DHANALAKSHMI COLLEGE OF ENGINEERING, CHENNAI

DEPARTMENT OF ELECTRONICS AND COMMUNICATION ENGINNERING

EC6403 ELECTRO MAGNETIC FIELDS

UNIT-I: STATIC ELECTRIC FIELD

PART-A (2 Marks)

1. Convert the given rectangular coordinate A(x=2, y=3, z=1) into the corresponding cylindrical coordinate.

(N-10)

The Cartesian (rectangular) co-ordinates (x, y, z) can be converted into cylindrical coordinates

(r, ,z).

Given: x = 2, y = 3, z = 1

=0.07

2. What is an electric dipole? Write down the potential due to an electric dipole. (N-10), (M-12)

Dipole or electric dipole is defined as two equal and opposite point charges are separated by a

very small distance.

Potential due to an electric dipole

3. State Divergence theorem. (M-11),(N-12), (N-09), (M-09), [M/J – 07], (M-13)

The volume integral of the divergence of a vector field over a volume is equal to the surface

integral of the normal component of this vector over the surface bounding the volume.

4. What is the conservative field? (N-10)

The sign indicates integral over a closed path. Such a field having property of

associated with it is called conservative field or lamellar field.

5. State Coulomb’s Law. (M-12), (N-12), (N-09)

Coulomb’s law states that the force between two very small charged objects separated by a large

distance compared to their size is proportional to the charge on each object and inversely proportional

to the square of the distance between them.

6. What is the physical significance of div D? (M-12)

The Divergence of a vector flux density is electric flux per unit volume leaving a small volume.

This is equal to the volume charge density.

7. Define -Stokes Theorem (N-12) , (M-12)

The line integral of a vector around a closed path is equal to the surface integral of the normal

component of its curl over any closed surface.

8. What are the features of Coulomb’s law? (M-11)

Coulomb’s law states that there exists a force between charged bodies and it is:

1) Proportional to the product of the two charges,

2) Inversely proportional to the distance between the charges.

3) The force also depends on the medium in which the charges are closed.

Mathematically, coulomb’s law is given by,

where k=constant of proportionality

, are two point charges (C)

r = distance between the charges (m)

9. A vector field is given by the expression in spherical coordinates. Determine F in Cartesian

form at a point, x = 1, y = 1 & z = 1 unit. (M-09)

Solution:

In general (in spherical coordinates)

In this example, the field varies only as a function of the radial distance of the point from the origin.

The point assumed to lie on a sphere with radius R. The components and are, therefore non-

existent.

At the point x = 1, y = 1, z = 1

Hence, at (1, 1, 1),

10. Define ─ Electric Dipole (N-10)

Dipole or electric dipole is defined as two equal and opposite point charges are separated by a

very small distance.

11. What is electric flux density?

Electric flux density or displacement density is defined as the electric flux per unit area.

12. Write the relation between potential and electric field.

The relation between potential and electric field is given by:

13. What is electric field intensity?

Electric field intensity is defined as the electric force per unit positive charge. It is denoted by E.

14. State Gauss’s law.

Gauss’s law state that an electric flux passing through any closed surface is equal to the total

charge enclosed by that surface.

15. Define ─ Electric Potential and Electric Field (M-13)

Electric potential: potential at any point as the work done in moving a unit position charge from infinity to

that point in an electric field.

Electric field: Electric field intensity is defined as the electric force per unit positive charge. It is denoted

by E.

Part – B (16 Marks) 1. (i) A point charge located at (1, −1, −3) m experiences a force

. Due to point charge at (3, −3, −2) m. Find the charge . (10)

(ii) Given that in spherical coordinates, evaluate both sides of divergence theorem

for volume enclosed by r = 4 m and . (6) (M-12)

2. (i) Derive the expression for potential due an electric dipole at any point P. Also find electric field intensity at

the same point. (10)

(ii) Two point charges, 1.5nC at (0, 0, 0.1) and −1.5nC at (0, 0, −0.1), are in free space. Treat

two charges as a dipole at the origin and find potential at P (0.3, 0, 0.4). (6) (M-12)

3. (i) Find the electric field intensity at a point P located at(0, 0, h)m due to charge of surface charge density

uniformly distributed over the circular disc . (10)

(ii) Determine the divergence and curl of the given field at

(1, 1, −0.2) and hence state the nature of the field. (6) (N-12)

4. (i) Point charges Q and –Q are located at (0, 0, d⁄2) and (0, 0, d⁄2). Show that the potential at a

point ) is inversely proportional to nothing that . (8)

(ii) Given a field , find the potential difference between

A (−7, 2, 1) and B (4, 1, 2). (8) (N-12)

5. (i) Assume a straight line charge extending along the z axis in a cylindrical coordinate system from

.Determine the electric field intensity at every point resulting from a uniform line charge

density . (8)

(ii) Consider an infinite uniform line charge of parallel to z axis at x = 1, y = 6. Find the electric field

intensity at the point P (0, 0, 5) in free space. (8) (N-11)

6. (i) The flux density within the cylindrical volume bounded by r = 2m, z = 0 and z = 5m is given by

.What is the total outward flux crossing the surface of the cylinder.

(8)

(ii) State and prove Gauss’s law for the electric field. Also find the differential form of Gauss law.

(8) (N-11)

7. (i) Given point P (−1, 4, 3) and vector express P and A in Cylindrical and Spherical

coordinates. Evaluate A at P in the Cartesian and Spherical systems. (8)

(ii) Determine D at (2, 0, 3) if there is a point charge at (3, 0, 0) and a line charge along

the y-axis. (8) (M-11)

8. (i) Determine stokes theorem. (8)

(ii) Show that in Cartesian coordinates for any vector A . (8) (M-11)

9. (i) Given the two points and .Find the

spherical Co-ordinates of A and Cartesian Co-ordinates of B. (8)

(ii) Find curl , if (8) (M-10)

10. (i) A circular disc of radius ‘a’m charged uniformly with a charge of . Find the electric field intensity

at a point ‘h’ meter from the disc along its axis. (8)

(ii) If volts, find and at P(6, ─2.5, 3). (8) (M-10)

UNIT- II: CONDUCTORS AND DIELECTRICS

PART – A (2 Marks)

1. State the Laplace’s equations in Cartesian, cylindrical and spherical coordinates. (N-09)

The laplace’s equations in

1) Cartesian form:

2) Cylindrical form:

3) Spherical form:

2. State ohms law at a point. (M-13)

Point form of ohm’s law states that the field strength within a conductor is proportional to the current density.

Where is conductivity of the material

3. Write the Poisson’s and Laplace’s equation. (M-09)

1) Passion Equation

Where

= Volume charge density

ε = Permittivity of the medium

= Laplacian operator

2) Laplace equation

4. Classify the magnetic materials. (N-08)

Magnetic materials can be classified into three groups according to their behavior.

There are

1) Diamagnetic

2) Paramagnetic

3) Ferromagnetic

5. Define the term capacitance between two conductors or capacitor. (N-08)

The capacitance of two conducting planes is defined as the ratio of magnetic of charge on either of the

conductor to the potential difference between conductors. It is given by

The unit of capacitance is Coulombs/volt or Farad.

6. Find the capacitance of a parallel plate capacitor having stored energy of with a voltage between the

plates 0f 5V. (N-08)

Solution:

Given:

Energy stored in a capacitor

Voltage between the plate V = 5V

Capacitance of a parallel plate

7. What are the boundary conditions between two dielectric media? (N-07)

1) The tangential component of electric field E is continuous at the surface. That is E is the same

just outside the surface as it is just inside the surface.

2) The normal component of electric flux density is continuous if there is no surface charge

density. Otherwise D is discontinuous by an amount equal to the surface charge density.

8. What is polarization? (N-07)

Polarization of a uniform plane wave refers to the time varying natural of the electric field vector at some

fixed point in space

or

Polarization is defined as dipole moment per unit volume.

9. Determine the capacitance of the parallel plate capacitance composed of tin foil sheets, 25 cm square for plates separated through a glass dielectric 0.5 cm thick with relative permittivity 6. (N-07), (M-13) Solution:

Substitute in the above function

10. What is Displacement current density? (N-09)

Displacement current is nothing but the current flow through the capacitor.

Displacement current density is given by

11. Write the continuity equation. (N-08)

The continuity equation is

(Integral form)

(Differential form)

12. What is mean by Circular Polarization?

If x and y component of electric field and have equal amplitude and phase difference, the

locus of the resultant electric field E is a circle and the wave is said to be circularly polarized.

13. Differentiate conductor and Dielectric.

Conductor:

If the valance band merges smoothly into a conduction band, then additional kinetic energy may be given to

the valance electrons by an external field, resulting in an electro flow. The solid is called a metallic

conductor.

Dielectric:

If the forbidden energy gap between valence band and conduction band of a material is high, it requires

large applied energy to conduct. The material is called a dielectric.

PART – B (16 Marks)

1. Derive the boundary conditions of the normal and tangential components of electric field at the Inter face of

two media with different dielectrics. (16) (N-08) , (M-14), (N-14)

2. Derive an expression for the energy stored and energy density in a capacitor. (N-14), (M-09)

3. Drive an expression for energy stored and energy density in an Electrostatic field (16) (N-14)

4. Derive an expression for the capacitance of two wire transmission line. (8)

5. Derive an expression for capacitance of co-axial cable. (8) (M-09), (N-06)

6. Find the expression for the cylindrical capacitance using Laplace equation. (16) (N-14)

7. Derive the boundary conditions of the normal and tangential components of magnetic field at the inter face

of two media with different dielectrics. (16) (N-14)

8. Derive the expression for co-efficient of coupling. (8)

9. Prove Laplace’s and Poisson’s equations. Also using the concept of magnetic vector potential, derive Biot

Savart’s law and amperes law? (M-10), (M-12)

10. Derive the expression for co-efficient of coupling. (8) Also using the concept of magnetic vector

potential, derive Biot Savart’s law and amperes law? (M-10), (M-12)

11. Derive an expression for the capacitance of a spherical capacitor with conducting shells of radius a and b.

(M-09), (N-06)

12. Derive the expression for the continuity equation of current in differential form and also derive the

expression for inductance of a solenoid with N turns and l metre length carrying a current of I amperes.

(N-11)

13. Derive the expression for the inductance of a toroidal coil (solenoid) with N turns, carrying current I and the

radius of the toroid R. Also considering a toroidal coil derive an expression for energy density. (16)

(N-12), (M-12), (M-09), (N-10)

14. A solenoid has an inductance of 20 mH If the length of the solenoid is increased by two times and the

radius is decreased to half of its original value, find the new inductance (M-09)

15. Derive the expression for potential energy stored in the system of n-point charges. (16) (D - 09)

16. Derive an expression for Poisson and Laplace equations and also Derive an expression for the

inductance of solenoid (M-10), (N-10), (M-14)

17. Derive the boundary conditions at an interface between two magnetic Medias. (M-10), (M-09), (N-06)

18. A small loop wire lays a distance z above the center of a large loop. The planes of the two loops are

parallel, and perpendicular to the common axis. Suppose current I flows in the big loop. Find the flux

through the little loop. Find the mutual inductance. (16) (M-14)

19. Solve the Laplace equation for the potential field in the homogenous region between the two

concentric conducting spheres with radius a and b and v=0 at r=b and v=vo at r=a; Find the capacitance

between the two concentric spheres. (8) (M-11)

20. A metallic sphere of radius 10cm has a surface charge density of 10nc. Calculate the energy stored in the

system. And also state and explain the electric boundary conditions between two dielectrics with

permittivity’s e1 and e2 (16) (N-11)

21. Derive the expression for the energy of a point charge distribution. Three point charges -1nc, 4nc, 3nc are

located at (0, 0, and 0) (0, 0, and 1) (1, 0, and 0) respectively, Find the energy in the system. (M-10)

22. Find the permeability of the material whose magnetic susceptibility is 49 also find, if the inner and outer

conductors of a co axial cable are having radii a and b respectively If the inner conductor is carrying

current I and outer conductor is carrying the return current I in the opposite direction. Derive the

expressions for the internal and external inductance (16) (M-11)

UNIT- III: STATIC MAGNETIC FIELD

PART –A (2 Marks)

1. Write the Lorentz's force equation for a moving charge. (N-10) ,(N-12)

The force on a moving charge particle arising from combined electric and magnetic fields is

obtained easily by superposition

This equation is known as the Lorentz force equation.

2. Find the magnetic field intensity at a point P(0.01, 0, 0)m, if the current a coaxial cable is 6A, which is along

the z axis and a = 3mm , b = 9mm & c = 11mm. (N-10)

Soln:

Given point is P (0.01, 0, 0) m =

a = 3mm; b = 9mm & c = 11mm

For the condition,

3. State Ampere's circuital law. (N-12),(N-09),(N-09)

Ampere’s circuital law states that the line integral of magnetic field intensity around a closed path is

exactly equal to the direct current enclosed by that path.

The mathematical representation is

4. A loop with magnetic dipole moment23

108 Amaz

, lies in a uniform magnetic field

2/4.02.0 mWbaaB

zx

. Calculate the torque. (M-11)

Soln:

Given magnetic dipole moment

Magnetic field

Torque is

T

5. List the applications of Ampere's circuital law. (N-10),(M-11)

Ampere's circuital law is used

1) to determine magnetic field due to a straight conductor carrying current.

2) to determine magnetic field due to a solenoid carrying current. and

3) to determine magnetic field due to a current in a toroid.

6. Distinguish between diamagnetic, paramagnetic and ferromagnetic materials. (M-12)

Diamagnetic material

1) The magnetic moment, intensity of magnetization and magnetic susceptibility are all negative while

magnetic permeability has value less than1

2) Repelled by a strong magnet

3) The magnetic susceptibility is independent of temperature

Paramagnetic material

1) The magnetic moment, intensity of magnetization and magnetic susceptibility are all positive while

magnetic permeability has value slightly greater than1

2) Attracted by a strong magnet

3) The magnetic susceptibility decreases with rise of temperature

Ferromagnetic material

1) The magnetic moment, intensity of magnetization and magnetic susceptibility are all positive and quite

large and magnetic permeability is of the order of hundreds and thousands.

2) The magnetic susceptibility decreases with rise of temperature

7. Define ─Magnetic Flux Density (N-12)

The total magnetic lines of force i.e. magnetic flux crossing a unit area in a plane at right angles to the

direction of flux is called magnetic flux density. It is denoted as Unit Wb/m2.

8. State Biot-Savart law. (N-09),(M-09),(M-13)

The BiotSavart law states that,

The magnetic field intensity produced at a point p due to a differential current element IdL is

1) Proportional to the product of the current I and differential length dL

2) The sine of the angle between the element and the line joining point p to the element.

3) And inversely proportional to the square of the distance R between point p and the element

9. What is solenoid? (N-08)

Solenoid is a cylindrically shaped coil consisting of a large number of closely spaced turns of insulated wire

wound usually on a non magnetic frame.

10. Mention the importance of Lorentz’s force equation. [M/J – 07]

The Lorentz’s force equation find its importance in

1) Determining electron orbits in the magnetron

2) Proton paths in the cyclotron

3) Plasma characteristics in a magnetohydrodynamic (MHD) generator

4) Charged particle motion in combined electric and magnetic fields.

11. A long conductor with current 5A is in coincident with positive z direction. If Find the

force per unit length. [A/M – 08]

Soln:

Given

Current I=5A

Length L=1m

Magnetic field

Force is

Part – B (16 Marks)

1. (i) Find the magnetic field intensity due to a finite wire carrying a current I and hence deduce an

expression for magnetic field intensity at the centre of a square loop. (8)

(ii) Derive the magnetic field intensity in the different regions of co-axial cable by applying

Ampere’s circuital law. (8) (M-12)

2. (i) Obtain the expression for scalar and vector magnetic potential. (8)

(ii) The vector magnetic potential in a certain region of

free space.

1. Show that = 0. (3)

2. Find the magnetic flux density and the magnetic field intensity at P (2, −1, 3).

(5) (M-12)

3. (i) Derive an expression for magnetic field intensity due to a linear conductor of infinite length carrying

current I at a distance point P. Assume R to be the distance between conductor and point P. Use Biot

Savart’s law. (8)

(ii) Derive an expression for magnetic field intensity on the axis of a circular loop of radius ‘a’ carrying

current I. (8)(N-12)

4. (i) At a point P(x, y, z) the components of vector magnetic potential are given as ,

and . Determine the magnetic flux density at the point P.

(8)

(ii) Given the magnetic flux density , find the total magnetic flux

crossing the strip defined by . (8)(N-12)

5. (i) Find the force on a wire carrying a current of 2 mA placed in the xy plane bounded by x = 1,

x = 3, y = 0 and y = 2 as shown in figure. The magnetic field is due to a long conductor, located

in y-axis, carrying a current of 15A as shown. (8)

(ii) A differential current element is located at the origin in free space. Obtain the expression

for vector magnetic potential due to the current element and hence find the magnetic field

intensity at the point . (8)(N-11)

6. (i) Find the maximum torque on an 85 turns, rectangular coil with dimension (0.2 × 0.3)m,

carrying a current of 5 Amps in a field B = 6.5T. (8)

(ii) A circular loop located on carries a direct current of 7A along . Find the

magnetic field intensity at (0, 0, −5). (4)

(iii) Using Ampere’s circuital law determine the magnetic field intensity due to a infinite long wire

carrying a current I. (4) (N-11)

7. Two equal point charges are placed on a line at distance ‘a’ apart, This line joining the charges is parallel to

the surface of an infinite conducting region which is at zero potential. The specific line is at a distance

from the surface of the conducting region. Show that the force between the charges is .

What happens to the force when sign of one the charges is reversed. (16) (M-11)

8. Conducting spherical shells with radii a = 8cm and b = 20cm are maintained at a potential difference of 100

V such that V(r = b) = 0 and V= (r = a) =70V. Determine V and E in the region between the shells. If

in the region determine the total charge induced on shells and the capacitance of the capacitor. (16) (M-11)

9. (i) State and explain Ampere’s circuital law. (8)

(ii) Find an expression for at any point due to a long, straight conductor carrying I amperes.

(8) (M-10)

10. A circular loop located on carries a direct current of 10A along . Determine H at (0,

0, 4) and (0, 0, ─4). (16) (N-09)

UNIT – IV: MAGNETIC FORCES AND MATERIALS

PART A (2 Marks) 1. Write down the expression for the torque experienced by a current carrying loop situated in a magnetic

field. (M-12)

The expression for the torque experienced by a current carrying loop in a magnetic field is given by

where I is the current

S is the area of current carrying loop B is the magnetic field

2. What is meant by magnetic moment? (N-12)(N-09)(M-09)

The Magnetic moment of a current loop is defined as the product of current through the loop and the area of

the loop, directed normal to the current loop.

3. What is meant by magnetic field intensity? (M-12)

Magnetic field intensity at any point in the magnetic field is defined as the force experienced by a unit north

pole of one Weber strength, when placed at that point. Unit: N/Wb (or) AT /m. It is denoted as .

4. What is Torque? (N-10)

The Moment of a force or torque about a specified point is defined as the vector product of the moment arm

and the force . It is measured in Nm.

5. Define mutual inductance. (M-09), (N-08)

The mutual inductance between two coils is defined as the ratio of induced magnetic flux linkage in one

coil to the current through in other coil.

where

= is number of turns in coil 2

=Magnetic flux links in coil 2

= Current through coil 1

6. Write the force on a current element.

The force on a current element Idl is given by

dF = I x B dl

= BI dl sinθ Newton

7. Define – magnetic vector potential It is defined as that quantity whose curl gives the magnetic flux density.

where A is the magnetic vector potential

drr

JA

V4

Web / m

8. Define – self inductance The self induction of a coil is defined as the ratio of total magnetic flux linkage in the circuit to the current through the coil.

where

is magnetic flux N is number of turns of coil i is the current.

9. Define – mutual inductance The mutual inductance between two coils is defined as the ratio of induced magnetic flux linkage in one coil to the current through in other coil.

where

N2 is number of turns in coil 2

12 is magnetic flux links in coil 2 i1 is the current through coil 1

10. What will be effective inductance, if two inductors are connected in (a) series and(b) parallel?

(a) For series L = L1 + L2 M2 + sign for aiding

(b) For Parallel L = MLL

MLL

221

2

21

- sign for opposition 11. Write the expression for inductance of a solenoid.

l

ANL

o

2

where N is number of turns A is area of cross-section l is length of solenoid is free space permeability

12. Write the expression for inductance of a toroid.

R

ANL

o

2

2

= R

rNo

2

22

where

N is number of turns r is radius of the coil R is radius of toroid is free space permeability

13. Write the expression for inductance per unit length of a co-axial transmission line.

L = a

bo

ln2 H/m.

Where a is the radius of inner conductor b is the radius of outer conductor. 14. What is the mutual inductance of two inductively tightly coupled coils with self inductance of 25mH and

100mH. L1 = 25 mH

L2 = 100 mH

M = K 21LL

= 10025 X = 50 mH

PART – B (16 Marks)

1. Write down the Poisson’s and Laplace’s equations. State their significance in electrostatic problems.

(4)

2. Two parallel conducting plates are separated by distance ‘d’ apart and filled with dielectric medium having

as relative permittivity. Using Laplace’s equation, derive an expression for capacitance per unit length

of parallel plate capacitor, if it is connected to a DC source supplying ‘V’ volts. (12)

3. Derive the expression for inductance of a toroidal coil carrying current. (8)

4. A solenoid is 50 cm long, 2 cm in diameter and contains 1500 turns. The cylindrical core has a diameter of

2 cm and relative permeability of 75. This coil is co-axial with second solenoid, also 50 cm long, but 3 cm

diameter and 1200 turns. Calculate L for the inner solenoid; and L for the outer solenoid. (8)

5. State and derive Poisson’s equation and Laplace equation . (16)

6. Obtain the expression for energy stored in magnetic field and also derive an expression for magnetic

energy density. (16)

7. Derive an expression for the capacitance of a parallel plate capacitor with two dielectric media. (8)

8. A parallel plate capacitor with a separation of 1 cm has 29 kV applied, when air was the dielectric used.

Assume that the dielectric strength of air as 30kV/cm. A thin piece of glass with with a dielectric

strength of 290 kV/cm with thickness 0.2 cm is inserted. Find whether glass or air will break.

(8)

9. Derive an expression for inductance of a solenoid with N turns and meter length carrying current I

amperes. (8)

10. Determine whether or not the following potential fields satisfy the Laplace’s equation. (8)

11. Considering a toroidal coil, derive an expression for energy density. (8)

12. Derive the boundary conditions at an interface between two magnetic medias. (8)

13. A parallel plate capacitor has an area of 0.8m2 , separation of 0.1mm with a dielectric for which

and field of 106 V/m. Calculate C and V. (8)

14. A solenoid with radius of 2 cms is wound with 20 turns per cm and carries 10mA. Find H at the centre of the

solenoid if its length is 10cm. If all the turns of the solenoid were compressed to make a ring of radius 2

cms, what would be H at the centre of the ring? (8)

15. Find the permeability of the material whose magnetic susceptibility is 49. (4)

16. The inner and outer conductors of a co-axial cable are having radii ‘a’ and ‘b’ respectively. I f the inner

conductor is carrying current ‘I’ and outer conductor is carrying the return current ‘I’ in the opposite direction.

Derive the expressions for the internal inductance and the external inductance. (12)

17. Solve the Laplace’s equation for the potential field in the homogeneous region between the two concentric

conducting spheres with radius ‘a’ and ‘b’ where b>a, V=0 at r = b, and V=V0 at r = a. Find the capacitance

between the two concentric spheres. (8)

18. Calculate the inductance of a solenoid of 200 turns wound tightly on a cylindrical tube of 6 cm diameter.

The length of the tube is 60 cm and the solenoid is in air. (8)

UNIT V –TIME VARYING FIELD AND MAXWELL’S EQUATIONS

PART – A (2 Marks)

1. Write down the Maxwell’s equations in integralphasor form. (M-11)

SS

dsEjdsDjJdlH .)()(.

SS

dsHjBdsjdlE ..

dvdsD

S

.

0.

S

dsB

where E- electric fieldintensity , D-electric flux density, B-magnetic flux density, H-magnetic field intensity, µ-permeability of medium, σ-conductivity of medium , Ɛ -permittivity of medium, J-current density and ρ-resistivity of medium

2. Write down the Maxwell’s equation in integral form. (M–13)

From Ampere’s Law

dst

DdlH

S

.

From Faraday’s Law

dst

BdlE

S

.

From Electric Gauss’s Law

0.

s

dsD

From Magnetic Gauss’s Law

0.

s

dsB

where D-electric flux density , B-magnetic flux density, E- electric field intensity, H-magnetic field intensity

3. Explain the significance of displacement current. Write the Maxwell’s equation in which it is used.

(N-12)

The displacement current iDthrough a specified surface is obtained by integration of the normal component of JD over the surface.

iD = S

DdsJ .

= dst

D

S

.

iD = dst

E

S

.

This is a current which directly passes through the capacitor. Maxwell’s equation

DC

JJH

= t

EE ( Differential form )

dst

DJdlH

SC

)(. ( Integral form )

whereE- electric field intensity, D-electric flux density, B-magnetic flux density, H-magnetic field intensity, µ-permeability of medium, σ-conductivity of medium, Ɛ -permittivity of medium, JC- conduction current density, ρ-resistivity of medium, JD-displacement current density

4. Explain why 0.B (N-12)

0.B states that there is no magnetic charges. The net magnetic flux emerging through any closed surface is zero.

5. Write down the Maxwell’s equation in integral form. (M–12)

From Ampere’s Law

dst

DJdlH

S

.

From Faraday’s Law

dst

BdlE

S

.

From Electric Gauss’s Law

vs

dvdsD .

From Magnetic Gauss’s Law

0.

s

dsB

where E- electric field intensity ,

D-electric flux density, B-magnetic flux density, H-magnetic field intensity, J-current density, ρ-resistivity of medium.

6. Write the fundamental postulate for electromagnetic induction and explain how its leads to Faraday’s Law.

(N–11) A changing magnetic flux (Φ) through a closed loop, produces anemf or voltage at the

terminals as given by

dt

dv

where the voltage is the integral of the electric field E around the loop. For uniform magnetic field Φ = B.A where B is the magnetic flux density and A is the area of the loop.

dst

BdlEv .

This is Faraday’s law. It states that the line integral of the electric field around a stationary loop equals the surface integral of the time rate of change of the magnetic flux density B integrated over the loop area.

7. Write down the Maxwell’s equation in point form. (M–10) From Ampere’s Law

t

DH

From Faraday’s Law

t

BE

From Electric Gauss’s Law 0.D

From Magnetic Gauss’s Law 0.B

whereE- electric field intensity , D-electric flux density, B-magnetic flux density, H-magnetic field intensity.

8. What is meant by Displacement current density? (M– 09)

Displacement current is the current flowing through the Capacitor.

9. Write down the general, integral and point form of Faraday’s law. (N-10)

emfdt

dv ( General )

dst

BdlE . ( Integral )

t

BE ( Point form )

whereE- electric field intensity , B-magnetic flux density, H-magnetic flux intensity, Φ-magnetic flux

10. Write down the general, integral and point form of Faraday’s law in phasor form. [A/M – 09]

EjDjJH )(

HjBjE

D.

0.B whereE- electric field intensity ,

D-electric flux density, B-magnetic flux density, H-magnetic flux intensity, σ-conductivity of medium , Ɛ -permittivity of medium, J-current density, ρ-charge density ω- angular frequency

11. Distinguish between transformer emf and motional emf. (N-09) The emf induced in a stationary conductor due to the change in flux linked with it, is called

transformer emf or static induced emf.

emf = - dst

B. eg. Transformer.

12. Write down the Maxwell’s equations in point phasor forms. (M–10)

EjDjJH )(

HjBjE

D.

0.B whereE- electric field intensity, D-electric flux density, B-magnetic flux density , H-magnetic field intensity, µ-permeability of medium, σ-conductivity of medium , Ɛ -permittivity of medium, J-current density,

ω- angular frequency, ρ-resistivity of medium

13. Explain why .0E (N-11)

In a region in which there is no time changing magnetic flux, the voltage around the loop would be zero. By Maxwell’s equation

t

BE =0

where E-electric field intensity , B-magnetic flux density

14. Explain why 0. D (M-09)

In a free space there is no charge enclosed by medium . The volume charge densityis zero. By Maxwell’s equation

0.v

D

where D-magnetic flux density,

v-volume charge density

15. State Ampere’s circuital law. Must the path of integration be circular? (M-11) The integral of the tangential component of the magnetic field strength around a closed path is equal to the current enclosed by the path.

IdlH .

The path of integration must be enclosed one. It must be any shape and it need not be circular alone.

PART- B (16 Marks)

1. State Maxwell’s equations and obtain them in differential form. Also derive them for harmonically varying

field. (16) (N-13)

2. State Maxwell’s equations and obtain them in integral and differential form.

(16) (N-13)

3. State and prove pointing theorem. (8) (N-12)

4. Derive the expression for total power flow in coaxial cable. (8) (N-12)

5. State and prove pointing theorem. Write the expression for instantaneous ,

average and complex poynting vector. (16) (N-13)

6. Write the inconsistency of Ampere’s law. Is it possible to construct a generator

of EMF which is constant and does not vary with time by using

EM induction principle? Explain. (16) (M-13)

7. Derive and explain the Maxwell’s equations in point and integral form using

Ampere’s circuital law and Faraday’s law. (16) (M-12)

8. Compare the field theory and circuit theory. (8) (M-12)

9. The conduction current flowing through a wire with conductivity σ=3x107 s/m and the relative permeability

εr=1 is given by Ic=3sin ωt (mA). If ω=108 rad/sec, find the displacement current.

(8) (M-13)

10. An electric field in a medium which is source free is given by E=1.5cos(108 t-βz)ax V/m. Find magnetic flux

density,magnetic field intensity and electric flux density . Assume εr=1 , μr = 1, σ = 0.

(8) (M-13)