Instructions:
- The marks are indicated in the right-hand margin.
- There are NINE questions in this paper.
- Attempt FIVE questions in all.
- Question No. 1 is compulsory.
Questions
Choose the correct answer of the following (any seven) :
The mathematical perception of the gradient is said to be
The permeability of a dielectric material in air medium will be
The frequency in rad/sec of a wave with velocity of that of light and phase constant of 20 units (in GHz) is
One end of a lossless transmission line having the characteristic impedance of 75 ohm and length of 1 cm is short circuited. At 3 GHz, the input impedance at the other end of the transmission line is
A transmission line is distortionless, if
In the design of a single-mode step-index optical fiber close to upper cutoff, the single-mode operation is NOT preserved, if
Which parameters cannot be computed from the Smith chart?
In a microwave test bench, why is the microwave signal amplitude modulated at 1 kHz?
The number of modes in a waveguide having a V number of 10 is
The magnetization is defined by the ratio of
Attempt any two parts of the following :
(i) What do you mean by transverse nature of EM wave? Explain mathematically. (ii) What are degenerative and dominant modes?
Write Maxwell's equations in free space for time varying fields both in differential and integral forms.
The energy density of an electromagnetic wave is proportional to the square of the amplitude of the electric (or magnetic) field. Prove it.
Attempt any two parts of the following :
Describe Poynting vector and Poynting theorem. What are their applications?
Assuming that , show that Maxwell's equations imply the following complex valued version of Poynting's theorem where .
Describe Kramers-Kronig dispersion relations.
Attempt any two parts of the following :
(i) With the help of diagram, explain spherical coordinate system. (ii) Write boundary condition for electric and magnetic fields.
A transmission is lossless and 25 m long. It is terminated with a load of at a frequency of 10 MHz. The inductance and capacitance of the line are and . Find input impedance at the source and at the midpoint of the line.
What is relation between conduction current density and displacement current density ?
Attempt any two parts of the following :
Compare circuit theory and field theory in detail.
Discuss reflection of plane wave at the interface of conductor for oblique interface.
Develop an expression for the potential difference at any point between spherical shells in terms of the applied potential employing Laplace's equation.
Attempt any two parts of the following :
(i) Write about reflection by a perfect dielectric at normal incidence. (ii) Write about conductors and dielectrics in detail.
Attempt any two parts of the following :
Derive the expressions for magnetic field intensity due to toroidal coil and circular coil.
Derive the expressions for energy stored and energy density in magnetic field.
(i) Derive the expressions for self-inductance of two-wire transmission line. (ii) Derive the expressions for force between two current-carrying conductors.
Attempt any two parts of the following :
(i) Define and explain Biot-Savart law. (ii) Find at the centre of an equivalent triangular loop of side 4 m carrying current of 5A.
Define uniform plane wave propagation. Discuss and derive its properties.
Write short notes on the following : (i) Magnetic vector potential (ii) Helmholtz equations
Instructions:
- The marks are indicated in the right-hand margin.
- There are NINE questions in this paper.
- Attempt FIVE questions in all.
- Question No. 1 is compulsory.
- Symbols and notations carry their usual meanings.
Q.1 Choose the correct answer of any seven of the following:
In free space, the Poisson equation becomes
Poisson equation can be derived from which of the following equations?
Poynting vector gives the
Using volume integral, which quantity can be calculated?
Electric flux density in electric field is referred to as
Which of the following correctly states Gauss law?
Find the power reflected in a transmission line, when the reflection coefficient and input power are 0.45 and 18 W respectively.
In a waveguide, which of the following conditions is true always?
The phase and group velocities do not depend on which of the following?
Q.2 Attempt any two parts of the following:
A rectangular waveguide with dimensions of $ 3 \text{ cm} \times 2 \text{ cm} $ operates at 10 GHz. Find: (i) cut-off frequency $ (f_c) $; (ii) cut-off wavelength $ (\lambda_c) $; (iii) guided wavelength $ (\lambda_g) $; (iv) phase constant $ (\beta_g) $.
What do you mean by transmission line? Derive an expression for transmission line equations.
Determine the expression for average power of Poynting vector.
Q.3 Attempt any two parts of the following:
(i) Define quality factor. Give its relation with attenuation factor. (ii) Define reflection coefficient and VSWR. Also write their interrelation.
(i) Compare wave impedance and characteristic impedance. (ii) Define tangent loss.
Derive the field components when wave is propagating inside a rectangular waveguide with TM mode propagation.
Q.4 Attempt any two parts of the following:
Derive an expression for input impedance when transmission line is terminated with any load impedance.
What is equipotential surface? Explain Poynting vector and average Poynting vector.
State and prove Ampere's work law as $ \nabla \times \vec{H} = J $.
Q.5 Attempt any two parts of the following:
Derive the Gauss divergence theorem and Stokes' theorem along with their significances.
Explain the wave between parallel planes. Derive the expression for the attenuation in parallel plane guide.
Derive the expressions for the reflection and refraction of the waves by the perfect dielectric.
Q.6 Attempt any two parts of the following:
Find the reflection and transmission coefficient for the interface between air and freshwater $ \epsilon+ j180 $ in the case of perpendicular incidence.
Derive the relationship between the following: (i) Standing-wave ratio and magnitude of reflection coefficient. (ii) Standing-wave ratio and the reflection coefficient.
(i) Write the condition for a line to be distortionless. (ii) Define the term 'phase velocity'.
Q.7 Attempt any two parts of the following:
What is polarization of wave? Discuss the properties of S- and P-polarized light. Explain why P-polarized light is also called as TM-polarized light.
Explain the term 'standing-wave ratio' related to transmission line. What will be the values of input impedances when output impedances are (i) open-circuited and (ii) short-circuited?
Explain why TEM wave does not propagate in waveguide.
Q.8 Attempt any two parts of the following:
A transmission line has a characteristic impedance of 100 ohms and is terminated in a load impedance of $ 200 + j180 $ ohms. Find the voltage reflection coefficient.
What is the penetration depth in current penetration in copper at a frequency of $ 10^4 $ MHz, if the resistivity is $ 1.7 \times 10^{-6} \Omega $ cm?
What are the satisfactory conditions for low-loss transmission lines?
Q.9 Attempt any two parts of the following:
A uniform plane wave propagating in a medium has $ E = 2e^{-\alpha z}\sin(10^8t - \beta z)a_y $. If the medium is characterized by $ \epsilon_r = 1 $, $ \mu_r = 20 $ and $ \sigma = 3 \text{ mhos/m} $, then find $ \alpha $, $ \beta $ and $ \vec{H} $.
In a non-magnetic medium $ E = 4 \sin(2\pi \times 10^7 - 0.8x)a_z \text{ V/m} $. Find (i) the time-average power carried by the wave; (ii) the total power crossing $ 100 \text{ cm}^2 $ of plane $ 2x + y = 5 $.
What is the boundary condition for metal dielectric interface?
Instructions:
- The marks are indicated in the right-hand margin.
- There are NINE questions in this paper.
- Attempt FIVE questions in all.
- Question No. 1 is compulsory.
- Symbols and notations carry their usual meanings.
Q.1 Choose the correct answer of any seven of the following:
In free space, the Poisson equation becomes
Poisson equation can be derived from which of the following equations?
Poynting vector gives the
Using volume integral, which quantity can be calculated?
Electric flux density in electric field is referred to as
Which of the following correctly states Gauss law?
Find the power reflected in a transmission line, when the reflection coefficient and input power are 0.45 and 18 W respectively.
In a waveguide, which of the following conditions is true always?
The phase and group velocities do not depend on which of the following?
Q.2 Attempt any two parts of the following:
A rectangular waveguide with dimensions of $ 3 \text{ cm} \times 2 \text{ cm} $ operates at 10 GHz. Find: (i) cut-off frequency $ (f_c) $; (ii) cut-off wavelength $ (\lambda_c) $; (iii) guided wavelength $ (\lambda_g) $; (iv) phase constant $ (\beta_g) $.
What do you mean by transmission line? Derive an expression for transmission line equations.
Determine the expression for average power of Poynting vector.
Q.3 Attempt any two parts of the following:
(i) Define quality factor. Give its relation with attenuation factor. (ii) Define reflection coefficient and VSWR. Also write their interrelation.
(i) Compare wave impedance and characteristic impedance. (ii) Define tangent loss.
Derive the field components when wave is propagating inside a rectangular waveguide with TM mode propagation.
Q.4 Attempt any two parts of the following:
Derive an expression for input impedance when transmission line is terminated with any load impedance.
What is equipotential surface? Explain Poynting vector and average Poynting vector.
State and prove Ampere's work law as $ \nabla \times \vec{H} = J $.
Q.5 Attempt any two parts of the following:
Derive the Gauss divergence theorem and Stokes' theorem along with their significances.
Explain the wave between parallel planes. Derive the expression for the attenuation in parallel plane guide.
Derive the expressions for the reflection and refraction of the waves by the perfect dielectric.
Q.6 Attempt any two parts of the following:
Find the reflection and transmission coefficient for the interface between air and freshwater $ \epsilon+ j180 $ in the case of perpendicular incidence.
Derive the relationship between the following: (i) Standing-wave ratio and magnitude of reflection coefficient. (ii) Standing-wave ratio and the reflection coefficient.
(i) Write the condition for a line to be distortionless. (ii) Define the term 'phase velocity'.
Q.7 Attempt any two parts of the following:
What is polarization of wave? Discuss the properties of S- and P-polarized light. Explain why P-polarized light is also called as TM-polarized light.
Explain the term 'standing-wave ratio' related to transmission line. What will be the values of input impedances when output impedances are (i) open-circuited and (ii) short-circuited?
Explain why TEM wave does not propagate in waveguide.
Q.8 Attempt any two parts of the following:
A transmission line has a characteristic impedance of 100 ohms and is terminated in a load impedance of $ 200 + j180 $ ohms. Find the voltage reflection coefficient.
What is the penetration depth in current penetration in copper at a frequency of $ 10^4 $ MHz, if the resistivity is $ 1.7 \times 10^{-6} \Omega $ cm?
What are the satisfactory conditions for low-loss transmission lines?
Q.9 Attempt any two parts of the following:
A uniform plane wave propagating in a medium has $ E = 2e^{-\alpha z}\sin(10^8t - \beta z)a_y $. If the medium is characterized by $ \epsilon_r = 1 $, $ \mu_r = 20 $ and $ \sigma = 3 \text{ mhos/m} $, then find $ \alpha $, $ \beta $ and $ \vec{H} $.
In a non-magnetic medium $ E = 4 \sin(2\pi \times 10^7 - 0.8x)a_z \text{ V/m} $. Find (i) the time-average power carried by the wave; (ii) the total power crossing $ 100 \text{ cm}^2 $ of plane $ 2x + y = 5 $.
What is the boundary condition for metal dielectric interface?
Instructions:
- The marks are indicated in the right-hand margin.
- There are NINE questions in this paper.
- Attempt FIVE questions in all.
- Question No. 1 is compulsory.
Q.1 Write short answer of the following questions (any seven):
Explain the importance of a unit vector.
State divergence theorem and give its mathematical form.
Define propagation constant.
What do you understand by homogeneous and isotropic medium?
Write down Maxwell's equation in free space.
What is dissipation factor of dielectric?
What is displacement current? Give its expression.
What is rotational and irrotational vector field?
Are all the four Maxwell's equations independent? Explain briefly.
Explain briefly the significance of skin depth.
Q.2 Solve all questions :
Express the position vector $ r = xa_x + ya_y + za_z $ in the spherical coordinate system.
State and prove the Gauss's theorem. Explain why it is called the divergence theorem.
Justify that the net electric field within a conductor is always zero.
Q.3 Solve both questions :
Let the spherical surfaces $ r = 4 $ cm and $ r = 9 $ cm be separated by two perfect dielectric shells, $ \epsilon_{R1} = 2 $ for $ 4 < r < 6 $ cm and $ \epsilon_{R2}=5 $ for $ 6 < r < 9 $ cm. If $ E_1=(2000/r^2)a_r \text{ V/m} $, find (a) $ E_2 $; (b) the total electrostatic energy stored in each region.
Derive the Laplace equation from Gauss's law in electrostatics.
Q.4 Solve both questions :
Derive the Maxwell's curl equation for time varying electric fields.
The magnetic field intensity in a certain conducting medium is $ H = xy^2a_x + x^2za_y - y^2z a_z \text{ A/m} $. Calculate the current density at point $ P(2, -1, 3) $. What is $ dp_v/dt $ at P?
Q.5 Solve both questions :
What is the limitation of Ampere's circuital law? Explain the correction done by Maxwell to Ampere's law by explaining continuity equation.
For a current distribution in free space $ A = (2x^2y+yz)a_x + (xy^2-xz^3)a_y - $ $ (6xyz-2x^2y^2)a_z \text{ Wb/m} $. Calculate magnetic flux density.
Q.6 Solve both questions :
A plane electromagnetic wave described by its magnetic field is given by the expression, $ \vec{H} = H_0\sin(kz-\omega t)\hat{y} $. Determine the corresponding electric field and the time average Poynting vector. If it is incident on a perfect conductor and is totally reflected what would be the pressure exerted on the surface? Determine the surface current generated at the interface.
Derive a wave equation for non-dissipative medium making use of Maxwell equations and field vectors E and H.
Q.7 Solve both questions :
A uniform plane wave is incident on the interface of two perfect dielectric media with relative
permittivities of $ \epsilon_1 $ and $ \epsilon_2 $. The electric field E is parallel
to the plane of incidence. Show that reflection coefficient $ \Gamma = E_{r}/E_{i} $ and
transmission coefficient $ \tau = E_{t}/E_{i} $ are given by
$ \Gamma =
\frac{\sqrt{\epsilon_2}\cos\theta_1 -
\sqrt{\epsilon_1}\cos\theta_2}{\sqrt{\epsilon_2}\cos\theta_1 +
\sqrt{\epsilon_1}\cos\theta_2} $ ; $ \tau =
\frac{2\sqrt{\epsilon_2}\cos\theta_1}{\sqrt{\epsilon_2}\cos\theta_1 +
\sqrt{\epsilon_1}\cos\theta_2} $
where $ \theta_1 $ and $ \theta_2 $ are
angles of incidence and refraction, respectively.
State Poynting's theorem. What is Poynting vector?
Q.8 Solve both questions :
What is the boundary condition? Derive the law of refraction of the electric field at a dielectric-dielectric boundary free of charge conditions.
The transmission line is excited by a voltage source $ V_0 $coswt at $ z=0 $. What are the voltage and current distributions if the line is short circuited at $ z=l $?
Q.9 Solve both questions :
Find the input impedance of the distortion-less transmission line at radio frequencies in both open-circuited and shorted cases.
A 100 ohm line with air dielectric is terminated by a load of $ 75+j40 $ ohm and is excited at 1GHz by a matched generator. Find the position of a single matching stub of 100 ohm impedance on the line and determine the length of the stub.
Instructions:
- The marks are indicated in the right-hand margin.
- There are NINE questions in this paper.
- Attempt FIVE questions in all.
- Question No. 1 is compulsory.
Q.1 Write short answer of the following questions (any seven):
Explain the importance of a unit vector.
State divergence theorem and give its mathematical form.
Define propagation constant.
What do you understand by homogeneous and isotropic medium?
Write down Maxwell's equation in free space.
What is dissipation factor of dielectric?
What is displacement current? Give its expression.
What is rotational and irrotational vector field?
Are all the four Maxwell's equations independent? Explain briefly.
Explain briefly the significance of skin depth.
Q.2 Solve all questions :
Express the position vector $ r = xa_x + ya_y + za_z $ in the spherical coordinate system.
State and prove the Gauss's theorem. Explain why it is called the divergence theorem.
Justify that the net electric field within a conductor is always zero.
Q.3 Solve both questions :
Let the spherical surfaces $ r = 4 $ cm and $ r = 9 $ cm be separated by two perfect dielectric shells, $ \epsilon_{R1} = 2 $ for $ 4 < r < 6 $ cm and $ \epsilon_{R2}=5 $ for $ 6 < r < 9 $ cm. If $ E_1=(2000/r^2)a_r \text{ V/m} $, find (a) $ E_2 $; (b) the total electrostatic energy stored in each region.
Derive the Laplace equation from Gauss's law in electrostatics.
Q.4 Solve both questions :
Derive the Maxwell's curl equation for time varying electric fields.
The magnetic field intensity in a certain conducting medium is $ H = xy^2a_x + x^2za_y - y^2z a_z \text{ A/m} $. Calculate the current density at point $ P(2, -1, 3) $. What is $ dp_v/dt $ at P?
Q.5 Solve both questions :
What is the limitation of Ampere's circuital law? Explain the correction done by Maxwell to Ampere's law by explaining continuity equation.
For a current distribution in free space $ A = (2x^2y+yz)a_x + (xy^2-xz^3)a_y - $ $ (6xyz-2x^2y^2)a_z \text{ Wb/m} $. Calculate magnetic flux density.
Q.6 Solve both questions :
A plane electromagnetic wave described by its magnetic field is given by the expression, $ \vec{H} = H_0\sin(kz-\omega t)\hat{y} $. Determine the corresponding electric field and the time average Poynting vector. If it is incident on a perfect conductor and is totally reflected what would be the pressure exerted on the surface? Determine the surface current generated at the interface.
Derive a wave equation for non-dissipative medium making use of Maxwell equations and field vectors E and H.
Q.7 Solve both questions :
A uniform plane wave is incident on the interface of two perfect dielectric media with relative
permittivities of $ \epsilon_1 $ and $ \epsilon_2 $. The electric field E is parallel
to the plane of incidence. Show that reflection coefficient $ \Gamma = E_{r}/E_{i} $ and
transmission coefficient $ \tau = E_{t}/E_{i} $ are given by
$ \Gamma =
\frac{\sqrt{\epsilon_2}\cos\theta_1 -
\sqrt{\epsilon_1}\cos\theta_2}{\sqrt{\epsilon_2}\cos\theta_1 +
\sqrt{\epsilon_1}\cos\theta_2} $ ; $ \tau =
\frac{2\sqrt{\epsilon_2}\cos\theta_1}{\sqrt{\epsilon_2}\cos\theta_1 +
\sqrt{\epsilon_1}\cos\theta_2} $
where $ \theta_1 $ and $ \theta_2 $ are
angles of incidence and refraction, respectively.
State Poynting's theorem. What is Poynting vector?
Q.8 Solve both questions :
What is the boundary condition? Derive the law of refraction of the electric field at a dielectric-dielectric boundary free of charge conditions.
The transmission line is excited by a voltage source $ V_0 $coswt at $ z=0 $. What are the voltage and current distributions if the line is short circuited at $ z=l $?
Q.9 Solve both questions :
Find the input impedance of the distortion-less transmission line at radio frequencies in both open-circuited and shorted cases.
A 100 ohm line with air dielectric is terminated by a load of $ 75+j40 $ ohm and is excited at 1GHz by a matched generator. Find the position of a single matching stub of 100 ohm impedance on the line and determine the length of the stub.
Instructions:
- The marks are indicated in the right-hand margin.
- There are NINE questions in this paper.
- Attempt FIVE questions in all.
- Question No. 1 is compulsory.
Q.1 Fill in the blanks any seven of the following:
VSWR of a matched load is ___________
Normal component of magnetic flux density is ___________ across the boundary.
Intrinsic impedance of lossless medium is ___________
If the current flowing in the two wires is in the same direction, then there will be force of ___________
Divergence of velocity of water well within the surface of water is ___________
The product of short-circuited impedance and open-circuited impedance of a transmission line is ___________
The input impedance of a resonant section of transmission line is given by ___________
Brewster angle is given by ___________
Surface impedance of a good conductor is ___________
Power loss in a plane conductor is ___________
Q.2 Solve both questions :
Discuss uniqueness theorem in detail.
Find the capacitance between two spheres whose separation d is very much larger than their radii R.
Q.3 Solve both questions :
Express Laplacian operator in curvilinear coordinate system and find $ \nabla^2V $, where $ V=10\; r\; \sin^2\theta\; \cos\phi $.
Discuss boundary condition at the interface between two dielectric mediums and find the magnitude of $ \vec{D} $ and $ \vec{E} $ in one medium as compared to another medium.
Q.4 Solve all questions :
Find the energy stored in a magnetic field.
Discuss Stokes' law.
Discuss magnetic vector potential.
Q.5 Solve all questions :
In a medium characterized by $ \sigma=0 $, $ \mu=\mu_0 $ and $ \epsilon_r=4 $, $ \vec{E} $ is given by $ \vec{E}=20\sin(10^8t-\beta z)\hat{y} \text{ V/m} $. Calculate $ \vec{H} $ and $ \beta $.
Discuss continuity equation for current.
Discuss analogies between electric and magnetic fields.
Q.6 Solve both questions :
Find the ratio of $ \vec{E} $ and $ \vec{H} $ in a uniform plane wave in a lossless medium.
Using $ \nabla\cdot\vec{D}=\rho $, Ohm's law and equation of continuity, show that if at any instant a charge density $ \rho $ existed within a conductor, it would decrease to $ 1/e $ times this value in a time $ \epsilon/\sigma $ sec. Calculate this time for copper conductor.
Q.7 Solve this question :
The electric field strength of a uniform plane electromagnetic wave in free space is $ 1 \text{ V/m} $ and the frequency is 300 MHz. If a very large thick flat copper plate is placed normal to the direction of wave propagation, determine (a) the electric field strength and magnetic field strength at the surface of the plate, (b) the depth of penetration, (c) the conduction current density at the surface, (d) conduction current density at a distance of 0.01 mm below the surface, (e) the linear current density, $ I_s $, (f) the surface impedance, (g) the power loss per square meter of surface area. For copper $ \sigma=5.8\times 10^7 $, $ \epsilon=\epsilon_0 $ and $ \mu=\mu_0 $.
Q.8 Solve both questions :
Explain Poynting theorem.
A short vertical transmitting antenna erected on the surface of a perfectly conducting earth produces effective field strength $ E_{eff} = E_{\theta \text{ eff}} = 100\sin\theta \text{ mV/m} $ at points a distance of one mile from the antenna. Compute the Poynting vector and total power radiated.
Q.9 Solve both questions :
Find the quality factor of a resonant transmission line section.
Discuss quarter wave line as transformer.
Instructions:
- The marks are indicated in the right-hand margin.
- There are NINE questions in this paper.
- Attempt FIVE questions in all.
- Question No. 1 is compulsory.
Q.1 Fill in the blanks any seven of the following:
VSWR of a matched load is ___________
Normal component of magnetic flux density is ___________ across the boundary.
Intrinsic impedance of lossless medium is ___________
If the current flowing in the two wires is in the same direction, then there will be force of ___________
Divergence of velocity of water well within the surface of water is ___________
The product of short-circuited impedance and open-circuited impedance of a transmission line is ___________
The input impedance of a resonant section of transmission line is given by ___________
Brewster angle is given by ___________
Surface impedance of a good conductor is ___________
Power loss in a plane conductor is ___________
Q.2 Solve both questions :
Discuss uniqueness theorem in detail.
Find the capacitance between two spheres whose separation d is very much larger than their radii R.
Q.3 Solve both questions :
Express Laplacian operator in curvilinear coordinate system and find $ \nabla^2V $, where $ V=10\; r\; \sin^2\theta\; \cos\phi $.
Discuss boundary condition at the interface between two dielectric mediums and find the magnitude of $ \vec{D} $ and $ \vec{E} $ in one medium as compared to another medium.
Q.4 Solve all questions :
Find the energy stored in a magnetic field.
Discuss Stokes' law.
Discuss magnetic vector potential.
Q.5 Solve all questions :
In a medium characterized by $ \sigma=0 $, $ \mu=\mu_0 $ and $ \epsilon_r=4 $, $ \vec{E} $ is given by $ \vec{E}=20\sin(10^8t-\beta z)\hat{y} \text{ V/m} $. Calculate $ \vec{H} $ and $ \beta $.
Discuss continuity equation for current.
Discuss analogies between electric and magnetic fields.
Q.6 Solve both questions :
Find the ratio of $ \vec{E} $ and $ \vec{H} $ in a uniform plane wave in a lossless medium.
Using $ \nabla\cdot\vec{D}=\rho $, Ohm's law and equation of continuity, show that if at any instant a charge density $ \rho $ existed within a conductor, it would decrease to $ 1/e $ times this value in a time $ \epsilon/\sigma $ sec. Calculate this time for copper conductor.
Q.7 Solve this question :
The electric field strength of a uniform plane electromagnetic wave in free space is $ 1 \text{ V/m} $ and the frequency is 300 MHz. If a very large thick flat copper plate is placed normal to the direction of wave propagation, determine (a) the electric field strength and magnetic field strength at the surface of the plate, (b) the depth of penetration, (c) the conduction current density at the surface, (d) conduction current density at a distance of 0.01 mm below the surface, (e) the linear current density, $ I_s $, (f) the surface impedance, (g) the power loss per square meter of surface area. For copper $ \sigma=5.8\times 10^7 $, $ \epsilon=\epsilon_0 $ and $ \mu=\mu_0 $.
Q.8 Solve both questions :
Explain Poynting theorem.
A short vertical transmitting antenna erected on the surface of a perfectly conducting earth produces effective field strength $ E_{eff} = E_{\theta \text{ eff}} = 100\sin\theta \text{ mV/m} $ at points a distance of one mile from the antenna. Compute the Poynting vector and total power radiated.
Q.9 Solve both questions :
Find the quality factor of a resonant transmission line section.
Discuss quarter wave line as transformer.
Instructions:
- The marks are indicated in the right-hand margin.
- There are NINE questions in this paper.
- Attempt FIVE questions in all.
- Question No. 1 is compulsory.
Questions
Choose the correct alternatives for any seven of the following :
A Gaussian surface for application is
Which one of the following statements is not characteristic of a static magnetic field?
Electric field in a region containing space charges can be found using
In a transmission line, electromagnetic energy is transported by
In a certain region, the electric field , potential , there must be
The work done by the force in giving a charge a displacement of is
Which of the following is a mathematically incorrect expression?
The flux through each turn of a 100 turn coil is , where is in seconds. The induced e.m.f. at is
Which is the major factor for determining whether a medium is free space, lossless dielectric, loss dielectric or good conductor?
Answer the following :
Find the divergence and curl of the following vectors .
Given the point . Express in cylindrical and spherical coordinates.
A point charge of is located at the origin while plane carries charge . Find at .
A thin ring of radius is placed on the plane so that its centre is at . If the ring carries along , find at .
Answer the following :
Derive the following equations : (i) (ii)
Determine the self-inductance of a coaxial cable of inner radius and outer radius .
Find the force on a straight conductor of length carrying a current of in the direction, where the field is .
Answer the following :
State and explain the significance of Helmholtz's theorem.
Write Lorentz force equation. Hence obtain the expression of force acting on a straight conduction of length in a uniform magnetic field .
Answer the following :
Explain the following : (i) Divergence of a vector field (ii) Gradient of a scalar field
Consider the volume current density distribution in cylindrical coordinates as $J(r, \phi, z) = 0, \quad 0 < r < a$ $J(r, \phi, z) = J_0 (r/a) a_z, \quad a < r < b$ $= 0, \quad b < r < \infty$ Find the magnetic field intensity in various regions.
Answer the following :
Derive wave equation for lossy dielectric medium.
What is propagation constant?
Derive the expression for intrinsic impedance for lossy dielectric medium.
Answer the following :
Transform a vector into cylindrical coordinate.
State the expression of divergence for three-coordinate system.
In electrostatic field problem, the electric field is given by , where is the scalar field potential. If in spherical coordinate, find .
Answer the following :
Determine the charge densities due to each of the following electric flux densities : (i) (ii)
A spherical charge distribution is given by $\rho = \rho_0 \left( \frac{r}{a} \right), \quad r < a$ $0, \quad r > a$ Find and everywhere.
Write short notes of the following :
Stokes theorem
Green's theorem
Helmholtz theorem
Laplace and Poisson's equation
Instructions:
- The marks are indicated in the right-hand margin.
- There are NINE questions in this paper.
- Attempt FIVE questions in all.
- Question No. 1 is compulsory.
Questions
Choose the correct alternatives for any seven of the following and explain :
The electric field on equipotential surface is :
Electric field in a region containing space charge can be found using :
Electrostatic field is :
One Weber is equal to :
Two thin parallel wires carry currents along the same direction. The force experienced by one due to the other is :
The magnetic field at any point on the axis of a current-carrying circular coil will be :
To apply Gauss's law, the Gaussian surface should be chosen in such a way that field is :
Gradient of a scalar function results in a :
Answer the following:
Derive an expression for electric field due to surface (sheet) charge uniformly distributed over an infinite plane having density C/m².
State and explain the following : (i) Stokes theorem (ii) Helmholtz's theorem
Deduce boundary condition of electric field for Dielectric-Dielectric boundary.
Deduce an expression for magnetic field intensity due to an infinitely long current-carrying conductor carrying current . Use Biot-Savart law.
Answer the following:
Derive an expression for Lorentz force on a moving charge in an electromagnetic field.
What are conduction and displacement currents?
From the concept of displacement current derive an expression for modified Ampere's law.
Write and explain differential and integral forms of Maxwell's equations.
Answer the following:
A plane polarized wave is travelling along Z-axis. Show graphically the variation of and with Z. Show that for the wave.
Develop the analogy between the uniform plane EM waves and the transmission line.
A uniform transmission line has constants , , and . At , find (i) impedance (ii) dB attenuation in
Answer the following:
Establish the relation . The symbol used has usual meaning.
What do you mean by linearly polarized plane E.M. waves in free space?
What do you mean by depth of penetration in such medium? If the penetration depth is at . what will this be at ?
Answer the following:
Write down general procedure for solving Poisson's and Laplace's equation.
Deduce an expression of energy density in electrostatic field.
What is meant by the following? (i) Transformer and motional e.m.f. (ii) Electric potential and potential gradient
Answer the following:
Find curl at the origin, where .
Show that (i) (ii)
It is required to hold four equal point charges each in equilibrium at the corners of a square. Find the point charge which will do this if placed at the centres of the square.
Answer the following:
The magnetic field component of a plane wave in a lossless dielectric is . Find: (i) (ii) the wavelength and wave velocity (iii) the wave impedance (iv) the polarization of the wave (v) the corresponding electric field component
Develop the analogy between the uniform plane EM waves and the electric transmission line.
Answer the following:
What are skin effect and skin depth?
Show that in case of semi-infinite solid conductor, the depth is given by where & have their usual meaning.
What is polarization of electro-magnetic wave?
Explain the significance of poynting vector.
Instructions:
- The marks are indicated in the right-hand margin.
- There are NINE questions in this paper.
- Attempt FIVE questions in all.
- Question No. 1 is compulsory.
Questions
Fill in the blanks (any seven) :
The curl of a gradient of a scalar quantity is ____.
Energy density in the magnetic field is ____.
Normal component of electric flux density is ____ across the interface between two dielectric media.
The relation of depth of penetration in good conductor is given by ____.
The direction of magnetic vector potential is same as the direction of ____.
VSWR for a matched termination is ____.
If the standing wave of voltage slopes down towards the termination, then the terminating reactance will be ____.
Quarter wave section is an ____.
Uniform plane waves are ____ waves.
Two conductors carrying current in opposite direction experience ____ force.
Answer the following:
Derive an expression for potential due to a long pair of parallel wires.
Deduce the equation for equipotential surfaces for parallel line charges.
Find the capacitance of parallel cylindrical conductors having equal radii and separation between their axes as .
Answer the following:
Find the conductor properties and boundary conditions.
A point charge is located at a distance above an infinite conducting plane. Using the method of images, find the displacement density normal to the plane and hence surface charge density. Also obtain total charge on the infinite conducting plane.
Answer the following:
Obtain curl of a vector and interpret it.
Prove Stokes' theorem.
Discuss ampere force law.
Answer the following:
Obtain two Maxwell's equations which deviate from steady-state condition.
Using , Ohm's law and the equation of continuity, show that if at any instant a charge density existed within a conductor, it would decrease to times this value in time seconds.
Answer the following:
Discuss the propagation in a conducting medium and hence obtain the expression for attenuation constant and phase-shift constant .
Find the values of and for good conductor and good dielectric.
Answer the following:
Find out the reflection coefficient for perfect conductor in the case of normal incidence.
The electric field of a uniform plane electromagnetic wave in free space is and frequency is . If a very large thick flat copper plate is placed normal to the direction of wave propagation, determine— (i) and at the surface of plate; (ii) depth of penetration; (iii) conduction current density at the surface; (iv) conduction current density at a distance of below the surface; (v) linear current density, $J_s$; (vi) surface impedance; (vii) power loss per square metre of surface area. [Take : .]
Answer the following:
Discuss instantaneous, average and complex Poynting vectors.
Obtain power loss in a plane conductor.
A short vertical transmitting antenna erected on the surface of a perfectly conducting earth produces an effective field strength, at points a distance one mile from the antenna. Compute Poynting vector and total power radiated.
Answer the following:
Discuss UHF line as circuit element and obtain input-input resistance of the line for resonant length.
Discuss quarter wave line as a transformer.
A lossless transmission line has a characteristic impedance of and is one-quarter wavelength long. What will be the voltage at the open-circuited receiving end, if sending end is connected to a generator which has a internal impedance and generated voltage of ?
Instructions:
- The marks are indicated in the right-hand margin.
- There are NINE questions in this paper.
- Attempt FIVE questions in all.
- Question No. 1 is compulsory.
Questions
Fill in the blanks (any seven) :
Divergence of a curl of a vector is ____.
Energy density in the electrostatic field is ____.
The value of relative permeability is slightly less than one for ____ and slightly greater than one for ____.
Tangential component of electric field is ____ across the interface between two dielectric media.
Surface impedance of good conductor is just equal to ____.
For uniform plane wave field and field has ____ in the direction of propagation.
VSWR varies from ____ to ____.
Short circuited quarter wave section and open end half-wave section is analogous to ____.
If the standing wave of voltage slope is up towards the termination, then the reactance will be ____.
The quality factor of a resonant section of transmission line is equal to the ratio of ____ per unit length to ____ per unit length.
Answer the following:
For a two-dimensional system , determine , when .
Find out the divergence of vector and interpret it by giving physical examples.
State and prove divergence theorem.
Answer the following:
State and prove uniqueness theorem.
Find the capacitance of two spheres, whose separation is very much larger than their radii . Hence show that the capacitance of sphere above an infinite ground plane is independent of the height above the plane when .
Answer the following:
Describe magnetic vector potential.
Explain Ampere force law.
Find the magnetic field inside a solid conductor carrying a direct current and hence obtain total magnetic flux per unit length within the conductor.
Answer the following:
Obtain continuity equation for time-varying field.
Explain inconsistency of Ampere circuital law.
The electric vector of a electromagnetic wave in free space is given by the expression . Using Maxwell's equation for free space condition, determine magnetic vector .
Answer the following:
Find the component of and in the direction of the propagation for uniform plane wave.
Establish the relation between and in a uniform plane wave.
Show that the function satisfies the wave equation provided that the wave velocity is given by .
Answer the following:
Find the reflection coefficient by perfect dielectric for parallel polarization and hence obtain Brewster angle.
Discuss surface impedance.
Answer the following:
State and prove Poynting theorem.
Discuss Smith chart.
Answer the following:
Find the quality factor of a resonant transmission line section.
Find the voltage step up in quarter wave line.
Instructions:
- The marks are indicated in the right-hand margin.
- There are TEN questions in this paper.
- Attempt any FIVE questions.
Q.1 Solve both questions :
Find the potential distribution due to a long pair of parallel wires of negligible cross-section and having equal and opposite line charge density. Also obtain equipotential surfaces produced by them.
Find the capacitance of two parallel cylindrical conductors having their radii as a and separation between their axes as b.
Q.2 Solve all questions :
State uniqueness theorem and prove it.
Explain conductor properties and obtain boundary conditions.
For a two-dimensional system in which $ r=\sqrt{x^2+y^2} $ determine $ \nabla^2V $ when $ V=\frac{1}{r} $.
Q.3 Solve all questions :
Find the energy density in the magnetic field.
Find the magnetic field inside a solid conductor carrying a direct current, and hence obtain total magnetic flux per unit length within the conductor.
Prove Stokes' theorem.
Q.4 Solve both questions :
Obtain two Maxwell's equations which deviate from steady-state field.
The electric field of electromagnetic wave is given by $ E_x=0=E_z $, $ E_y=A\cos\omega(t-\frac{z}{c}) $. Using Maxwell's equation in free space, find the magnetic vector $ \vec{H} $.
Q.5 Solve both questions :
Find the ratio of $ \vec{E} $ and $ \vec{H} $ in a uniform plane wave.
Discuss the wave propagation in conducting medium and obtain the value of $ \alpha $ and $ \beta $.
Q.6 Solve this question :
Derive the reflection coefficient of perfect dielectric for oblique incidence in the case of parallel polarization. Obtain Brewster angle.
Q.7 Solve both questions :
State Poynting theorem and prove it.
A short vertical transmitting antenna erected on the surface of a perfectly conducting earth produces effective field strength $ E_{\text{eff.}} = E_{\theta \text{eff.}} = 100\sin\theta \frac{\text{mu}}{\text{m}} $ at points at a distance of one mile from the antenna. Compute the Poynting vector and total power radiated.
Q.8 Solve both questions :
Discuss UHF line as circuit element and hence find the input impedance of short-circuited quarter-wave line.
Discuss quarter-wave line as transformer.
Q.9 Solve both questions :
Discuss Smith chart and its uses.
Design a necessary matching unit to join without impedance mismatch the two different sections of transmission line whose impedances are 75 ohm and 50 ohm.
Q.10 Solve this question :
Find the field component of TM wave in parallel plane guide and hence discuss TEM wave.
Instructions:
- The marks are indicated in the right-hand margin.
- There are TEN questions in this paper.
- Attempt any FIVE questions.
Q.1 Solve both questions :
Find the potential distribution due to a long pair of parallel wires of negligible cross-section and having equal and opposite line charge density. Also obtain equipotential surfaces produced by them.
Find the capacitance of two parallel cylindrical conductors having their radii as a and separation between their axes as b.
Q.2 Solve all questions :
State uniqueness theorem and prove it.
Explain conductor properties and obtain boundary conditions.
For a two-dimensional system in which $ r=\sqrt{x^2+y^2} $ determine $ \nabla^2V $ when $ V=\frac{1}{r} $.
Q.3 Solve all questions :
Find the energy density in the magnetic field.
Find the magnetic field inside a solid conductor carrying a direct current, and hence obtain total magnetic flux per unit length within the conductor.
Prove Stokes' theorem.
Q.4 Solve both questions :
Obtain two Maxwell's equations which deviate from steady-state field.
The electric field of electromagnetic wave is given by $ E_x=0=E_z $, $ E_y=A\cos\omega(t-\frac{z}{c}) $. Using Maxwell's equation in free space, find the magnetic vector $ \vec{H} $.
Q.5 Solve both questions :
Find the ratio of $ \vec{E} $ and $ \vec{H} $ in a uniform plane wave.
Discuss the wave propagation in conducting medium and obtain the value of $ \alpha $ and $ \beta $.
Q.6 Solve this question :
Derive the reflection coefficient of perfect dielectric for oblique incidence in the case of parallel polarization. Obtain Brewster angle.
Q.7 Solve both questions :
State Poynting theorem and prove it.
A short vertical transmitting antenna erected on the surface of a perfectly conducting earth produces effective field strength $ E_{\text{eff.}} = E_{\theta \text{eff.}} = 100\sin\theta \frac{\text{mu}}{\text{m}} $ at points at a distance of one mile from the antenna. Compute the Poynting vector and total power radiated.
Q.8 Solve both questions :
Discuss UHF line as circuit element and hence find the input impedance of short-circuited quarter-wave line.
Discuss quarter-wave line as transformer.
Q.9 Solve both questions :
Discuss Smith chart and its uses.
Design a necessary matching unit to join without impedance mismatch the two different sections of transmission line whose impedances are 75 ohm and 50 ohm.
Q.10 Solve this question :
Find the field component of TM wave in parallel plane guide and hence discuss TEM wave.