Plane Waves Part II. 1. For an electromagnetic wave incident from one medium to a second medium, total reflection takes place when

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1 Plane Waves Part II. For an electromagnetic wave incident from one medium to a second medium, total reflection takes place when (a) The angle of incidence is equal to the Brewster angle with E field perpendicular to the plane of incidence. (b) The angle of incidence is equal to the Brewster angle with E field parallel to the plane of incidence. (c) The angle of incidence is equal to the critical angle with the wave moving from the denser medium to a rarer medium (d) The angle of incidence is equal to the critical angle with the wave moving from a rarer medium to a denser medium [GATE 987: 2 Marks] Soln. For an electromagnetic wave incident from one medium to a second medium, total (internal) reflection takes place when wave moves from a denser to rarer medium and angle of incidence should be greater than or equal to the critical angle. The critical angle θ c, sin(θ c ) ε 2 ε sin(θ i ) ε 2 ε Where θ i is the angle of incidence? As sin θ i or sin θ c can not be greater than, ε > ε 2 Option (c) 2. In a good conductor the phase relation between the tangential components of electric field E t and the magnetic field H t is as follows (a) E t and H t are in phase (c) H t leads E t by 90 0 (b) E t and H t are out of phase (d) E t leads H t by 45 0 [GATE 988: 2 Marks] Soln. For a good conductor σ ωε Intrinsic impedance

2 η E H jωμ σ + jωε η ωμ σ 450 E leads H by 45 0 Option (d) 3. The skin depth of copper at a frequency of 3 GHz is micron (0-6 meter). At 2 GHz, for a non magnetic conductor whose conductivity is /9 times that of copper, the skin depth would be (a) 9 4 microns (b) 9/4 microns (c) 4/9 microns (d) 9 4 microns [GATE 989: 2 Marks] Soln. For a good conductor, skin depth δ πfμσ For copper δ f 3 GHz For non - magnetic with μ μ 0 πf μσ 0 6 m σ 2 σ 9 at f 2 2 GHz δ 2 πf 2 μσ 2 δ fσ for the same μ δ 2 δ f σ f 2 σ 2 3 σ 2 σ 9 Option (b)

3 4. The electric field component of a uniform plane electromagnetic wave propagating in the Y direction in a lossless medium will satisfy the equation (a) 2 E y y 2 (b) 2 E y x 2 μ 2 E y t 2 μ 2 E y t 2 (c) (d) 2 E X y 2 E X 2 +E Z 2 H X 2 +H Z 2 μ 2 E X t 2 μ [GATE 99: 2 Marks] Soln. In a uniform plane EM wave propagating in the y direction, the components of E and H in the direction of propagation Y E y, H y are zero. E and H should be function of Y and t satisfying second order partial differential equation δ 2 E x δ y 2 με 2 E x t 2 E and H are related as E z H x η, E x H z η E H E x 2 + E2 z H 2 x + H2 η z η Intrinsic impedance of the medium η 0 μ Option (c) and (d)

4 5. A material is described by the following electrical parameters at a frequency of 0 GHz σ 0 6 mho, μ μ m 0 and 0 0. The material at this frequency is considered to be ( 0 36π 0 9 F m) (a) Good conductor (b) Good dielectric (c) Neither a good conductor nor a good dielectric (d) Good magnetic material [GATE 993: 2 Marks] Soln. For a good conductor σ ω f 0 GHz, σ 0 6 mho/m 0 0 or σ ω Option (a) π π 0 2π π 0 8 2π π A plane wave is incident normally on a perfect conductor as shown in i i figure. Here E x, H y and P i are electric field, magnetic field and Poynting vector respectively for the incident wave. The reflected wave should have E X i X P i Z H Y i

5 (a) E x r E x i (b) H y r H y i (c) P r P i (d) E r i x E X [GATE 993: 2 Marks] Soln. The tangential component of E is continuous at the surface. That is it is the same just outside the surface as it is just inside the surface. As E is zero within a perfect conductor, tangential component just outside the conductor at z 0, E x i + E x r Tangential component just inside the conductor at z 0+ 0 E x r E x i The amplitude of E x i is reversed on reflection, but H x r H x i P r P i The average power flow is zero indicating a standing wave in the incident reflected medium Option (a) & (c) 7. A uniform plane wave in air is normally incident on an infinitely thick slab. If the refractive index of glass slab is.5, then the percentage of the incident power that is reflected from the air glass interface is (a) 0% (b) 4% Soln. Medium Medium 2 n (c) 20% (d) 00% [GATE 996: 2 Marks] n 2. 5 μ μ 0 μ 2 μ 0 If r n, n 2 are refractive indices and v, v 2 are the velocities n n 2 v 2 v μ ε μ 2 ε 2. 5

6 2 for μ μ 2 μ 0 ε ε Reflection coefficient, E r η 2 η E i η 2 +η impedances of medium and medium 2 respectively η and η 2 are the intrinsic η μ ε, η 2 μ 2 ε 2 μ μ 2 μ 0 E r E i E r 2 E i P r E r 2 4% P i E i 2 25 Option (b) Soln. 8. Some unknown material has a conductivity of 0 6 mho/m and a permeability of 4π 0 7 H/m. The skin depth for the material at GHz is (a) 5.9 μm (b) 20.9 μm σ 0 6 mho/m μ μ 0 4π 0 7 H/m (c) 25.9 μm (d) 30.9 μm [GATE 996: 2 Marks]

7 f GHz skin depth δ πfμσ m 50 2π 0 4 π Option (a) 5. 9 μm π 0 9 4π μm 9. A uniform plane wave in air impinges at 45 0 angle on a lossless dielectric material with dielectric constant ε r. The transmitted wave propagates in a 30 0 direction with respect to the normal. The value of ε r is (a).5 (c) 2 (b).5 (d) 2 [GATE 2000: 2 Marks] Soln. At the interface between air and lossless dielectric angle of incidence θ i 45 0 angle of refraction θ r According to Snell s law sin θ 2 sin θ v 2 v v 2 is the velocity of wave in medium 2 v is the velocity of wave in medium v 2 μ 2 2, v μ μ μ 2 μ 0, 2 r

8 sin θ 2 sin 300 sin θ sin 45 0 μ μ 2 2 sin sin r 2 Option (d) r 0. A uniform plane electromagnetic wave incident normally on a plane surface of a dielectric material is reflected with a VSWS of 3. What is the percentage of incident power that is reflected? (a) 0% (b) 25% Soln. VSWR 3 Let S 3 S + K K K S S E r E i K P r E r 2 P i E i 2 K2 4, k is the reflection coefficinet 25% Option (b) (c) 50% (d) 75% [GATE 200: 2 Marks]

9 . A plane wave is characterized by E (0.5x + y e jπ/2 )e jωt jkz. This wave is (a) Linearly polarized (c) Elliptically polarized (b) Circularly polarized (d) Un polarized [GATE 2002: 2 Marks] Soln. The wave E (0. 5x + y e jπ/2 )e jωt jkz represents a wave travelling in the positive Z direction. Taking the real part of E E x (z, t) 0. 5 cos(ωt kz) E y (z, t) cos (ωt + π 2 kz) sin(ωt kz) ( E x 0. 5 ) 2 + ( E y ) 2 This is the equation of an ellipse. The wave is elliptically polarized Option (c) 2. Distilled water at mho C is characterized by σ.7 0 m 78 0 at a frequency of 3 GHz. Its loss tangent tan δ is (a) (b) Soln. For distilled water σ mho/m and (c) (d) (78 0 ) [GATE 2002: 2 Marks] 78 0, f 3 GHz, π Loss tangent tan δ σ ω σ ω π 0 9 2π

10 Jd δ J conduction current density σe J d displacement current density jω E Option (a) J 3. Medium has the electrical permittivity ε.5 ε 0 farad/m and occupies the region to left of x 0 plane. Medium 2 has the electrical permittivity ε ε 0 farad / m and occupies the region to the right of x 0 plane. If E in medium is E (2 u x 3 u y + u z ) volt/m, the E 2 in medium 2 is (a) (2 u X 7.5 u Y u Z ) volt m (b) (2 u X 2 u Y u Z ) volt m (c) (.2 u X 3 u Y + u Z ) volt m (d) (.2 u X 2 u Y u Z ) volt m [GATE 2003: 2 Marks] Soln. The interface between medium and 2 is x 0 or y z plane. 2 X < X > X 0 Y Z plane E (2 u x 3 u y + u z ) volt/m The y and z components of E are same as the y and z components of E2 as the tangential components of E are continuous.

11 E t E t2 The normal components of D are continuous D N D N2 E N 2 E N2 E X 2 E X2 or E X2 E X E (. 2 u x 3 u y + u z ) volt/m Option (c) 4. A uniform plane wave traveling in air is incident on the plane boundary between air and another dielectric medium with ε r 4. The reflection coefficient for the normal incidence is (a) Zero (b) (c) (d) [GATE 2003: 2 Marks] Soln. For normal incidence of a uniform plane wave at air dielectric interface, reflection coefficient K, r 4 K E r E i η 2 η η 2 + η η 2 η η 2 η + μ μ 2 μ 0, η 2 μ 0 2

12 η μ 0 K Where, 0 and 2 r 0 r K r K Option (d) 5. If the electric field intensity associated with a uniform plane electromagnetic wave traveling in a perfect dielectric medium is given by E(z, t) 0 cos(2π 0 7 t 0. πz) volt/m, the velocity of the traveling wave is (a) m/sec (b) m/sec (c) m/sec (d) m/sec [GATE 2003: 2 Marks] Soln. For a uniform plane wave in a perfect dielectric medium σ 0 E(z, t) 0 cos(2π 0 7 t 0. πz) v/m The E field represents wave travelling in the positive z direction with ω 2π 0 7 rad/ sec β 0. π rad/sec Velocity of propagation V ω β 2π 07 0.π

13 2 0 8 m/sec Option (b) 6. A plane electromagnetic wave propagating in free space is incident normally on a large slab of loss less, non magnetic, dielectric material with ε > ε 0. Maxima and minima are observed when the electric field is measured in front of the slab. The maximum electric field is found to be 5 times the minimum field. The intrinsic impedance of the medium should be (a) 20 π Ω (b) 60 π Ω Soln. E max 5 E min in medium (c) 600 π Ω (d) 24 π Ω [GATE 2004: Marks] VSWR, S E max E min 5 Reflection coefficient K S S free space 2 loss less σ 0, μ μ 0 σ 0 0 Incident wave Reflected wave Non magnetic μ μ 0 dielectric > 0 K η 2 η η η 3η 2 η 3 2η 2 η + 2

14 η 2 η 5, η 2 5η η μ 0 0 4π π 0 9 Option (c) (20π)Ω η π 600 π Ω 7. A medium is divided into regions I and II about x 0 plane, as shown in the figure below. An electromagnetic wave with electric field E 4a x + 3a y + 5a z is incident normally on the interface from region I. The electric filed E 2 in region II at the interface is Region I Region II σ 0, μ μ 0 σ 2 0, μ 2 μ 0 r 3 r2 4 E E 2 X 0 Soln. (a) E 2 E (b) 4 a x a y.25 a z E 4a x + 3a y + 5a z (c) 3 a x + 3 a y + 5 a z (d) 3 a x + 3 a y + 5 a z [GATE 2006: 2 Marks] The Y and Z components of E are same as the Y and Z components of E2 as the tangential components of E are continuous at the boundary. E t E t2 E y2 E y, E z2 E z

15 The normal components of D is continuous at the boundary D n D n2 ε E x ε 2 E x2 or E x2 ε E x E 2 3a x + 3a y + 5a z Option (c) 8. When a plane wave traveling in free space is incident normally on a medium having ε r 4.0, the fraction of power transmitted into the medium is given by (a) 8/9 (b) /2 (c) /3 (d) 5/6 [GATE 2006: 2 Marks] Soln. Normal incidence from free space on a medium is shown in figure E t E i 2η 2 η 2 + η 2 + η η 2 2 free space μ μ 0 r 4 μ 2 μ 0 E i P i 0 η η 0 E r E t P t P r

16 η η 2 μ 2 2 μ μ μ 2 μ 0 r 2 E t 2 E i P i Incident power E i 2 η P t Transmitted power E t 2 η 2 P t 2 E t η 4 P 2 i E i η Option (a) Soln. 9. The H field (in A/m) of a plane wave propagating in free space is given by H x 5 3 cos(ωt βz) + y 5 sin (ωt βz + π ). The time average η 0 η 0 2 power flow density in Watts is (a) η 0 00 (b) 00 η 0 (c) 50η 0 2 (d) 50 η 0 H x 5 3 cos(ωt βz) + y 5 sin (ωt βz + π η 0 η 0 2 ) [GATE 2007: Marks] H x 5 3 η 0, H y 5 η 0

17 H T H 2 x + H 2 y (5 3)2 η η2 0 5 η 0 ( 3) η η 0 Time average power flow density P avg 2 η 0H T 2 2 η 00 0 η η 0 watts Option (d) 20. A plane wave having the electric field component E i 24 cos(3 0 8 t βy) a z V/m and traveling in free space is incident normally on a lossless medium with μ μ 0 and ε 9 ε 0 which occupies the region y 0. The reflected magnetic field component is given by (a) 0π cos(3 08 t + y) a x A/m (b) 20π cos(3 08 t + y) a x A/m (c) 20π cos(3 08 t + y) a x A/m (d) 0π cos(3 08 t + y) a x A/m [GATE 200: 2 Marks] Soln. E iz 24 cos(3 0 8 t βy) v/m ω rad/sec

18 β ω V V V 0 for free space m/sec β r/m η η 0 20π E iz H ix H ix 24 cos(3 08 t y) 20π cos(3 08 t y) 5π H r η η η 2 η 2 H i η + η η 2 + η 2 η η 2 μ 2 2 μ μ 2 μ μ 0 η η H r 3 H i H r 2 cos(3 08 t + y) a x A/m H r is reflected wave which travels in negative Y direction Option (a) 2. The electric and magnetic fields for a TEM wave of frequency 4 GHz in a homogeneous medium of relative permittivity ε r and relative permeability μ r are given by

19 E E P e j(ωt 280πy) u z V / m H 3 e j(ωt 280πy) u x V / m Assuming the speed of light in free space to be m/s, the intrinsic impedance of free space to be 20 π, the relative permittivity ε r of the medium and the electric field amplitude E P are (a) ε r 3, E P 20π (b) ε r 3, E P 360π (c) ε r 9, E P 360π (d) ε r 9, E P 20π [GATE 20: 2 Marks] Soln. E and H, the wave is traveling in the Y direction Phase shift constant β 280π η E H E P 3 Velocity of wave V ω β V 2π π 0 8 m/s V μ Velocity for free space V μ r μ 0 0 μ r r V 3 08 r 0 8, r 9 μ r r μ m/sec η μ μ rμ 0 μ 0 r 0 0 r

20 μ r η 20π r E P 3 20π r E P 20π V/m Option (c) 22. If the electric field of a plane wave is E (z, t) x 3 cos(ωt kz ) y 4 sin(ωt kz ) (mv/m) The polarization state of the plane wave is (a) Left elliptical (b) Left circular Soln. Equation for electric field of plane wave is (c) Right elliptical (d) Right circular [GATE 204: 2 Mark] m) E (z, t) x. 3 cos(ωt kz ) y. 4 sin(ωt kz ) (mv/ Here wave is propagation in +z direction. Above equation can be written for z 0 E (0, t) x. 3 cos(ωt ) y. 4 sin(ωt ) x. 3 cos(ωt ) + y. 4 sin(ωt ) E(0, t) x. 3 cos(ωt ) + y. 4 cos(ωt ) Ey component is leading Ex by 05 0 As in the given figure the resultant E vector rotates in clockwise direction as shown. Since the direction of propagation is +z direction. If we look the wave in +z direction then it will look moving counter clockwise or left hand elliptically polarized ( E x E y )

21 y E x z Hint: For right handed coordinate system. In the given figure y E 2 E z E x E - E2 - E - amplitude of wave linearly polarized in x direction amplitude of wave linearly polarized in y direction Resultant vector If E2 leads E, then E vector moves (clockwise) as indicated If E leads E2 then it moves in counter clockwise. To observe the wave polarization, we have to look towards the direction of propagation of wave. (Refer: Antennas by Kraus)

22 23. The electric field of a plane wave propagation in a lossless non magnetic medium is given by the following expression E (z, t) a x5 cos(2π 0 9 t + βz) + a y3 cos (2π 0 9 t + βz π 2 ) The type of the polarization is (a) Right hand circular (b) Left hand elliptical Soln. Given (c) Right hand elliptical (d) Linear [GATE 205: 2 Mark] E (z, t) a x5 cos(2π 0 9 t + βz) + a y3 cos (2π 0 9 t + βz π 2 ) Let ω 2π 0 9 E (z, t) a x5 cos(ωt + βz) + a y3 cos (ωt + βz π 2 ) Here, Ey is lagging Ex by 90 0 The resultant vector moves as shown (opposite to leading case) as shown in the given figure E y E z E x z Direction of propagation as per the equation is z direction Thus, the resultant vector E moves in counter clock clockwise if we look into z direction Thus, the wave is left hand elliptically polarized (since unequal amplitude) Option (b)

Plane Waves GATE Problems (Part I)

Plane Waves GATE Problems (Part I). A plane electromagnetic wave traveling along the + z direction, has its electric field given by E x = cos(ωt) and E y = cos(ω + 90 0 ) the wave is (a) linearly polarized