MHz is Aluminum σ = 38. M s/m and μ r = 1. Solution: Given data σ = 38.M s/m = 38. x 10 6 s/m μ r = 1 f x x For Good conductor, Skin depth 1. x10.1x 4x10 = = x 10-5 m..38.x At what frequencies may earth be considered perfect, if σ = 6 x 10-3 s/m, μ r = 1 and ε r = This is the boundary line between dielectric and conductor 1 3 6X x x 36 x109x10 f 3 108x 10 f f = 108 x 10 6 Hz if frequency is greater than 108MHz, it acts as dielectric. 18. A uniform plane wave in free space is described by E = 100e -(πz/3) a x. Determine the frequency and wave length. E=100 e -(πz/3) ax 6 1

35 47 Β = = 3 6m 8 e 3x10 f 50MHz Write Helmholtz s equation. E E 0 j ( j ) 0. Define Pointing Vector Pointing vector is defined as rate of flow of energy of a wave as it propagates. It is the vector product of electric field and magnetic field. P = E x H 1. Write the expression for average power flow in electromagnetic field and average pointing vector. Average power VI Wav = COS Average Pointing vector Pav = 1 / Real part of [ E x H* ]. Write the complex Pointing vector in rectangular co-ordinates. Px = ½ [ Ey Hz* - EzHy* ] 3. Define Slepian Vector Slepian vector is defined as a vector which defines at every point such that its flux coming out of any volume is zero. ( S. ) 0. Slepian vector is given by, S x( H) V is electric potential H is magnetic field intensity 4. State Pointing theorem.

36 The vector product of electric field intensity at any point is a measure of the rate of energy flow per unit area at that point. P = E x H 5. State Snell s law. When a wave is travelling from one medium to another medium, the angle of incidence is related to angle of reflection as follows. sin sin i 1 ( 1 0) t 1 48 i is angle of incidence is angle of refraction t 1 is dielectric constant of medium 1 is dielectric constant of medium 6. What is meant by Brewster angle? Brewster angle is an incident angle at which there is no reflect wave for parallely polarized wave. tan 1 1 is dielectric constant of medium 1 is dielectric constant of medium 1 7. Define Surface Impedance Surface impedance is defined as the ratio of tangential component of electric field at the surface of a conductor to the linear current density. Etan Zs J s is propagation constant is conductivity medium

37 8. Write the expression for plane electromagnetic waves propagating in a dielectric media in a direction x with respect to origin ( 0, 0, 0) The equation for plane electromagnetic waves propagating in a dielectric medium is given by Ey 1 Ey t x (or) Hy 1 Hy t x 9. In a time varying situation how is a good conductor and lossy dielectric defined? For good conductor >>1 f α and β are large i.e., the wave is attenuated greatly as it propagates through the conductor. For lossy dielectric, dielectric current becomes complex, ' '' <<1 and Loss tangent of the medium is defined as '' tan ' 30. What is meant by total internal reflection? When a wave is incident from the denser medium to rarer medium at an angle equal to or greater than the critical angle, the wave will be totally internally reflected back. This phenomenon is called total internal reflection. 31. Write the expression for pointing theorem in point form. 49

38 50 Pointing theorem in point form is given by 1. PE [ H E ] t 3. Write the expression for pointing theorem in integral form. The expression for pointing theorem in integral form is given by S 1 P. ds E [ H E ] t t V V 33. What is normal incidence and oblique incidence? Normal incidence: When a uniform plane wave incidences normally to the boundary between the media, then it is known as normal incidence. Oblique incidence: When a uniform plane wave incidences obliquely to the boundary between the two media, then it is known as oblique incidence. 34. Define Standing Wave Ratio Standing wave ratio is defined as the ratio of maximum to minimum amplitudes of voltage. E 1 1s max S = E 1 1s min