AKSHAYA COLLEGE OF ENGINEERING AND TECHNOLOGY
AKSHAYA COLLEGE OF ENGINEERING AND TECHNOLOGY Dept. of Electrical and Electronics Engineering EE1302- ELECTROMAGNETIC THEORY Prepared by: R.Subramanian QUESTION BANK UNIT I – VECTOR FUNCTIONS AND CO-ORDINATE SYSTEMS PART A 1. What are the sources of electro magnetic fields? 2. Transform a vector A=yax-xay+zaz into cylindrical coordinates. 3. How the unit vectors are defined in cylindrical co-ordinate systems? 4. What is the physical significance of the term “divergence of a vector field”? 5. Define scalar triple product and state its characteristics. 6. Give the relation between Cartesian and cylindrical co-ordinate systems 7. Define curl. 8. What is unit vector? What is its function while representing a vector? 9. Which are the differential elements in Cartesian co-ordinate system? 10. What is the physical significance of divergence? 11. Define Surface Integral. 12. Sketch a differential volume element in cylindrical co-ordinates resulting from differential changes in three orthogonal co-ordinate directions. 13. What is volume charge density? 14. Define Line Integral. 15. Calculate the total charge enclosed by a circle of 2m sides, centred at the origin and with the edge s parallel to the axes when the electric flux density over the cube is D=10x3/3ax(C/m2) 16. State stoke’s theorem. 17. Give practical examples for diverging and curling fields. 18. State Divergence theorem and mention the significance of the theorem. 19. A vector field F = (1/ r) ar in spherical co-ordinates. Determine F in Cartesian form at a point x =1, y =1 and z = 1. 20. Given A= 10 ay + 3az and B = 5ax + 4ay , find the projection of A on B 21. Prove that curl gradΦ = 0. 22. Verify that the vectors A = 4ax – 2ay + 2az and B = -6ax + 3ay – 3az are parallel to each other. 23. What are different co-ordinate systems? 24. The temperature in an auditorium is given by T= x2 + y2 – z. A mosquito located at (1, 1, 2) desires to fly in such a direction that it will get warm as soon as possible. In what direction it must fly? PART B 1. (i) Explain the electric field distribution inside and outside a conductor (ii) Explain the principle of electrostatic shielding. (iii) Draw the equipotent lines and E lines inside and around a metal sphere. 2. i) State and prove Divergence theorem. ii) State and prove Stokes theorem. 3. Given A = ax + ay, B = ax + 2az and C = 2 ay + az . Find (A X B) X C and compare it with A X (B X C). Using the above vectors, find A.B X C and compare it with A X B.C. 4. Find the value of the constant a, b, c so that the vector E = (x + 2y + az) ax + (bx – 3y –2) ay + (4x + cy + 2z) az is irrotational. 5. Using the Divergence theory, evaluate ∫∫ E.ds = 4xz ax – y2 ay + yz az over the cube bounded by x = 0; x = 1; y = 0; y = 1; z = 0; z = 1. 6. Explain the spherical co-ordinate system? 7. (i) Use the cylindrical coordinate system to find the area of the curved surface of a right circular cylinder where r = 20 m, h = 5m and 30º 120 º. (ii)State and explain Divergence theorem. 8. (i) Derive the stoke’s theorem and give any one application of the theorem in electromagnetic fields (ii)Obtain the spherical coordinates of 10ax at the point P (x = -3, y = 2, z = 4). 9. Find the charge in the volume 0 x 2m 0 y 2m 0 z 2m If ρ = 10 x2 y z μC/m3. 10. (i) Determine the constant c such that the vector F = (x + ay)i + (y + bz)j + (x + cz)k will be solenoidal. (ii) Given A 2r cos I r r I in cylindrical co-ordinate. For the contour shown in Fig.Q-32, verify Stoke’s theorm. Fig.Q-32. 11. Verify Stoke’s theorem for a vector field F = ρ2cos2 Φaρ+zsinΦaz around the path L defined by 0≤ ρ ≤ 3, 0≤ Φ ≤ 45o and z = 0. 12. i) What are the major sources of Electromagnetic fields (any five)? ii) What are the positive and negativeeffcts of EM fields on living things? iii) What are the E and H field limits for public exposure? iv)Give any one example to reduce the effect of EM field. 13. Verify the divergence theorem for a vector field A = xy2 ax+y3 ay+y2z az and the surface is a cuboid defined by 0<x<1, 0<y<1, 0<z<1. 14. Given that F = x2 y ax - yay. Find ∫ F. dl for the closed path shown in figure and also verify Stoke’s theorem. 15. i) Given A = 5ax and B = 4ax + t ay ; Find ‘t’ such that the angle between A and B is 45˚ ii) Using the Divergence theorem, evaluate ∫∫ A.dS = 2xy ax + y2 ay + 4yz az over the cube bounded by x = 0; x = 1; y = 0; y = 1; z = 0; z = 1. 16. i) Determine divergence and curl of the vector A = x2 ax + y2 ay + y2 az. ii) Determine the gradient of the scalar field at P (√2, π / 2, 5) defined in cylindrical co-ordinate system as A= 25ρ sinφ. 17. a) Determine divergence and Curl of the following vector fields. i) P= x2yz ax + xz az. ii) Q = ρsinΦaρ + ρ2z aΦ + zcosΦ az 2 iii) T = 1 / r cosθ ar + r sinθ cosθ aθ+ cosθ aΦ b) Fnd the gradient of the following scalar fields i) V = e-z sin2x coshy ii) U = ρ2z cos 2Φ iii) W = 10r sin2θ cosΦ. 18. i) Show that the vector field A is conservative if A possesses one of the following two properties. 1) The line integral of the tangential component of A along a path extending from point P to point Q is independent of the path.2) The line integral of the tangential component of A along a closed path s zero. ii) If A = ρcosΦaρ + sinΦaΦ , evaluate ∫ A.dl around the path shown in figure. Confirm this using Stoke’s theorem. 19. i) A vector field is given by the expression F = (1/ρ ) aρ in cylindrical co-ordinates and F = (1/r) ar in spherical co-ordinates. Determine F in each case in the Cartesian form at a point (1, 1, 1). ii) If a scalar potential is given by the expression Φ = xyz, determine the potential gradient and also prove that vector F = grad Φ is irrotational. 20. i) Given two points A (2, 3,-1) and B (4, 25˚, 120˚). Find the spherical and cylindrical co- ordinates of point A and Cartesian and cylindrical co-ordinates of point B. ii) Find the curl of H at P (2, Π/6, 0), where H = 2ρ cosφ aρ - 4ρ sinφ aφ + 3az. UNIT II – ELECTRIC FIELDS PART A 1. State coulomb’s law. 2. What are the different types of charges? 3. State Gauss Law. 4. Draw the equipotential lines and electric field lines for a parallel plate capacitor. 5. Define dielectric strength. What is the dielectric strength of co-axial cable? 6. Write and explain the coulomb’s law in vector form. 7. Define electric field intensity at a point. 8. What is the electric field around a long transmission line? 9. Sketch the electric field lines due to an isolated point charge Q. 10. A uniform line charge with PL = 5 µc/m lies along the x-axis. Find E at (3,2,1). 11. What are the various types of charge distributions, give an example of each. 12. Define Dielectric strength of a material. Mention the same for air. 13. Write and explain the coulomb’s law in vector form. 14. Using Gauss’s law, derive the capacitance of a coaxial cable. 15. Write down Poisson’s and Laplace’s equation. 16. Calculate the total charge enclosed by a cube of 2m side, centered at the origin and with the edges parallel to the axes when the electric flux density over the cube is D = 10x3 / 3 ax C / m2. 17. Define dielectric strength of a material and mention the same for air. 18. The electric potential near the origin of the system is V = ax2 + by2 + cz2. find the electric field at (1, 2, 3). 19. What are symmetrical charge distributions? 20. Define dipole moment. 21. An infinite line charge charged uniformly with a line charge density of 20 nC / m is located along z- axis. Fine E at (6, 8, 3) m. 22. Define electric potential and potential difference. 23. Using Gauss’s law, derive the capacitance of the co-axial cable. 24. Derive Poissons equation. 25. Two point charges q1 and Q2 are located at (1, 2, 0) and (2, 0, 0) respectively. Find the relation between Q1 and Q2 such that the total charge at the point P (-1, 1, 0) will have no x- component). 26. Verify the following potential satisfy Laplace’s equation V= 15 x2 yz – 5 y3z. 27. A spherical capacitor consists of an inner conducting sphere of radius Ri and an outer conductor with a spherical inner wall of radius Ro. The space in between is filled with dielectric of permittivity ε. Determine the capacitance. 28. What is the capacitance of co-axial cable? 29. Write the continuity equation. 30. Why Gauss’s law can’t be applied to determine the electric field due to finite line charge? 31. A uniform surface charge of σ = 2 μC / m2 is situated at z=2 plane. What is the value of flux density at P (1, 1, 1) m. UNIT II PART B 1. State and explain the experimental law of coulomb? 2. State and prove Gauss’ law and write about the applications of Gauss law? 3. State and explain Gauss’s law. Derive an expression for the potential at a point outside a hollow sphere having a uniform charge density 4. (i) A circular disc of radius ‘a’, m is charged uniformly with a charge density of σ C/m2 Find the electric field intensity at a point ‘h’, m from the disc along its axis. (ii) A circular disc of 10 cm radius is charged uniformly with a total charge of 10-6c. Find the electric intensity at a point 30 cm away from the disc along the axis. 5. A line charge of uniform density q C / m extends from the point (0, -a) to the point (0, 1) in the x-y plane. Determine the electric field intensity E at the point (a, 0). 6. Define the electric potential, show that in an electric field, the potential difference b between two points a and b along the path, Va – Vb = - E.dl a 7. What is dipole moment? Obtain expression for the potential and field due to an electric dipole.Two point charges Q1 = 4nC1, Q = 2nC are kept at (2, 0, 0) and (6, 0, 0).Express the electric field at (4, -1, 2) 8. Derive the electric field and potential distribution and the capacitance per unit length of a coaxial cable. 9. Explain in detail the behavior of a dielectric medium in electric field. 10. i)Discuss Electric field in free space, dielectric and in conductor. ii) Determine the electric field intensity at P ( -0.2,0,-2.3) due to a point charge of 5 nC at (0.2, 0.1, -2.5) in air. 11.(i) Derive the electrostatic boundary conditions at the interface of two deictic media. (ii) If a conductor replaces the second dielectric, what will be the potential and electric field inside and outside the conductor? 12. (i) Derive the expression for scalar potential due to a point charge and a ring charge. (ii) A total charge of 100 nC is uniformly distributed around a circular ring of 1.0m radius. Find the potential at a point on the axis 5.0 m above the plane of the ring. Compare with the result where all charges are at the origin in the form of a point charge. 13. (i) Derive the expression for energy density in electrostatic fields. (ii) A capacitor consists of squared two metal plates each 100 cm side placed parallel and 2 mm apart. The space between the plates is filled with a dielectric having a relative permittivity of 3.5. A potential drop of 500 V is maintained between the plates. Calculate i) the capacitance, ii) the charge of capacitor, iii) the electric flux density, iv) the potential gradient 14. A uniformly distributed line charge, 2m long, with a total charge of 4 nC is in alignment with z axis, the mid point of the line being 2 m above the origin. Find the electric field E at a point along X axis 2 m away from the origin. Repeat for concentrated charge of 4 nC on the z axis 2 m from the origin, compare the results. 15. (i) Define the potential difference and absolute potential. Give the relation between potential and field intensity. (ii) Two point charges of +1C each are situated at (1, 0, 0) m and (-1, 0, 0) m. At what point along Y axis should a charge of -0.5 C be placed in order that the electric field E = 0 at (0, 1, 0) m? 16. If V = [2 x2y + 20z – 4 / (x2 + y2)] volts, find E and D at P (6,-2.5, 3). 17. Derive an expression for capacitance of a spherical capacitor with conducting shells Of radius ‘a’ and ‘b’. 18. Obtain an expression for energy stored and and energy density in a capacitor. 19. Conducting spherical shells with radii a = 10 cm and b= 30 cm are maintained at potential difference of 100 V such that V(r = b) = 0 and V(r = a) = 100 V. Determine V and E in region between shells. 20. A total charge of 10-8 C is distributed uniformly along a ring of radius 5m. Calculate the potential on the axis of the ring at a point 5m from the centre of the ring. 21. Two parallel plates with uniform surface charge densities equal and opposite to each other have an area of 2 m2 and distance of separation of 2.5 mm in free space. A steady potential of 200 V is applied across the capacitor formed. If a dielectric of width 1 mm and relative permittivity 2 is inserted into this arrangement what is the new capacitance formed? 22. i) Derive Poisson’s and Laplace’s equation and explain their significance in field theory. ii) Three concentrated charges of 0.25 μC are located at the vertices of an equilateral triangle of 10 cm side. Find the magnitude and direction of the force one charge due to the other two charges. 23. A positive charge Q is located at the centre of a spherical conducting shell of inner radius Ri and outer radius Ro. Determine E and V as function of radial distance R. 24. i) Write a note on dielectrics. ii) Find the electric field intensity at the point (0, 0, 5) m due to Q1 = 0.35μC at (0, 4, 0) and q2 =-0.55 μC at (3, 0, 0) m. 25. The electric flux density is given as D= r/4 ar nC / m2 in free space. Calaculate E at r = 0.25 m, the total charge within the sphere of r = 0.25m and the total flux leaving the sphere of r = 0.35m. 26. An infinitely long uniform line charge is located at y=3, z=5. If ρL= 30 nC/m. Find field intensity E at: i) origin ii) P (0, 6,) and iii) Q (5, 6, 1). UNIT III - MAGNETIC FIELDS PART A 1. State Biot–Savart’s law. 2. Distinguish magnetic scalar potential and magnetic vector potential. 3. Plane y=0 carries a uniform current of 30 az mA/m. Calculate the manetic field intensity at (1, 10,-2) m in rectangular coordinate system. 4. Plot the variation of H inside and outside a circular conductor with uniform current density. 5. What is vector A? 6. State Ampere’s Law. 7. What is the relation between magnetic field density B and vector potential A? 8. State the significance of E and H. Give an example of this. 9. What is magnetic boundary condition? 10. Draw the magnetic field pattern in and around a solenoid. 11. What is H due to a long straight current carrying conductor? 12. Calculate inductance of a ring shaped coil having a mean diameter of 20 cm wound on a wooden core of 2 cm diameter. The winding is uniformly distributed and contains 200 turns. 13. A conductor located at x=0.5 m , y=0 and 0<z<2.0 m carries a current of 10 A in the az direction. Along the length of the conductor B=2.5 ax T. Find the torque about the x axis. 14. What do you mean by magnetic moment? 15. Define mutual inductance. 16. Plot the variation of H inside and outside a circular conductor with uniform current density. 17. A long straight wire carries a current I = 1 A. At what distance s the magnetic field H = 1 A/m. 18. Write the expression for magnetic force when charge particle moves in a magnetic field. 19. State Ampere’s circuital law. 20. Write down the magnetic boundary conditions. 21. Define magnetic moment and magnetic permeability. 22. Draw the magnetic field pattern inside and outside the circular conductor with uniform current density. 23. What is the relation between magnetic flux density B and vector potential A? 24. Compare steady current and steady state current. 25. What is Lorentz law of force and writethe equation.? 26. Calculate h at (3,-6,2) due to a current element of length 2 mm located at the origin in free space that carries current 16 mA in +Y direction. 27. An infinitely long straight conductor with circular cross section of radius ‘b’ carries steady current I. determine the magnetic flux density inside the conductor. 28. A small circular loop of radius 10 cm is centered at origin and placed on the Z = 0 plane. If the loop carries a current of 1 A along aΦ. Calculate magnetic moment of the loop. 29. Define magnetic susceptibility. 30. What do you mean by magnetization? 31. State the boundary conditions of magnetic media. 32. State the modified form of expression curl H = ▼x H = J, if the contour does not enclose any current, then how is vector H expressed with scalar magnetic potential. 33. What is solenoid? 34. Classify the magnetic materials. UNIT III PART B 1. (i) Use Biot – Savart’s law to find magnetic field intensity for finite length of conductor at a point P on Y – axis. (ii)A steady current of ‘I’ flows in a conductor bent in the form of a square loop of side ‘a’. Find the magnetic field intensity at the centre of the current loop. iii) Find the magnetic field intensity at the centre of a square of sides equal to 5m and carrying 10 A current. 2. (i) when a current carrying wire is placed in an uniform magnetic field, show that torque acting on it is T= m X B (ii) A magnetic circuit comprising a toroid of 5000 turns and an area of 6cm2 and mean radius of 15 cm carries a current of 4A. Find the reluctance and flux given μr = 1. 3. Calculate B due to a long solenoid and a thin toroid. 4. (i) Derive for force and torque in a magnetic field using motor as an example. (ii) Find the torque about the y axis for the two conductors of length l, carrying current in opposite directions, separated by a fixed distance w, in the uniform magnetic field in x direction. 5. (i) Explain magnetization in magnetic materials and explain how the effect of magnetization is taken into account in the calculation of B/H. (ii) Find H in a magnetic material a. When µ = 0.000018 H/m and H = 120 A/m. b. When B = 300 µT and magnetic susceptibility = 20. 6. a) Derive the magnetic force between two parallel conductors carrying equal current in the (i) Same direction (ii) opposite direction b) Two wires carrying currents in the same direction of 5000 A and 10000 A are placed with their axes 5 cm apart. Calculate the force between them. 7. (i) Find the field intensity at a point due to a straight conductor carrying current I as shown in Fig.Q-7. Fig.Q-7 (ii) Find H at the centre of an equilateral triangular loop of side 4 m carrying current of 5 A. 8. (i) Derive the expression for co-efficient of coupling in terms of mutual and self inductances. (ii) An iron ring with a cross sectional area of 3 cm2 and a mean circumference of 15 cm is wound with 250 turns wire carrying a current of 0.3 A. The relative permeability of the ring is 1500. Calculate the flux established in the ring. If a saw cut of width 2mm is made in the above ring, find the new value of flux in the circuit. 9. Develop an expression for magnetic field intensity inside and outside a solid cylindrical conductor of radius ‘a’, carrying a current I with uniform density. Sketch the variation of the field intensity. 10. Derive H due to a circular current loop and extend the same to compute H due to a long solenoid. 11. i)State and prove Ampere’s circuital law. ii) State and explain Biot- Savart’s law 12. i) Obtain an expression for magnetic vector potential. ii) Give a brief note on magnetic materials. 13. At a point P (x,y,z) the components of vector magnetic potential A are given as Ax = (4x + 3y+2z); Ay = (5x + 6y +3z) and Az = (2x + 3y +5z). Determine B at point P. 14. Derive the boundary conditions between two magnetic media. 15. A solenoid has an inductance of 20 mH. If the length of the solenoid is increased by two times and the radius is decreased to half of its original value, find the new inductance. 16. Write short notes on Magnetic vector potential, Biot- Savart’s law, Lorentz law of force and Magnetic energy density. 17. An iron ring with a cross sectional area of 8 cm2 and a mean circumference of 120 cm is wound with 480 turns wire carrying a current of 2 A. The relative permeability of the ring is 1250. Calculate the flux established in the ring. 18. A uniform cylindrical coil of 2000 turns is 60 m long and 5 cm diameter. If the coil carries a current of 10 mA, find the magnetic flux density at the centre of the coil, on the axis at one end of the coil and on the axis halfway between centre and one end of the coil. 19. A circular loop located on x2 + y2 = 9, z=0 carries a direct current of 10 A along aΦ. Determine H at (0, 0, 4) and (0, 0, -4). 20. A small current loop L1 with magnetic moment 5 az Am2 is located at the origin while another small loop current L2 with magnetic moment 3 ay Am2 is located at (4, -3, 10). Determine the torque on L2. 21. i) Find the maximum torque on an 85 turns, rectangular coil with dimension (0.2 x 0.30) m, carrying current of 5A in a field B= 6.5 T ii) Derive an expression for magnetic vector potential. 22. i) Derive an expression for the inductance of solenoid. ii) Derive the boundary conditions at an interface between two magnetic media. 23. Determine the force per unit length between two long parallel wires A and B separated by 5 cm in air and carrying currents of 40 A in the same direction and inthe opposite direction. 24. Derive the expression for curl H = J. 25. Explain the concepts of scalar and vector magnetic potential. UNIT IV- MAXWELL’S EQUATIONS PART A 1. Write point form or differential form of Maxwell’s equation using faraday’s law. 2. Explain why X E = 0. 3. Write the Maxwell’s equation from faraday’s law both in integral and point forms. 4. Compare field theory with circuit theory. 5. What is motional emf? 6. Mention the limitation of circuit theory. 7. Write down the expression for the emf induced in the moving loop in static B field. 8. State Faraday’s law. 9. Distinguish between transformer EMF and motional EMF. 10. What is displacement current density? 11. Time varying field is not conservative.Prove it. 12. Write the EMF equation for moving conducting loop in a time varying field. UNIT IV PART B 1. (i) Summarize Maxwell’s equation for time varying fields in integral and differential form. (ii) Compare the magnitude of conduction current density and displacement current density in a good conductor in which σ = 107 S/m, Єr = 1 when E = 1sin 120πt. Comment on the result. 2. (i) Derive Maxwell’s equation for E and H. (ii) Explain (a) Motional emf. (b) Transformer emf. 3. Explain the different methods of emf induction with necessary governing equations and with suitable examples. 4. (i) Write short notes on Faradays laws of electromagnetic induction. (ii) What do you by displacement current? Write down the expression for the total current density. 5. Explain the relationship between the field theory and circuit theory using a simple RLC series circuit. Also explain the limitations of the circuit theory. 6. i) A straight conductor of length 40 cm moves perpendicularly to its axis at a velocity of 50 m/s in a uniform magnetic field of flux density 1.2 T. Evaluate the emf induced in the conductor if the direction of motion is - normal to the field - parallel to the field - at an angle 60o to the orientation of the field. ii) A circular cross section conductor of radius 2mm carries a current ic = 2.5 sin (5x 108t) µA.What is the amplitude of displacement current density if σ = 35 MS/m and εr = 1. 7. Differentiate conduction and displacement current and derive the same .Explain the need of displacement current in Maxwell’s equations. 8.i) Do the fields E = Em sinx sint ay and H = Em/μo cos x cos t ay , satisfy Maxwell’s equations. ii) Find the amplitude of displacement current density in the air near car antenna where the field strength of EM signal E = 80 cos (6.277 x 108 t – 2.092 y ) az V/m. 9.From the fundamental laws , derive the Maxwell’s equations and the need for the Maxwell’s contributions in both differential and integral form. 10.i) Discuss the relation between field theory and circuit theory. (8) ii) In free space H = 0.2 cos (ωt – βx ) az A/ m. Find the total power passing through a circular disc of radius 5 cm. 11. State and explain Faraday’s law of electromagnetic induction. Hence derive the expressions for statically and dynamically induced emfs. 12. A circular loop of N turns of conducting wire lies in the xy – plane with its centre at the origin of a magnetic field specified by B = Bo cos (πr / 2b ) sin ωt where ‘b’ is the radius of the loop and ω is the angular frequency. Find the mf induce din the loop. 13. i ) derive modified form of Ampere’s circuital law in intehral and differential forms, ii) Find the amplitude of displacement current density inside a capacitor where εr = 600 and D = 3 x 10-6 (sin ( 6 x 106 t – 0.3464x ) az C/m2 . 14. i) A square coil loop area 0.01 m2 and 50 turns is rotated about its axis at right angles to a uniform magnetic field B = 1 T. Calculate the instantaneous value of EMF induced in the coil when its plane is at right angles to the field, at 45 deg to the field and in the plane of the field. Speed of rotation is 1000 rpm. ii) A conducting cylinder of radius 5 cm, height 20 cm rotates at 600 rev / sec in a radial field B = 0.5 T. Sliding contacts at the top and bottom are connected to Voltmeter. Find the induced voltage. 15. What is displacement current? Show that the displacement current in the dielectric Parallel plate capacitor is equal to the conduction current in the leads. 17. i) The conduction current flowing through a wire with conductivity σ = 3 x 107 S/m and εr =1 is given by Ic = 3 sin ωt mA. If ω = 108 rad / sec. Find the displacement current. ii) The magnetic field intensity in free space is given as H = Ho sinθ ay A/m, where θ = ωt = βz and β is a constant. Determine the current density vector J. UNIT V- ELECTROMAGNETIC WAVES PART A 1. 2. 3. 4. State wave equation in phasor form. Given E(z,t) = 100 sin (wt-βz)ay(V/m) in free space, sketch E and H at t=0. Define Poynting vector. Find the average power loss/volume for a dielectric having, εr=2 and tanδ = 0.005, if E = 1.0 kV/m at 500 MHz. 5. Calculate the skin depth and wave velocity at 2 MHz in Aluminum with conductivity 40 MS/m and µr=1. 6. Given E = Emsin (wt-βz)ay in free space, sketch E and H at t=0. 7. Derive an expression for ‘loss tangent’ in an insulating material and mention the practical significance of the same. 8. A medium has constant conductivity of 0.1 mho/m, µr=1, εr=30. When these parameters do not change with the frequency, check whether the medium behaves like a conductor or a dielectric at 50 kHz and 10 GHz. 9. In free space E(z,t) = 100 sin (wt-βz)ax (V/m). Find the total power passing through a square area of side 25 mm, in the z=0 plane. 10. Define skin depth. 11. Find the velocity of a plane wave in a lossless medium having µ=10, εr=20. 12. What do you mean by “depth of penetration”? 13. For a lossy dielectric material having µr=1, εr=48, σ=20 s/m. Calculate the propagation constant at a frequency of 16 GHz. 14. What is the velocity of electromagnetic wave in free space and in lossless dielectric? 15. Define a Wave. 16. What is skin effect? UNIT V PART B 1. (i)Discuss the parameters : , , , , , , Vphase and V group. ii) Define Brewster angle and discuss the Brewster angle and degree of polarization. 3. What is Poynting vector? Explain. Derive pointing theorem. 3. (i) Explain when and how an electromagnetic wave is generated. (ii) Derive the electromagnetic wave equations in free space and mention the types of solutions. 4. A plane wave propagating through a medium with µr=2, εr=8 has E = 0.5 sin (108 t-βz)az (V/m). Determine (i) β (ii) The loss tangent (iii) wave Impedance (iv) wave velocity (v) H field 5. A plane traveling wave has a peak electric field intensity E as 6 kV/m. If the medium is lossless with µr=1, εr=3, find the velocity of the EM wave , peak POYNTING vector, impedance of the medium and the peak value of the magnetic field H. Derive all the formulae used. 6. Determine the amplitude of the reflected ad transmitted E and H at the interface of two media with the following properties. Medium 1: µr=1, εr=8.5,σ = 1. Medium 2: free space. Assume normal incidence and the amplitude of E in the medium 1 at the interface is 1.5 mV/m. Derive all the formulae used. 7. (i) Derive the electromagnetic wave equation in frequency domain and the propagation constant and intrinsic impedance. (ii) Explain the propagation of EM waves inside the conductor. 8. Define and derive skin depth. Calculate skin depth for a medium with conductivity 100 mho/m, µr=2, εr=3 at 50Hz, 1MHz and 1 GHz. 9. In free space E (z,t) = 100 cos (wt-bz)ax(V/m). Calculate H and plot E and H waveforms at t=0. 10. Derive the transmission and reflection coefficients at the interface of two media for normal incidence. Discuss the above for an open and a short circuited line. 11. A free space –silver interface has E (incident) = 100 V/m in the free space side. The frequency is 15 MHz and the silver constants are µr=1, εr=1, σ=61.7 MS/m. Determine E (reflected), E (transmitted) at the interface. 12. Calculate intrinsic impedance η, propagation constant γ and wave velocity υ for a conducting medium in which σ = 58 S / m, µr=1, εr=1 at a frequency of 100 MHz.