TUTORIAL SHEET -1Q1) Point charges 1 mC and -2mC are located at (3.2,-1) and (-1,-1, 4) respectively. Calculate electric force on a -10 nC charge located at (0, 3, 1) and electric field at that point. Q2) Point charges 5nC and -2nC are located at (2, 0, 4) and (-3, 0, 5) respectively. a) Determine the force on 1nC point charge located at (1,-3, 7). b) Find electric field E at (1,-3, 7). Q3) Two charges Q1=2µC and Q2=5µC are located at (-3,7,-4) and (2,4,-1) respectively. Determine the force on Q2 due to Q1 and force on Q1 due to Q2. Q4) A point charge Q=10nC is at the origin in free space. Find electric field at P(1,0,1).Also find electric flux density at P. 2 Q5) If electric flux density D is given by D = [(2 y + z )a x + 4 xya y + xa z ] µC / m 2 .Find volume charge density at (0,0,0)and (-1,0,4). Q6) Find the potential and volume charge density at P(0.5,1.5,1) in free space given the potential field a) V = 2 x 2 − y 2 − z 2 volts b) V=6ρφz volts c) V= 5(2r 2 − 7) cosθcosφ volts d) V=3x-y volts Q7) Let D = 2 ρz 2 a ρ +ρcos 2 ϕa z .Evaluate a) ∫D.ds S b) ∫∇• Ddv V over the region defined by 0 ≤ ρ ≤ 5,−1 ≤ z ≤ 1,0 ≤ φ ≤ 2π . Q8) electric field is given 2 D = 2 ρ ( z + 1) cos ϕa ρ − ρ( z + 1) sin ϕaϕ + ρ cos φa z µC/m a) Find charge density b) Calculate charge enclosed by volume 0<ρ<2, 0<φ<π/2,0<z<4. c) Confirm gauss law by finding the net flux through the surface of volume in ‘b’. 2 In a certain region the by Q9) In free space V = x 2 y ( z + 3) volts. Find a) E at (3, 4, -6) b) the charge within the cube 0<x, y ,z<1. TUTORIAL SHEET -2 Eθ and Eφ at P (1.2.0. → b) Find | E | at P (3.-4. D = y 2 z 3 a x + 2 xyz 3 a y + 3 xy 2 z 2 a z Q5. Q3) A point charge QA=1µC is at A(0. 2.−1 ≤ x ≤ 2. Let a) Find E at r=0.0.-1. 1). . 0 ≤ y ≤ 2 .3) b) ρv at P → c) E at P.Q1) Find the total charge inside each of the volume indicated 2 −0. 1).1 x sin πy.6 a) ρV = 10 z e b) ρV = 4 xyz 2 . Q6.4).2.2m b) find total charge with in sphere r=0. x2 + y2 → r→ 3 a) Show that ∇⋅ B = 0 . d) Does V satisfy Laplace’s equation? Q7.0 ≤ y ≤ 1. 0 ≤ z ≤ 1 in the direction away from origin.0 ≤ ρ ≤ 2.Determine the magnetic filed produced by current element at (1. b) Where should a 30nC point charge be located so that E is zero at the origin.0 ≤ φ ≤ π / 2. Let a) Find total electric flux passing the surface defined by x=3.3). b) Find the current density J .3 ≤ z ≤ 3. nC/m 2 in free space. 2.5 ) fills region 1( x ≤ 0 ) while region-2( x ≥ 0 ) is a free space.1) and QB=-1µC is at B (0.2m. 20( xa x + ya y ) Q9) A certain magnetic field intensity is given by H = A/m in free space. If ε = ε 0 and V= 8x2yz find a) V at P(2. pC/m2 in free space. a) Find E at the origin.0 ≤ z ≤ 3 Q2) In Free space let Q1=10nC be at P1(0. Find E r. D = a r Q4. A homogenous dielectric ( ε r = 2.-1). c) Find the total charge enclosed in an incremental sphere having radius of 2 µm centered at P (3.0) and Q2=20nC be at P2(0. a) If D1=12ax -10ay+4az nC/m2 find D2 and θ2 b) if E2=12 volt/m and θ2 =600 find E1 and θ1.3). Take θ1 and θ2 as the angles which E1 and E2 makes with normal to the interface.0. Q8) The vector magnetic potential A due to direct current in a conductor in free space is given by A = ( x 2 + y 2 )a z µWb/m2. 0<y<3m at t=1microseconds. a) Determine J at (5. by . z) if Ax =Ay = 0 and Az = 0 at P(1.01xa z mT.1). b) Find current passing through x = − c) Show that ∇ ⋅ B = 0. z<2. Q10) In a certain conducting region H = yz ( x 2 + y 2 )a x − y 2 xza y + 4 x 2 y 2 a z A/m.0<x<20m.-3) 1. Find a) the magnetic flux passing through the surface defined z=0.1.2.c) Find A (x.0 <y. y. TUTORIAL SHEET -3 Q1) Given the magnetic flux density B = 6 cos 10 6 t sin 0. H and k.e Magnetic field c) B i. Find JD and E. µ =1. Find β and corresponding E using Maxwell’s equation. H = 5 cos(1011 t − 4 y ) a z A/m.e Magnetic flux Density d) ω. frequency of elctromagnetic field wave. Calculate a) JD b)H c)ω 7 Q6) In air E = sin θ cos(6 × 10 t − βr )aφ V/m. t ) Q5) In free space E = 20 cos(ωt − 50 x)a y V/m.01z ) a y V/m.25ε the magnetic field of wave is 8 H = 0. ε = ε ε r and µ = µ . displacement current density b) H i. Q3) material of infinite extent with 9 ε = 2 ×10 F / m. If all the fields vary sinusoidally use maxwell’s equation to find D. B. Q9) In free space H = ρ(sin φa ρ + 2 cos φaφ ) cos(4 ×10 6 t ) A/m. . Q10) An antenna radiates in free space and 12 sin θ H = cos(2π ×10 8 t − βr )a θ mA/m. Make use of the maxwell’s equation to find a) εr b) H ( z .If E = 800 sin(10 6 t −0. Find a) J D and D b) εr Q8) In a certain Region with σ=0. r Q7) In a charge free region for which σ=0. −10 −−5 Assume a homogenous Q4) A certain material has σ=0 and µ r=1. and σ=0.b) The value of closed line integral of E around the perimeter of the surface specified above at t=1microseconds.25 ×10 H / m. Find β and H. Q2) In free space E = 20 cos( wt −50 x)a y V/m calculate a) JD . r Find corresponding E in terms of r and θ.Let E = 400 cos(10 t − kz ) a y V/m.6 cos βx cos10 ta z A / m .µ= µ and ε = 6. TUTORIAL SHEET -4 1. Calculate the radiation Resistance of £/20 dipole in free space. The radiation resistance of an antenna is 80 ohm and loss resistance is 10ohm. if the power gain is 20? 2.What is the Directivity. . Two vertically oriented half wave dipoles are spaced 1. it is required to place a null at an angle 33.Calculate its radiation resistance. An antenna has a loss resistance 10ohm. Calculate the power radiated by this antenna and its radiation resistance.5 £apart to form an array . Calculate the maximum effective aperture of microwave antenna which has Directivity of 900. Also calculate the angle at which the mail beam is placed for this phase distribution. 7. Find out radiation resistance of l/8 wire dipole in free space. A maximum current carried by λ / 40 antenna is 125 amp.Calculate the progressive phase shifts to be applied to the elements. TUTORIAL SHEET -5 1. Calculate the maximum Effective aperture of an antenna which is operating at wavelength of two meters and a directivity of 100. when the dipoles are fed with: 8.Calculate half power bandwidth of the major lobes of the array in horizontal plane.3. In the radiation pattern of a 3 element array of isotropic radiators equally spaced at distances l/4. 5. power gain of 20 and directivity of 22. 6. Calculate the radiation resistance of λ / 10 wire dipole in free space. . a)equal and in phase current (broadside array) and 9. 2.56 degree of the endfire direction . 4. b)equal current but with a phase difference of 540 between the two current(End fire Array) 10. for the following two cases. TUTORIAL SHEET -6 1. The rms current of antenna is 40 amperes at 550 kHz. III.5Ω. Calculate: I. Drive an expression for the vector potential Az at a point P located at a large distance r from a half wave dipole placed along the Z. Explain the effective length of an antenna as a radiator of electromagnetic energy.8Ω. Show that the directivity of isotropic antenna is unity. the radiation and induction fields have equal amplitude at distance. The total resistance of an antenna having effective height of 62 meters is 48. 7. Radiation power from this antenna Efficiency of antenna 10. 4. In Hertzian dipole. 6.axis. 5. 9. The power radiated from the dipole antenna is maximum at right angle to the axis of the antenna. Compute the effective length of a half wave dipole. Radiation resistance of antenna II. 2. Justify the statement. What do you mean by the radiation resistance of an antenna? What is the nature of the current distribution in a base fed half wave vertical antenna erected just above a perfect earth? 8.3. Establish that . Calculate efficiency and total resistance of λ / 16 dipole antenna if the loss resistance is 1. c. .where the terms have their usual meaning. TUTORIAL SHEET -7 1. What is Radiation Resistance of an antenna? Show that the radiation resistance of Half Wave dipole is 73ohms. 7. The power delivered to an isotropic radiator is 1KW and antenna efficiency is 90%. The electric field in free space is given by E = 50 cos(108 t + βx)a y V/m a. 6. A directional antenna has an effective radiated power of 1. b. a) Find the Field strength at a distance of 10Km.088 amp rms. Find the direction of wave propagation.1kW. 4. Calculate β and time it takes to travel a distance of λ/2. The radiation resistance is 74Ω at resonance and the measured antenna current is 1. b) Find the Power Radiated. Find the electric field intensity at a distance of 100 km. Sketch the wave at 0. Show that the effective area and effective length of an antenna are related by . Find a) The antenna efficiency b) The antenna power loss c) The directive gain in decibels over an isotropic radiator. T/4& T/2 sec. 3. A transmitting antenna with an effective height of 100 meters has a current at the base 100 amperes (rms) at the frequency of 300 KHz. 5. when it is fed with a terminal input power of 90 watts. 1mho / m ) normal to x-axis at x=0. 5.y. ε =ε . Find (a) Reflection and transmission coefficient (b) E r and H r (c) E t and H t 6. The plane wave E = 50 sin (ωt .10ns) mho/m at 10 7. µ r =1 and H= .5 sin ( ω t-z) ay A/m. Let E s = 20 e −y z a x V/m then find (a) attenuation constant α (b) Phase constant β (c) Wave velocity u (d) Wavelength λ (e) Intrinsic impedance η (f) H s (g) E (2.write expressions for (a) E (x. If E = 400cos ωt ax V/m at y=0. Determine (a) β (b)Loss tangent (c) wave impedance .5. µ r = 4 and σ =10 -3 MHz. the plane wave propagating through the dielectric has the magnetic field component H= 10e-αx cos ( ω t –x/2) ay A/m Find E and α.y.y. In a lossless medium for which η = 60π. Determine the reflected wave Hr Er and the transmitted wave Ht. A lossy dielectric is characterized by ε r = 2. A lossy dielectric has an intrinsic impedance of 20 ∠ If at that frequency.0.30 Ω at a particular frequency.t) (b) E s(x. µ r = 2 has E =0. σ = 0. A plane wave propagating through a medium with ε r = 8.y. 3. a plane with H = 10 cos (10 8t-βx) ax mA/m is incident normally on a lossless medium (ε = 2ε . 4. ω and E. Calculate ε r.5e-z/3 sin(108 t.1cos ( ω t-z)ax +0. Et.4. Determine the skin depth and wave polarization.z) (c) H s(x.z. A uniform plane wave in free space is propagation in the -ay direction at a frequency of 10 MHz. µ = 8µ ) in region (z ≥ 0).5 x ) a y V/m in a lossless medium ( µ = 4µ . In free space (z ≤ 0).3.t) 8.z. ε = ε ) encounters a lossy medium( µ = µ .βz)ax V/m. 2.z) (d) H (x. Calculate the maximum Aperture of a λ/2 (Half wave) Antenna. Calculate the Effective length of an Antenna. The magnetic field component of a plane wave in a lossless dielectric is H=30 sin (2 π × 108t-5x)az mA/m (a) If µ r =1 find ε r (b) Calculate the wavelength and wave velocity (c) Determine the polarization of wave. .13 λ^2 and η=120π.βx)(ay+ az) V/m µ If =50 µ .(d) wave velocity (e) H field 9. Given Rr= 73 ohms. Ae(max)= 0. Calculate the maximum effective aperture of an antenna which is operating at a wavelength 0f 2 meters and has a directivity of 100. TUTORIAL SHEET -8 1. 3. In a certain medium E=10 cos(2 π × 107t. 2. 10. (e) Find displacement current density. (d) Find the corresponding electric field component. ε = 2ε and σ = 0 find β and H. . Calculate the FBR (Front to Back) ratio of an antenna in DB which radiates 3 KW in its most optimum direction and 500 Watts in the Opposite direction. A thin dipole antenna is λ/15 long. 7. The Noise figure of an amplifier at room temperature (T=290 K) is 0. power gain of 20 and directivity 22. Calculate the radiation resistance of an antenna which is radiating 1000 watts and drawing a current of 5 amps.2 DB. 10. 5. 9. Beamwidth and capture area for a parabolic antenna with a 6 meters diameter dish and dipole feed at a frequency of 10 GHz. find radiation resistance and the efficiency. Find the gain. Calculate its radiation resistance. Calculate the gain of an Antenna with a circular aperture of diameter 3 meters at a frequency of 5 GHz. 6. 8.4. An antenna has loss resistance 10 ohms.5 ohms. Find the equivalent temperature. If its loss resistance is 1. A maximum current carried by wire dipole in free space. In Hertzian dipole. 6. at 550 kHz.TUTORIAL SHEET -9 1. The total resistance of an atenna having effective height of 62 metres is 48. What do you mean by the radiation resistance of an antenna? What is the nature of the current distribution in a base fed half wave vertical antenna erected just above a perfect earth? 8. the radiation and induction fields have equal amplitude at distance. Show that the directivity of isotropic antenna is unity. 3.axis. 4. dipole antenna if the loss resistance . 5. 7. antenna is 125 amp. Explain the effective length of an antenna as a radiator of electromagnetic energy. Calculate the radiation resistance of 2. Calculate the power radiated by this antenna and its radiation resistance. Calculate effeciency and total resistance of is 1. Justify the statement. Drive an expression for the vector potential Az at a point P located at a large distance r from a half wave dipole placed along the Z. The rms current of antenna is 40 amp.8Ω. Calculate: a) Radiation resistance of antenna b) Radiation power from this antenna c) Efficiency of antenna 10. 9.5Ω. The power radiated from the dipole antenna is maximum at right angle to the axis of the antenna. (a) What is Radiation Resistance of an antenna ? (b) Show that the radiation resistance of Half Wave dipole is 73ohms. . 6. Find a) The antenna efficiency b) The antenna power loss c) The directive gain in decibels over an isotropic radiator. 5. Establish that Where the terms have their usual meaning. 7.1kW.TUTORIAL SHEET -10 1. 2. when it is fed with a terminal input power of 90 watts. A transmitting antenna with an effective height of 100 meters has a current at the base 8.088 amp rms. The power delivered to an isotropic radiator is 1KW and antenna effeiciency is 90%. 100 amp(rms) at the frequency of 300KHz. Compute the effective length of a half wave dipole. 3. A directional antenna has an effective radiated power of 1. Find the Field strength at a distance of 10Km. Find the electric field intensity at a distance of 100 km. Show that the effective area and effective length of an antenna are related by . 9. 10. Find the Power Radiated. 4. The radiation resistance is 74Ω at resonance and the measured antenna current is 1.