AKSHAYA COLLEGE OF ENGINEERING AND TECHNOLOGY

Dept. of Electrical and Electronics Engineering

EE1302- ELECTROMAGNETIC THEORY

Prepared by: R.Subramanian

QUESTION BANK

UNIT I – VECTOR FUNCTIONS AND CO-ORDINATESYSTEMS

PART A

  1. What are the sources of electro magnetic fields?
  2. Transform a vector A=yax-xay+zaz into cylindrical coordinates.
  3. How the unit vectors are defined in cylindrical co-ordinate systems?
  4. What is the physical significance of the term “divergence of a vector field”?
  5. Define scalar triple product and state its characteristics.
  6. Give the relation between Cartesian and cylindrical co-ordinate systems
  7. Define curl.
  8. What is unit vector? What is its function while representing a vector?
  9. Which are the differential elements in Cartesian co-ordinate system?
  10. What is the physical significance of divergence?
  11. Define Surface Integral.
  12. Sketch a differential volume element in cylindrical co-ordinates resulting from differential changes in three orthogonal co-ordinate directions.
  13. What is volume charge density?
  14. Define Line Integral.
  15. Calculate the total charge enclosed by a circle of 2m sides, centred at the origin and with the edge s parallel to the axes when the electric flux density over the cube is D=10x3/3ax(C/m2)
  16. State stoke’s theorem.
  17. Give practical examples for diverging and curling fields.
  18. State Divergence theorem and mention the significance of the theorem.
  19. A vector field F = (1/ r) arin spherical co-ordinates. Determine F in Cartesian form at a point x=1,y=1 and z= 1.
  20. Given A= 10 ay + 3az and B = 5ax + 4ay , find the projection of A on B
  21. Prove that curl gradΦ = 0.
  22. Verify that the vectors A = 4ax – 2ay + 2az and B = -6ax + 3ay – 3az are parallel to each other.
  23. What are different co-ordinate systems?
  24. The temperature in an auditorium is given by T= x2 + y2 – z. A mosquito located at (1, 1, 2) desires to fly in such a direction that it will get warm as soon as possible. In what direction it must fly?

PART B

  1. (i) Explain the electric field distribution inside and outside a conductor

(ii) Explain the principle of electrostatic shielding.

(iii) Draw the equipotent lines and E lines inside and around a metal sphere.

2. i) State and prove Divergence theorem.

ii) State and prove Stokes theorem.

3. Given A = ax + ay, B = ax + 2az and C = 2 ay + az . Find (A X B) X C and compare

it with A X (B X C). Using the above vectors, find A.B X C and compare it with

A X B.C.

4. Find the value of the constant a, b, c so that the vector E = (x + 2y + az) ax +

(bx – 3y –2) ay + (4x + cy + 2z) az is irrotational.

  1. Using the Divergence theory, evaluate ∫∫ E.ds = 4xz ax – y2 ay + yz az over the cube bounded by x = 0; x = 1; y = 0; y = 1; z = 0; z = 1.
  2. Explain the spherical co-ordinate system?
  3. (i) Use the cylindrical coordinate system to find the area of the curved surface of a

right circular cylinder where r = 20 m, h = 5m and 30º 120 º.

(ii)State and explain Divergence theorem.

8. (i) Derive the stoke’s theorem and give any one application of the theorem in electromagnetic fields

(ii)Obtain the spherical coordinates of 10ax at the point P (x = -3, y = 2, z = 4).

9. Find the charge in the volume

If ρ = 10 x2 y z μC/m3.

10. (i) Determine the constant c such that the vectorF = (x + ay)i + (y + bz)j +

(x + cz)k will be solenoidal.

(ii) Given in cylindrical co-ordinate. For the contour shown

in Fig.Q-32, verify Stoke’s theorm.

Fig.Q-32.

11. Verify Stoke’s theorem for a vector field F = ρ2cos2 Φaρ+zsinΦaz around the

path L defined by 0≤ ρ ≤ 3, 0≤ Φ ≤ 45o and z = 0.

12. i) What are the major sources of Electromagnetic fields (any five)?

ii) What are the positive and negativeeffcts of EM fields on living things?

iii) What are the E and H field limits for public exposure?

iv)Give any one example to reduce the effect of EM field.

13.Verify the divergence theorem for a vector field A = xy2ax+y3ay+y2zaz and

the surface is a cuboid defined by 0<x<1, 0<y<1, 0<z<1.

14. Given that F = x2 y ax- yay. Find ∫ F. dl for the closed path shown in figure

and also verify Stoke’s theorem.

15. i) Given A = 5ax and B = 4ax + t ay; Find ‘t’ such that the angle between A and

Bis 45˚

ii)Using the Divergence theorem, evaluate ∫∫ A.dS = 2xyax+ y2ay + 4yz az over the

cube bounded by x = 0; x = 1; y = 0; y = 1; z = 0; z = 1.

16. i) Determine divergence and curl of the vector A = x2ax+ y2ay + y2az.

ii)Determine the gradient of the scalar field at P (√2, π / 2, 5) defined in

cylindrical co-ordinate system as A= 25ρ sinφ.

17. a) Determine divergence and Curl of the following vector fields.

i) P= x2yz ax + xz az. ii) Q = ρsinΦaρ+ ρ2z aΦ+ zcosΦaz

iii) T = 1 / r2 cosθar + r sinθ cosθaθ+ cosθaΦ

b) Fnd the gradient of the following scalar fields

i) V = e-z sin2x coshy ii) U = ρ2z cos 2Φ iii) W = 10r sin2θ cosΦ.

18. i) Show that the vector field Ais conservative if Apossesses one of the following

two properties. 1) The line integral of the tangential component of A along a

path extending from point P to point Q is independent of the path.2) The line

integral of the tangential component of A along a closed path s zero.

ii) If A = ρcosΦaρ + sinΦaΦ, evaluate ∫ A.dl around the path shown in figure.

Confirm this using Stoke’s theorem.

19. i)A vector field is given by the expression F = (1/ρ ) aρin cylindrical

co-ordinatesand F = (1/r) arin spherical co-ordinates. Determine F in each case

in the Cartesian form at a point (1, 1, 1).

ii) If a scalar potential is given by the expression Φ = xyz, determine the potential

gradient and also prove that vector F = grad Φ is irrotational.

  1. i) Given two points A (2, 3,-1) and B (4, 25˚, 120˚). Find the spherical and

cylindrical co- ordinates of point A andCartesian and cylindrical co-ordinates

of point B.

ii) Find the curl of H at P (2,Π/6, 0), where H= 2ρ cosφ aρ- 4ρsinφ aφ+3az.

UNIT II–ELECTRIC FIELDS

PART A

1. State coulomb’s law.

2. What are the different types of charges?

3. State Gauss Law.

4. Draw the equipotential lines and electric field lines for a parallel plate capacitor.

5. Define dielectric strength. What is the dielectric strength of co-axial cable?

6. Write and explain the coulomb’s law in vector form.

7. Define electric field intensity at a point.

8. What is the electric field around a long transmission line?

9. Sketch the electric field lines due to an isolated point charge Q.

10. A uniform line charge with PL = 5 µc/m lies along the x-axis. Find E at (3,2,1).

11. What are the various types of charge distributions, give an example of each.

12. Define Dielectric strength of a material. Mention the same for air.

  1. Write and explain the coulomb’s law in vector form.
  2. Using Gauss’s law, derive the capacitance of a coaxial cable.
  3. Write down Poisson’s and Laplace’s equation.
  4. Calculate the total charge enclosed by a cube of 2m side, centered at the origin and with the edges parallel to the axes when the electric flux density over the cube is

D = 10x3 / 3 axC / m2.

17. Define dielectric strength of a material and mention the same for air.

18. The electric potential near the origin of the system is V = ax2 + by2 + cz2. find the

electric field at (1, 2, 3).

19. What are symmetrical charge distributions?

20. Define dipole moment.

21. An infinite line charge charged uniformly with a line charge density of 20 nC / m

is located along z- axis. Fine E at (6, 8, 3) m.

22. Define electric potential and potential difference.

23. Using Gauss’s law, derive the capacitance of the co-axial cable.

24. Derive Poissons equation.

25. Two point charges q1 and Q2 are located at (1, 2, 0) and (2, 0, 0) respectively.

Find the relation between Q1 and Q2 such that the total charge at the point

P (-1, 1, 0) will have no x- component).

26. Verify the following potential satisfy Laplace’s equation V= 15 x2 yz – 5 y3z.

27. A spherical capacitor consists of an inner conducting sphere of radius Ri and an

outer conductor with a spherical inner wall of radius Ro. The space in between is

filled with dielectric of permittivity ε. Determine the capacitance.

  1. What is the capacitance of co-axial cable?
  2. Write the continuity equation.
  3. Why Gauss’s law can’t be applied to determine the electric field due to finite line charge?
  4. A uniform surface charge of σ = 2 μC / m2 is situated at z=2 plane. What is the value of flux density at P (1, 1, 1)m.

UNIT II

PART B

1. State and explain the experimental law of coulomb?

2. State and prove Gauss’ law and write about the applications of Gauss law?

3. State and explain Gauss’s law. Derive an expression for the potential at a point outside a

hollow sphere having a uniform charge density

4. (i) A circular disc of radius ‘a’, m is charged uniformly with a charge density of σ C/m2

Find the electric field intensity at a point ‘h’, m from the disc along its axis.

(ii) A circular disc of 10 cm radius is charged uniformly with a total charge of 10-6c.

Find the electric intensity at a point 30 cm away from the disc along the axis.

5. A line charge of uniform density q C / m extends from the point (0, -a) to the point (0, 1)

in the x-y plane. Determine the electric field intensity E at the point (a, 0).

6. Define the electric potential, show that in an electric field, the potentialdifference

between two points a and b along the path, Va – Vb = -

7. What is dipole moment? Obtain expression for the potential and field due to an electric

dipole.Two point charges Q1 = 4nC1, Q = 2nC are kept at (2, 0, 0) and (6, 0, 0).Express

the electric field at (4, -1, 2)

8. Derive the electric field and potential distribution and the capacitance per unit

length of a coaxial cable.

9.Explain in detail the behavior of a dielectric medium in electric field.

10. i)Discuss Electric field in free space, dielectric and in conductor.

ii) Determine the electric field intensity at P ( -0.2,0,-2.3) due to a point charge of

5 nC at (0.2, 0.1, -2.5) in air.

11.(i) Derive the electrostatic boundary conditions at the interface of two deictic media.

(ii) If a conductor replaces the second dielectric, what will be the potential andelectric

field inside and outside the conductor?

12. (i) Derive the expression for scalar potential due to a point charge and a ring charge.

(ii) A total charge of 100 nC is uniformly distributed around a circular ring of 1.0m

radius. Find the potential at a point on the axis 5.0 m above the plane of the ring.

Compare with the result where all charges are at the origin in the form of a point

charge.

13. (i) Derive the expression for energy density in electrostatic fields.

(ii) A capacitor consists of squared two metal plates each 100 cm side placed parallel and 2 mm apart. The space between the plates is filled with a dielectric having a relative permittivity of 3.5. A potential drop of 500 V is maintained between the plates. Calculate i) the capacitance, ii) the charge of capacitor, iii) the electric flux density, iv) the potential gradient

14. A uniformly distributed line charge, 2m long, with a total charge of 4 nC is in

alignment with z axis, the mid point of the line being 2 m above the origin. Find the

electric field E at a point along X axis 2 m away from the origin. Repeat for

concentrated charge of 4 nC on the z axis 2 m from the origin, compare theresults.

15. (i) Define the potential difference and absolute potential. Give the relation between

potential and field intensity.

(ii) Two point charges of +1C each are situated at (1, 0, 0) m and (-1, 0, 0) m. At what

point along Y axis should a charge of -0.5 C be placed in order that the electric

field E = 0 at (0, 1, 0) m?

16. If V = [2 x2y + 20z – 4 / (x2 + y2)] volts, find E and D at P(6,-2.5,3).

17. Derive an expression for capacitance of a spherical capacitor with conducting shells

Ofradius ‘a’ and ‘b’.

18. Obtain an expression for energy stored and and energy density in a capacitor.

19. Conducting spherical shells with radii a = 10 cm and b= 30 cm are maintained at

potential difference of 100 V such that V(r = b) = 0 and V(r = a) = 100 V. Determine

V and E in region between shells.

20. A total charge of 10-8 C is distributed uniformly along a ring of radius 5m. Calculate

the potential on the axis of the ring at a point 5m from the centre of the ring.

21. Two parallel plates with uniform surface charge densities equal and opposite to each

other have an area of 2 m2 and distance of separation of 2.5 mm in free space. A

steady potential of 200 V is applied across the capacitor formed. If a dielectric of

width 1 mm and relative permittivity 2 is inserted into this arrangement what is the

new capacitance formed?

22. i) Derive Poisson’s and Laplace’s equation and explain their significance in field

theory.

ii) Three concentrated charges of 0.25 μC are located at the vertices of an equilateral

triangle of 10 cm side. Find the magnitude and direction of the force one charge

due to the other two charges.

23. A positive charge Q is located at the centre of a spherical conducting shell of inner

radius Ri and outer radius Ro. Determine E and V as function of radial distance R.

24. i) Write a note on dielectrics.

ii) Find the electric field intensity at the point (0, 0, 5)m due to Q1 = 0.35μC at

(0, 4, 0) and q2 =-0.55 μC at (3,0,0)m.

25. The electric flux density is given as D= r/4 ar nC / m2 in free space. Calaculate E

at r = 0.25 m, the total charge within the sphere of r = 0.25m and the total flux

leaving the sphere of r = 0.35m.

26. An infinitely long uniform line charge is located at y=3, z=5. If ρL= 30 nC/m.Find

field intensity E at: i) origin ii) P (0, 6,) and iii) Q (5, 6,1).

UNIT III - MAGNETIC FIELDS

PART A

  1. State Biot–Savart’s law.
  2. Distinguish magnetic scalar potential and magnetic vector potential.
  3. Plane y=0 carries a uniform current of 30 az mA/m. Calculate the manetic field intensity at (1,10,-2) m in rectangular coordinate system.
  4. Plot the variation of H inside and outside a circular conductor with uniform current density.
  5. What is vector A?
  6. State Ampere’s Law.
  7. What is the relation between magnetic field density B and vector potential A?
  8. State the significance of E and H. Give an example of this.
  9. What is magnetic boundary condition?
  10. Draw the magnetic field pattern in and around a solenoid.
  11. What is H due to a long straight current carrying conductor?
  12. Calculate inductance of a ring shaped coil having a mean diameter of 20 cm wound on a wooden core of 2 cm diameter. The winding is uniformly distributed and contains 200 turns.
  13. A conductor located at x=0.5 m , y=0 and 0<z<2.0 m carries a current of 10 A in the az direction. Along the length of the conductor B=2.5 ax T. Find the torque about the x axis.
  14. What do you mean by magnetic moment?
  15. Define mutual inductance.
  16. Plot the variation of H inside and outside a circular conductor with uniform current density.
  17. A long straight wire carries a current I = 1 A. At what distance s the magnetic field H = 1 A/m.
  18. Write the expression for magnetic force when charge particle moves in a magnetic field.
  19. State Ampere’s circuital law.
  20. Write down the magnetic boundary conditions.
  21. Define magnetic moment and magnetic permeability.
  22. Draw the magnetic field pattern inside and outside the circular conductor with uniform current density.
  23. What is the relation between magnetic flux density B and vector potential A?
  24. Compare steady current and steady state current.
  25. What is Lorentz law of force and writethe equation.?
  26. Calculate h at (3,-6,2) due to a current element of length 2 mm located at the origin in free space that carries current 16 mA in +Y direction.
  27. An infinitely long straight conductor with circular cross section of radius ‘b’ carries steady current I. determine the magnetic flux density inside the conductor.
  28. A small circular loop of radius 10 cm is centered at origin and placed on the Z = 0 plane. If the loop carries a current of 1 A along aΦ. Calculate magnetic moment of the loop.
  29. Define magnetic susceptibility.
  30. What do you mean by magnetization?
  31. State the boundary conditions of magnetic media.
  32. State the modified form of expression curl H = ▼x H = J, if the contour does not enclose any current, then how is vector H expressed with scalar magnetic potential.
  33. What is solenoid?
  34. Classify the magnetic materials.

UNIT III

PART B

1. (i) Use Biot – Savart’s law to find magnetic field intensity for finite length of

conductor at a point P on Y – axis.

(ii)A steady current of ‘I’ flows in a conductor bent in the form of a square

loop of side ‘a’. Find the magnetic field intensity at the centre of the

current loop.

iii) Find the magnetic field intensity at the centre of a square of sides equal to 5m

and carrying 10 A current.

  1. (i) when a current carrying wire is placed in an uniform magnetic field,

show that torque acting on it is T=X

(ii) A magnetic circuit comprising a toroid of 5000 turns and an area of 6cm2

and mean radius of 15 cm carries a current of 4A. Find the reluctance

and flux given μr = 1.

3. Calculate B due to a long solenoid and a thin toroid.

4. (i) Derive for force and torque in a magnetic field using motor as an example.

(ii) Find the torque about the y axis for the two conductors of length l, carrying

current in opposite directions, separated by a fixed distance w, in the uniform

magnetic field in x direction.

5. (i) Explain magnetization in magnetic materials and explain how the effect of

magnetization is taken into account in the calculation of B/H.

(ii) Find H in a magnetic material

  1. When µ= 0.000018 H/m and H = 120 A/m.
  2. When B = 300 µT and magnetic susceptibility = 20.

6. a) Derive the magnetic force between two parallel conductors carrying equal

current in the (i) Same direction (ii) opposite direction

b) Two wires carrying currents in the same direction of 5000 A and 10000 A are

placed with their axes 5 cm apart. Calculate the force between them.

7. (i) Find the field intensity at a point due to a straight conductor carrying current I

as shown in Fig.Q-7.

Fig.Q-7

(ii) Find H at the centre of an equilateral triangular loop of side 4 m carrying

current of 5 A.

8. (i) Derive the expression for co-efficient of coupling in terms of mutual and self

inductances.

(ii) An iron ring with a cross sectional area of 3 cm2 and a mean circumference of

15 cm is wound with 250 turns wire carrying a current of 0.3 A. The relative

permeability of the ring is 1500. Calculate the flux established in the ring.

If a saw cut of width 2mm is made in the above ring, find the new value of flux

in the circuit.

9. Develop an expression for magnetic field intensity inside and outside a solid

cylindrical conductor of radius ‘a’, carrying a current I with uniform density.

Sketch the variation of the field intensity.

10. Derive H due to a circular current loop and extend the same to compute H due to a

long solenoid.

11. i)State and prove Ampere’s circuital law.

ii) State and explain Biot- Savart’s law

12. i) Obtain an expression for magnetic vector potential.

ii) Give a brief note on magnetic materials.

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

Ax = (4x + 3y+2z); Ay = (5x + 6y +3z) and Az = (2x + 3y +5z). Determine B at

point P.

14. Derive the boundary conditions between two magnetic media.

15. 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.

16. Write short notes on Magnetic vector potential, Biot- Savart’s law, Lorentz law of

force and Magnetic energy density.

17. An iron ring with a cross sectional area of 8 cm2 and a mean circumference of

120 cm is wound with 480 turns wire carrying a current of 2 A. The relative

permeability of the ring is 1250. Calculate the flux established in the ring.

18. A uniform cylindrical coil of 2000 turns is 60 m long and 5 cm diameter. If the coil

carries a current of 10 mA, find the magnetic flux density at the centre of the coil, on

the axis at one end of the coil and on the axis halfway between centre and one end of

the coil.

19. A circular loop located on x2 + y2 = 9, z=0 carries a direct current of 10 A along aΦ.

Determine H at (0,0,4) and (0,0,-4).

20. A small current loop L1 with magnetic moment 5 azAm2 is located at the origin

while another small loop current L2 with magnetic moment 3 ay Am2 is located at

(4,-3,10). Determine the torque on L2.

21. i) Find the maximum torque on an 85 turns, rectangular coil with dimension

(0.2 x 0.30)m, carrying current of 5A in a field B= 6.5 T

ii) Derive an expression for magnetic vector potential.

22. i) Derive an expression for the inductance of solenoid.