ELECTROMAGNETIC FIELD THEORY

Choose the correct answer of the following (any seven) :

Attempt any two parts of the following :

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Attempt any two parts of the following :

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Attempt any two parts of the following :

Attempt any two parts 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?

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.

(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.

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 $.

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.

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'.

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.

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?

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?

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?

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.

(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.

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 $.

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.

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'.

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.

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?

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?

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.

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.

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.

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?

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.

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.

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?

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 $?

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.

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.

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.

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.

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?

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.

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.

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?

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 $?

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.

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 ___________

Discuss uniqueness theorem in detail.

Find the capacitance between two spheres whose separation d is very much larger than their radii R.

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.

Find the energy stored in a magnetic field.

Discuss Stokes' law.

Discuss magnetic vector potential.

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.

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.

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 $.

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.

Find the quality factor of a resonant transmission line section.

Discuss quarter wave line as transformer.

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 ___________

Discuss uniqueness theorem in detail.

Find the capacitance between two spheres whose separation d is very much larger than their radii R.

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.

Find the energy stored in a magnetic field.

Discuss Stokes' law.

Discuss magnetic vector potential.

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.

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.

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 $.

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.

Find the quality factor of a resonant transmission line section.

Discuss quarter wave line as transformer.

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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.

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} $.

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.

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} $.

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 $.

Derive the reflection coefficient of perfect dielectric for oblique incidence in the case of parallel polarization. Obtain Brewster angle.

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.

Discuss UHF line as circuit element and hence find the input impedance of short-circuited quarter-wave line.

Discuss quarter-wave line as transformer.

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.

Find the field component of TM wave in parallel plane guide and hence discuss TEM wave.

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.

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} $.

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.

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} $.

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 $.

Derive the reflection coefficient of perfect dielectric for oblique incidence in the case of parallel polarization. Obtain Brewster angle.

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.

Discuss UHF line as circuit element and hence find the input impedance of short-circuited quarter-wave line.

Discuss quarter-wave line as transformer.

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.

Find the field component of TM wave in parallel plane guide and hence discuss TEM wave.