emf UNIT I[1]

GNANAMANI COLLEGE OF ENGG / ECE / EC1253 / EMF

EC 1253 – ELECTROMAGNETIC FIELDS(FOR IV SEM –ECE STUDENTS)

PREPARED BY

Mr.M.SIVAPRAKASAN LECT/ECE

DEPARTMENT OF ECE

GNANAMANI COLLEGE OF ENGG,PACHAL

GNANAMANI COLLEGE OF ENGG / ECE / EC 1253 / EMF

UNIT I

STATIC ELECTRIC FIELD

TWO MARKS

1. What is a Scalar quantity?A Quantity which has magnitude only is called Scalar quantity. It is represented

by length. EG: Temperature, Mass, Volume and Energy.

2. What is a Vector quantity?A Quantity which has both magnitude and direction is called Vector quantity. It is

graphically represented by a line with an arrow to show magnitude and direction.EG: Force, Velocity, and Acceleration

3. Define Unit Vector?A Vector which has magnitude unity and defining the same direction as given

vector.

4. Give the properties of Vectors.1) Vector addition obeys commutative law A +Β = B +A

2) Vector addition obeys associative law A + (B +C) = (A + B) + C 3) - A is also a vector. It has same magnitude; its direction is 1800 away from Direction of A. A – B = A + (- B) 5. Define Scalar or Dot Product.

A.B = ABcos Φ 0≤ Φ ≤ Π where A = A and B = B and Φ angle between two vectors. It is denoted as A.B It is the product of magnitudes of A and B and the cosine of the angle between them.

6. Define Cross or Vector product.It is denoted as A ҳ B.It is a vector whose magnitude is equal to the product of

magnitudes of two vectors multiplied by the sine angle between them and direction perpendicular to plane containing A and B. A ҳ B = AB sin Φ 0 ≤ Φ ≤ Π Where A = A , B = B and Φ is the angle between two vectors.

7. Write the BAC- CAB rule A ҳ (B ҳ C) = B(A.C) – C(A.B).

8. Define Coordinate system and give its types. A system in which a vector can be described by its length, direction ,projections,angles or components is Coordinate system. There are three types in coordinate system. a) rectangular coordinate system: x, y, z

PREPARED BY : Mr.M.SIVAPRAKASAN , LECT/ECE 1

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b) Cylindrical coordinate system: r, Φ, z c) Spherical coordinate system: r, Ө, Φ

9. Give the conversion of cylindrical to Cartesian and Cartesian to cylindrical. cylindrical to Cartesian Cartesian to cylindrical. Given (r, Φ, z) Given (x,y,z) x = r cos Φ r = √x2 + y2

y = r sin Φ Φ= tan-1(y/x) z = z z = z

10. Give the conversion of Cartesian to spherical. Given (x , y , z ) r = √x2 + y2 + z2 r ≥ 0 Ө = cos-1(z / r) 0 ≤ Ө ≤ Π Φ = tan-1(y/x) 0 ≤ Φ ≤ 2 Π

11. Give the conversion of spherical to Cartesian Given (r, Ө, Φ) x = r sin Ө cos Φ y = r sin Ө sin Φ z = r cos Ө12. Sketch a differential volume element in cylindrical coordinate resulting from differential changes in 3 orthogonal [NOV 2002]

13. Define Gradient. It is the space rate of change of a scalar field at a given time. When the operator

Operates on a scalar function the resultant is a vector called gradiantGrad V = ∂Vax/∂x +∂Vay/∂y+ ∂Vaz/∂z

14. Define divergenceIt is the spatial derivative of a vector field. The dot product of Del and any vector

is called divergence.D = ∂Dx/∂x +∂Dy/∂y+ ∂Dz/∂z

15. Define curlThe cross product of del and any vector is called curl

ax ay az


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X H = ∂/∂x ∂/∂y ∂/∂z Hx Hy Hz

16. Define divergence theorem. [NOV 2003] [APRIL 2005]The volume integral of the divergence of a vector field equal the total out ward

flux of vector through the surface that bounds the volume∫ .A dv = ∫∫ .A ds

17. Define stokes theoremThe surface integral of the curl of a vector field over an open surface is equal to

closed line integral of the vector along the contour bounding the surface∫ X A ds = ∫∫ .A dl

18. Define the term electrostaticsIt is the study of the effect of electric charges which are static or rest

19. What is static electric field?The electric field produced by static electric charge is time invariant i.e. it does

not vary with time, so called as static electric field

20. State coulomb’s lawIt states that force between two charges Φ1 and Φ2 is 1. directly proportional to the product of charges2. inversely proportional to square of the distance between them(R2)

F = K Q1 Q2 aR

R2

Where K=1/4Πξ0

Force acts along the line joining the two point charges, and aR is the unit vector along the line joining two charges

21. What are the types of charge distributors?There are 4 types of charge distributors, namely

a) Line chargeb) Point chargec) Surface charged) Volume charge

22. Define point chargePoint charge is one whose maximum dimension is very small in comparison with

any other length

23. Define line charge PREPARED BY : Mr.M.SIVAPRAKASAN , LECT/ECE 3

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If the charge is uniformly distributed along a line, it is called line charge. The line may be finite or infinite

24. Define surface chargeIf the charge is distributed uniformly over a two dimensional surface then it is

called surface charge

25. Define volume chargeIf the charge is distributed uniformly over a volume then it is called as a volume

charge

26. Define line charge densityIt is denoted as L. it is the ratio of total charge in coulomb to total length in

meters

27. Define surface charge densityIt is denoted as is defined as charge per unit area

ρS = total charge/total area in C/m2

28. Define volume charge densityIt is denoted as v and is defined as charge per unit volume ρV = total charge/total volume in C/m3

29. Define electric field intensityIt is defined as the force exerted per unit charge. It is a vector quantity

30. Give the formula for electric field intensity at a point due to ‘n’ number of point charges.

E = 1/4Пξ0

Where E-electric field intensity V/mQi-charges due to n number of chargesRi-radius of P

31. Write the formula for electric field intensity due to line charge.


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E = ∫ ρL.dl ậR

4Пξ0R2

E = F Q

ρL= Q = total charge in coulomb l total length in meters


Where E- Electric field intensity due to line chargeL- Line charge densityR- Distance between the point and small element line charge

32. Give the electric field intensity due to surface charge

Where E- Electric field intensity due to surface chargeρS- surface charge densityR- Distance between point and small element of surface charge

33. Write the electric field intensity due to volume charge

Where E- Electric field intensity due to volume chargeρV = volume charge densityR- Distance between point and small element of volume charge

34. Write the electric field intensity due to finite line charge

Where E- Electric field intensity due to finite line charge

ρl - line charge density r- Distance between origin and point P

35. Give the electric field intensity due to infinite line charge.

Where ρl - linear charge density r - distance between line charge and point


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ρL 1E = ---------- x ------------ 2Пξ0r √1+(r/a)2

E= ∫ ρVdv âR

4П ξ0R2

E= ∫ ∫ ρSds âR

4Пξ0R2

E = ρl

ậR

2Пξ0r


36. Give the electric field intensity for charged disc.

Where E – electric field intensity a - radius of disc.

37.Give the principle of superposition. [ NOV 2002 ] If a system consists of n point charges namely q1,q2 ……….qn, then the force on its charge is given by the vector sum of all the individual forces given by coulombs law. This is linear superstition.

38.Define Potential . Work done in moving a unit positive charge from infinity to any point is called potential. Unit is volts.

39. Define Electric scalar potential . It can be written as negative of gradient of potential .

40.Give the relation between electric field and potential. E = V V/m d Where E – Electric field intensity V – potential d – distance.

41. Give the potential difference due to infinite line charge. ρl ln rB

VAB = volts. 2П ξ0 rA

42. Give the potential due to electrical dipole.

volts. where V - potential due to electrical dipole d - distance between the two charges r - distance between point and the origin. ξ0 - permittivity of free space.


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E = ρl r a ậR

2 ξ0 (a2 +r2)3/2

V = Q dcos ө 4П ξ0 r2


43. Define Electric flux. It states that the electric flus through any surface equals the integral of the flux density over the surface. Ψ = ∫ ∫ D.ds Where Ψ - electric flux through the surface(coulomb) D - Flux density (c/m2)

44.State Gauss law. The electric flux through the surface is equal to the total charge enclosed by the surface(for any closed surface) ∫ ∫ D.ds = Q where D – flux density vector at any point P. ds – small vector area in the surface at point P. Q - total charge enclosed by the surface.

45. Define Equipotential surface. It is the surface in which all the points will have the same absolute potential.

46.Give the applications of Gauss law. To find the electric field intensity fro symmetrical or uniform charge

configurations. To find the electric field intensity for unsymmetrical or non uniform field.

47. Give the relationship between potential gradient and electric field. E = - V E = - ∂Vâx + ∂Vây +∂Vâz∂∂Vâx

Where E – electric field intensity V/m V – potential (V)

48.What is physical significance of div D. . D = ρv

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.


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∂Vâx + ∂Vây +∂Vâz

∂x ∂y ∂z


PROOF

49.Prove Div curl Ā = 0 āx āy āz

X Ā == ∂/∂x ∂/∂y ∂/∂z

Ax Ay Az

== āx ∂Az - ∂Ay āy ∂Az - ∂Ax āz ∂Ay - ∂ Ax

- + ∂y ∂z ∂x ∂z ∂x ∂y

. x Ā == ∂ āx . āx ∂ Az - ∂ Ay + ∂ āy . āy ∂Ax - ∂Az + ∂x ∂y ∂z ∂y ∂z ∂x

∂āz . āz ∂Ay - ∂Ax . ∂z ∂x ∂y

== ∂2Az - ∂2Ay + ∂2Ax - ∂2Az + ∂2Ay - ∂2Ax ∂y∂x ∂z∂x ∂z∂y ∂x∂y ∂x∂z ∂y∂z

. x Ā == 0

50. Prove curl grad V = 0 i.e x V = 0

x V = āx āy āz

∂/∂x ∂/∂y ∂/∂z

∂V ∂V ∂V


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∂x ∂y ∂z

== āx ∂ 2V - ∂2V - āy ∂

2V - ∂ 2V + āz ∂

2V - ∂ 2V

∂y∂z ∂y∂z ∂x∂z ∂x∂z ∂x∂y ∂x∂y x V == 0

51. Prove . V = V

V = ∂V āx + ∂V āy + ∂V āz

∂x ∂y ∂z

= ∂ āx + ∂ āy + ∂ āz ∂V āx + ∂V āy + ∂V āz .

∂x ∂y ∂z ∂x ∂y ∂z

= ∂2V + ∂2V + ∂2V =

2V

∂x2 ∂y2 ∂z2

52. Div(Ā + В) = . Ā + . B . (Ā + В) = āx . ∂ (Ā + В) +āy. ∂(Ā + В) + āz . ∂(Ā + В) ∂x ∂y ∂z = ∂Ax + ∂Bx + ∂Ay + ∂By + ∂Az + ∂Bz

∂x ∂x ∂y ∂y ∂z ∂z

. (Ā + В) = . Ā + . B

53. curl (Ā + В) = x Ā + x B x (Ā + В) = āx āy āz

∂ ∂ ∂ ∂ x ∂y ∂ z

Ax+Bx Ay+By Az+Bz


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. . .


= āx āy āz āx āy āz

∂ ∂ ∂ ∂ ∂ ∂ ∂ x ∂y ∂ z + ∂x ∂y ∂z

Ax Ay Az Bx By Bz

x (Ā + В) = x Ā + x B

54. Transform a vector Ā =y Īx – x Īy +z Īz. Into cylindrical coordinates.[ april 2003] The cartesian coordinates are (y,-x,z) The cylindrical coordinates are(r,Φ,z) Ax = Ax cos Φ +Aysin Φ Ar = y cos Φ – x sin Φ But x = r cos Φ and y = rsin Φ So, Ar = rsin Φcos Φ - rcos Φ sin Φ=0 AΦ = - Ax sin Φ + Ay cos Φ = -y sin Φ – x cos Φ = -[ rsin Φsin Φ +rcos Φcos Φ] = -r[sin2 Φ + cos2 Φ] = -r Az = zVector A in cylindrical coordinates is Ā = -r ĪΦ zĪ z

55. Find the nature of the field given by determining the divergence and curl Ā =30 Īx +2xy Īy +5xz2 Īz Ā =30 Īx +2xy Īy +5xz2 Īz Div Ā = . Ā = ∂ (30) + ∂ (2xy) + ∂ (5xz2) ∂ x ∂ y ∂ z

= 0 + 2x + +10xz Divergence exists . So,field is non-solenoidal.


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Curl Ā = x Ā

X Ā = Īx Īy Īz

∂ ∂ ∂ ∂ x ∂y ∂ z 30 2xy 5xz2

= Īx(0) - Īy [∂ (5xz2) – 0 ] +Īz [∂ 92xy) -0] ∂x ∂y X Ā = - 5z2 Īy + 2y Īz Curl exists. So field is rotational..

56. Find the E at (0, 3,4)in cartesian coordinates due to a point charge Q = 0.5μc at the origin.

R = 3āy + 4 āz R = √ 9+16 = 5 âR = R R = 3āy + 4 āz = 0.6 āy + 0.8 āz 5

E = Q = 0.5 x 10 – 6 ( 0.6 āy + 0.8 āz ) 4ΠЄ0 R2 4Π x 1 x 10 -9 x 25

36Π E =180 V/m in the direction âR

57.A uniform line charge infinite in extent ,with ρl =20nc/m lies along z axis.Find E at (6,8,3) m.

In cylindrical coordinates r = √ 62 + 82 = 10 m Field is constsnt with Z

E = ρl = 20 x 10 -9 âR = 36 âR V/m 2ΠЄ0 R 2Π x 1 x 10 -9 x10 36 Π


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58. Two small identical conducting spheres have charge of -1nc and 2nc respectively if they are brought in contact and then separated by 4 cm. What is the force between them. q1q2

Force F = N 4ΠЄ0 R2 When both charges are brought in contact , Q = q 1 + q 2

= (-1 +2)/2 = 0.5nc 2 F = 0.5 x 10-9

4Π x 1 x 10-9 x 16x 10-4

36 Π

F = 0.2813 104 N

59. A uniform electric field. Is directed along +ve x axis and its magnitude is 30 V/m. What is the potential difference between two points P1 and P2 on x axis at a distance x1 = 5 cm ,x2 = 20 cm from the origin

If a positive test charge is moved from P2 to P1 work is done against the field an dso potential rises from P2 to P1 and is equal to E x (x2 – x1) =30 x 0.15 =4.5 V .P1 is at higher p[otential than P2 by 4.5 V.

60. Given that D=10x āx c/m2 determine the flux crossing a 1 m2 area that is normal to x axis at x=3 m.

We know Φ =DA = [30][1] =30c.

8mark questions

PREPARED BY : Mr.M.SIVAPRAKASAN , LECT/ECE

O 5 Cm P1 20cm P2 E 30 V/m

12

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1) Explain the three coordinate system2) Write the conversion formulas for different

coordinatesystem.3) State and prove Divergence theorem4) State and prove Stoke’s theorem5) Determine the electric field intensity of an

finite&infinite long, straight, line charge of uniform density fL in air. (Nov. 2003).

6) A Circular disc of radius ‘a’ is uniformly charged with a charge density of coulombs / M2. Find the electric field intensity at a point ‘h’ from the disc along its central axis.

7) Find the electric field intensity at a point p due to an Electric dipole. Hence define Electric Dipole moment. (April 2003).

8) Define Divergence, Gradient, Curl and haplacian in Cartesian, cylindrical and spherical coordinate system with mathematical expressions. (April 2004,).

9) State the principle of super position as applied to electricpotential and derive a general expression for the resultant potential due to point, line and surface, volume charges composing the systems (April 2004).

10) Derive the potential due to uniformly infinite line charge of charge density L c/m. 10 MARK QUESTIONS

11) Derive the potential due to an electric dipole.

12)Find the Laplacian of for r = f (x,y,z) where

. (Note Laplacian of scalar)13) Find Laplacian of Vector point function

14) Showthatvector is

irrotational and find its scalar potential. 15)

Determine the electric field Intensity at P (-0.2, 0, -2.3) due to point charge of 5nc at Q (0.2, 0.1, -2.5) in air. All dimensions are in metres (Nov. 2003).


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16)Given in cylindrical coordinates. For

the contour shown in fig, verify Stoke’s Theorem. (Nov. 2002)

17)Given a field . Find the

potential difference VAB given A (-7, 2, 1) B (4, 1, 2) [Nov. 2002].

18) i) State Divergence Theorem (April 2003) – Refer 2 Marks

(ii) A Vector field is given in

spherical co-ordinates. Evaluate both sides of divergence theorem for the volume enclosed between r=1 and r=2.

19 Derive the boundary conditions at the charge interface of two dielectric media.

20 Derive the electric field and potential distribution and the capacitance per unit length of a co-axial cable.

UNITII

Two Marks

1. Define Biot –Savart Law?The magnetic flux density produced by a current element at any point in a

magnetic field is proportional to the current element and inversely proportional to the square of the distance between them.

or


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where, dB is magnetic flux density by a current element I dl is the current element R is the distance between point and current element.

2. Write Biot –Savart Law in Vector form?

where

- Unit Vector,

dB - Magnetic flux density,Idl - Current element.

3. Give the Magnetic field intensity due to a finite and infinite wire carrying a current.

For a Finite wire

For a infinite wire

Where I - Current thro’ wireD - Distance between point P and Wire.

4. Give the Magnetic field Intensity due to a circular coil.The magnetic field intensity due to a circular coil is

where I - Current thro’ coil


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a - Radius of coild - Distance from centre of coil to point P

5. State Ampere’s LawMagnetic field Intensity around a closed path is equal to the current enclosed by

the path

6. What is the relation between magnetic flux density and magnetic field Intensity.

B = H (Tesla)B = 0 r H.

WhereB - Magnetic flux density (Tesla)H - Magnetic Field Intensity (A/m)0 - Permeability of free spacer - Relative permeability of medium

7. What is magnetic dipole moment.Magnetic dipole moment is the product of current and area of loop. Its direction

is normal to loop

M = IA Where M is magnetic dipole moment I is the current A is the area

8. Define Magnetic Flux Density It is flux per unit area B = /A Wb/m2 or Tesla.

- Flux (Wb)A - Area (m2)

9. Give Lorentz force Equation

It is expressed as follows.

where F - Lorentz force - Charge of the moving charge


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v -Velocity of chargeB -Magnetic field.

10. What is the expression for the torque experienced by a current carrying loop, placed in a magnetic field. [April 2003]

Torque experienced by a current carrying loop, placed in a magnetic field is given by T = BIA cos m = IA

T = mB cos or T = m x B where

m - magnetic moment -A – m2.

A - Area (m2)

11. What is the difference between scalar and vector magnetic potential.[Nov 2003]Magnetic scalar potential is a quantity whose negative gives the magnetic

intensity.

or

Scalar potential Magnetic Vector potential is a quantity whose curl gives the magnetic flux

density

Vector Potential

12. Can a magnetic field exist in a good conductor if it is static or time varying? ExplainYes, magnetic field exist in a good conductor if the field is static or Time Varying.For a good conductor, conductuvity is high and current exists.

But Static or time varying Magnetic field exists.

13. A very along and thin, straight wire located along Z axis carries a current I in Zdn. Find Magnetic field Intensity at any point in free space. [Nov 2002]The Magnetic fi8eld intensity at any point P due to current carrying conductor is

given by


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If the conductor is infinitely long, 1 = 2 = 0

14. Give the force between two parallel conductors

where

F - Force between two conductors

I1 - Current flowing in 1st conductor

I2 - Current flowing in2nd conductor

D - Distance between two conductors.

15. Give the formula for magnetic field due to square loop at a distance h.

Where a - Side of square looph - Distance b/w point P and centre of coil.H - Magnetic field due to a square loop.

16. Write Magnetic field of square loop at centre of coilMagnetic field at centre of coil is

17 Determine the force between two parallel conductors of length 1m separated by 50 cm in air and carrying currents of 30A. i. in the same dn b). in opposite dn.

Solution:Given

I = 30 A, l = 1m,


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d = 0.5 m.a). Force of attraction:

=

b). Force of Repulsion:

18. What is the maximum torque on a square loop of 1000 turns in a field of intensity of 1 Tesla. The loop has 10 cm sides and carries 3A. What is magnetic moment of loop?

Solution:N = 1000 a = 0.1 mI = 3AB = 1 TeslaArea = 0.01m2

Torque = IAB= 3 x 0.01 x 1= 0.03 N-m

Magnetic Moment = IA= 3 x 0.01= 0.03 Amp. m2

19. Write Magnetic field Intensity due at the centre of coil (circular)

Magnetic field Intensity at centre of circular coil radius a (m) with current I is


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20. A circular coil of radius 10 Cm is made up of 100 turns. It carries a current of 5A. Compute Magnetic field Intensity at centre of coil.

Solution:

Given a = 10 x 10-2 mN = 100 turnsI = 5A

21. Calculate the magnetic flux density due to a circular coil fo 100 amp turns and area of 70 cm2 on the axis of coil at distance 10 cm from the centre.

Solution:Given

NI = 100 ATArea = d = 0.1m

Magnetic flux Density B =

=

= 103.7 x 10-6 Tesla.B = 103.7T.

22. Two wires carrying in the same dn of 500A and 800 A are placed with this axes 5cm apart. Calculate Force b/w them.

SolutionGiven

I1 = 500 AI2 = 800 AR = 5 x 10-2m.


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23. Find the maximum torque on a n 100 turn rectangular coil, 0.2 m by 0.3m, carrying a current of 2A in the field of flux density 5 Wb/m2?

Solution:Given

N = 100A = 0.2 x0.3 = 0.06 m2

I = 2AB = 5 Wb/m2

Tmax = NIAB= 100 x 2 x 0.06 x 5

Tmax = 60 N-m

24. Determine the force per unit length between two long parallel wires separated by 5 Cm in air and carrying current of 40A in the same direction.

Force/ length =

25. State Gauss Law for Magnetic field.The total magnetic flux passing through any closed surface is equal to zero.

26. Calculate if the vector potential

Solution:

= x Ā

= 0 0


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S.NO CONTENTS8 MARK QUESTIONS

PAGE NO

1. Derive Magnetic Field intensity at any point due to finite length wire carrying current I.

2. Derive Magnetic Field Intensity at any point due to infinite wire carrying current I.

3. Derive the magnetic field intensity along the axis of circular loop carrying current I

4. Derive the equation for force between two parallel conductors.[APRIL 2004]

5. Explain Magnetic Force on a moving charge (or) Derive Lorentz Force Equation.

6. Derive Torque on a Rectangular loop carrying current I[APRIL 2004]

7. A wire carrying a current of 100A is bent into the form of a circle of diameter 10cm. Calculate a). flux density at the centre of the coil. B). Flux density at appoint on the axis of the coil and 12 cm from it.

8. Derive scalar Magnetic potential and vector magneticpotential.

9. Find the magnetic flux density at the center of a square loop with side W carrying a current I.[NOV2003]

10. State and explain Ampere’s circuital law and Biot savart law using mathematical expression for finding magnetic flux density due to current I. Explain the merit and demerit of these laws.10 MARK QUESTIONS

11. Develop an expression for magnetic field intensity both inside and outside a solid cylindrical conductor of radius ‘a’ carrying a current I with uniform density and sketch the variation of field intensity as a function of distance from the conductor axis.


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12. A single phase circuit comprise two parallel conductors A and B, 1 cm in diameter and spaced 1m apart. The conductors carry currents of +100 amps and -100 amps. Determine the field intensity at the surface of each conductor and also in the space exactly midway between A and B.

13. A steady direct current I amps flows in a wire bent in the form of rectangle of length l and breadth b. Assume Z axis passing thro the centre of rectangle find the magnetic field intensity H at any point on the axis.(or)Derive an expression for magnetic field intensity at any point due to rectangular coil.

14. Determine the electric field intensity at P(-0.2, 0, -2.3) due to a point charge of +5 nC at Q (0.2, 0.1, -2.5) in air. All dimensions are in metres.

15. A circular disc of radius ‘a’ m is charged uniformly with a charge density of ρs C/m^2. Find the electric field at a point ‘h’ m from the disc along its axis.

16. Derive an expression for electric field intensity E due to an uniformly charged infinitely long straight line with constant charge density in C/m.

17. The potential exits in air is V = 10 y ( x^3 + 5 ) volt. (i) Find E at surface y = 0

(ii) Show that the surface y = 0 is an equipotential surface. (iii) If surface y = 0 is a conductor, find the total charge in the region 0 < x < 2, y = 0, 0 < z < 1. Assume V > 0 in the region outside the conductor.

18. Derive the boundary conditions at the charge interface of two dielectric media.

19. Conducting concentric spherical shells with radii a and b are at potentials Vo and 0 respectively. Determine the capacitance of the capacitor.

20. Derive the electric field and potential distribution and the capacitance per unit length of a co-axial cable.


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UNIT III

TWO MARKS

1. Write the poisson’s and laplace equation.

Poisson’s equation is

▼2V = - ρ/ Є ∂ 2V + ∂ 2V + ∂ 2V ∂x2 ∂y2 ∂z2 where ρ – volume charge density Є – permittivity of the medium ▼2 – laplacian operator

Laplace equation is 2V = 0 ∂ 2V + ∂ 2V + ∂ 2V = 0 ∂x2 ∂y2 ∂z2

2. Obtain Poisson’s equation from Gauss’s law? Gauss law in point form is ▼.D = ρ where D – electric flux density ρ - volume charge density but D = Є E therefore Є ▼.E = ρ ▼. E = ρ/ Є but E = - ▼V PREPARED BY : Mr.M.SIVAPRAKASAN , LECT/ECE 24

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- ▼ . ▼V = ρ/ Є

▼2V = - ρ/ Є This is poisson’s equation

3. What is displacement current? It is the current that flows through the capacitor.Displacement current density is given by JD = ∂ D

A/m2

∂t

4. What is a capacitor? A capacitor is an electric device which consists of two conductors separated by a dielectric medium which can store equal and opposite charges .

5. Express laplace equation in Cartesian coordinate system[ april 2003] Laplace equation ▼2V = 0 In Cartesian coordinate system ∂ 2V + ∂ 2V + ∂ 2V = 0 ∂x2 ∂y2 ∂z2

6 . Express laplace equation in cylindrical coordinate system[ april 2003]In cylindrical coordinate system,

1 ∂ ρ∂2V 1 ∂ 2V ∂2V + + = 0 ρ∂ρ ∂ρ2 ρ2 ∂Ф2 ∂z2

7. Define Magnetic dipole [ NOV 2003] A small bar magnet with pole strength m and length l may be trated as small bar magnet. A small current carrying loop is called a magnetic dipole.

8. What is magnetic dipole moment? It is the product of current and area of the loop .Its direction is normal to the loop. m = I *A where m – magnetic dipole moment I – current in the loop A – Area of the loop.

9.Define magnetization PREPARED BY : Mr.M.SIVAPRAKASAN , LECT/ECE 25

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It is defined as ratio of magnetic dipole moment to unit volume. M = Magnetic dipole moment = m Volume V

10. Define magnetic susceptibility. It is the ratio of magnetization to magnetic field intensity. Xm = M H

11 What is the relation between relative permeability and susceptibility. μA = 1 + Xm

where μA - relative permeability Xm - susceptibility

12. What are the different types of magnetic materials According to their behaviour ,magnetic materials are classified as diamagnetic ,paramagnetic and ferromagnetic materials.

13. Define magnetic flux?It is define das ratio of magnetomotive force to reluctance of magnetic circuit. Magnetic flux = mmf reluctance

13. Define mmf?

Magnetomotive force of a magnetic circuit is equal to the line integral of magnetic field H around the closed circuit. Mmf = ∫ H.dl = NI amp-turns.

14. Define Reluctance and pemeance? It is defined as ratio of total mmf of magnetic circuit to flux through it. Reluctance = mmf Magnetic flux Permeance is the revesal of reluctance. P = 1/R

15. What is the capacitance of a parallel plate capacitor? The capacitance of a parallel plate capacitor is C = A Є0 Єr farad

d Where PREPARED BY : Mr.M.SIVAPRAKASAN , LECT/ECE 26

26


A – area of the capacitor d – distance between the parallel plates

16. State the boundary conditions at the interface between two perfect dielectrics

a) The tangential component of electric field is continuous Et1 = Et2 b)The normal component of electric flux density is continuous. Dn1 = Dn2

17. Define Boundary conditions.The conditions existing at the boundary of the two media when field passes from one medium to other are called boundary conditions.

18.Write down the magnetic boundary conditions

a)The normal components of flux density B is continuous across the boundary.

Bn1 = Bn2

b)The tangential component of field intensity H is continuous across the boundary Ht1 = Ht2

19) Define Polarization The product of charge Q and length l between dipole is dipole moment.Dipole moment per unit volume is called polarization,a vector field directed from –Q to +Q . P = Ql/Al = Q/A ul where ul is unit vector directed –Q to +Q.

20)Write the point form of ohm’s lawOhm’s law at a point states that the field strength within a conductor is

proportional to the current density. J α E J = σE

21) Give the expression for capacitance of parallel plate capacitor with 2 dielelectrics medium

Capacitor of a capacitor with 2 dielectrics is

C = A Є0

d1 + d2

Єr1 Єr2

Where A – area of the capacitor D1 – distance between one platenof capacitor and one dielectric D2 – distance between other plate of capacitor and second dielectric. Єr1 Єr2 - relative epermittivity

22) Write down the capacitance for two concentric spheres. PREPARED BY : Mr.M.SIVAPRAKASAN , LECT/ECE 27

27


The capacitance fro two concentric spheres of radius a and b is C = 4π Є ab F b – a

23) Write the capacitance for 2 wire parallel transmission line The capacitance for 2 wire parallel transmission line of radius a is C = π Є ln (d/a) where d - distance between two transmission line

24) Give the inductance of solenoid.The inductance of solenoid

L = μN2A H l

where l - length N - number of turns A - area of cross section

25) Give inductance of toroid.The inductance of toroid is

L = μN2r2 H 2R where N – number of turns R – radius of the toroid r - radius of the ring.

26) Give the inductance for coaxial transmission line .The inductance per unit length for coaxial transmission line is

L = μ ln b/a H/m d 2R where L/d – Inductance per unit length a – radius of inner transmission line b - radius of outer transmission line

27) Give the Inductance for two wire transmission line The inductance per unit length for two wire transmission line is

L = μ ln D/a H/m d π


28


where L/d – Inductance of the two wire transmission line D – distance between two transmission lines a – radius of the transmission line

28) Give energy and energy density in a capacitor. Energy in a capacitor with Q charge and capacitance C is

1 Q2 1 1 W = = CV2 = Joules 2 C 2 2

29) Write the continuity of current equation:

30) Define conduction current densityIt is the product of conductivity and Electric field intensity.

31) Define energy and energy density in an inductor.Energy stored in inductor with current I is

Energy Density in Inductor per unit volume is

32) Define self inductance.The self induction of a coil is defined as ratio of total magnetic flux linkage to the

circuit thro’ the coil

where - Magnetic flux (Wb)N - Number of turnsI - Current thro’ the coil.

33) Define Mutual inductance. PREPARED BY : Mr.M.SIVAPRAKASAN , LECT/ECE 29

29


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

N2 = Number of turns in coil 2. Magnetic flux linkage in coil 12

I1= Current through coil 1.

34) Give the relation between mutual inductance and self inductance.

where M - Mutual inductance L1 - Self inductance of coil 1,L2 - Self inductance of coil 2,K - Coupling coefficient.

35). Define Hysteresis: (April / 2003)The phenomenon which causes magnetic flux density (B) ot lag behind magnetic

field intensity is called Hysteresis.

36) A parallel plate air core capacitor is flowing with 10V across the plates. If the separation between the plate is now increased to double the original value, calculate the new value of the voltage across the capacitor.

If d doubles, New capacitance


30


V1 = 2V= 2 x 10 = 20 V.

37) A parallel plate capacitor with d = 1m and plate area 0.8 m2 and a dielectric relative permittivity of 2.8. A dc volt of 500 v is applied between the plates. Find the capacitance and energy stored.

Capacitance

d = 1m, A = 0.8 m2

c = 19.833 pF

Energy stored

E = 2.48 x 10—6 J.

38) Find the resistance of a 1 Km length of is wire which has a conductivity and r radius of 1 x 10 -3

m.

Resistance

Area l = 1 x 103 m.

39) Given the potential function V = 4x+3y in free space, find energy density.V = 4x + 3y.


31


Magnitude of E is

Energy Density

40) Find the Polarization in a dielectric medium with if D = 5 x 10-8 C/m2

Polarization

then

But

then

p = 3 x 10-8C/m2

41 Two coils A and B with 800 and 1200 turns respectively are having common magnetic circuit. A current of 0.5 A in coil A reduces a flux of 3mwb and 80% of flux links with coil B. Calculate L1, L2 and M.

Coil A Coil BN1 = 800 N2 = 1200


32


I1 = 0.5 A

= 0.5 x 3 x 10-3 Wbh = 80% = 0.8

42 A and B are two coils with 1000 and 1500 turns respectively and lying in parallel planes, 50% of flux produced by coil A links with coil B. The hysteresis curve indicates that a current of 3A in the coil A produces 0.3 mWb while the same current in coil B produces 0.6 mWb. Calculate Mutual Inductance between them and self inductance of coils.

Coil A Coil B

NA = 1000 NB = 1500

IA = 3A IB = 3A

Since the flux linkage is 50% K = 0.5

Mutual Inductance

=


33


M = 0.0866 H.

43) Evaluate the loop inductance 1 Km of a single phase transmission circuit compressing two parallel conductor spaced 1m apart and with diameters 0.5 cm and 0.8 cm .

Solution:

d = 1m

r = 1 m.

Inductance of Transmission lines,

= 10-7 [1+23.026]

= 24.026 x 10-7 H/ m.

= 24.026 x 10-4 H/ Km.

L = 2.4 m H / Km.

44) Two coils of self inductances of 0.5 H and 0.8 H with negligible resistance are connected in series. It their mutual inductancy is 0.2H. Determine the effective inductance of the combination.

L1 = 0.5 H L2 = 0.8 H M = 0.2 H.

L = L1 + L2 L(aiding) = 0.5 + 0.8

= 1.7 H.

L (opposing) = 0.5 + 0.8 - 2 x 0.2= 0.9 H.

S.NO 8 MARKS


34


1 Obtain poisons & Laplace’s from Gauss’s Law [November 2002]

2 Derive capacitance of a parallel plate capacitor having tow dielectric media.

3 Derive an expression for capacitance of an isolated sphere.

4 Derive an expression for capacitance of concentric spheres. [November 2003] [April 2005] (or)

A spherical capacitor consists of inner conducting sphere of radius Ri and an outer conductor with spherical inner wall of Ra radius. Space is filled with dielectric Determine capacitance, [Use, ‘a’ as Ri and

‘S’ as Ra]5 Derive an expression for capacitance of coaxial cables

[Cylinders] (or)A cylindrical capacitor consists of an inner conductor of radios ‘a’ and ‘b’. The space between the conductors is filled with a dielectric of permittivity

and the length of the capacitor is L. Determine the capacitance of this capacitor [November 2003]

6 Derive an general expression for capacitance of parallel conductors i.e Two wire Transmission line.( or) A very long two wire transmission line each wire of radius ‘a’ separated by a distance ‘d’ conducting ground. Assuming both ‘d’ and ‘h’ is longer than ‘a’, find capacitance / unit length. [November 2004]

7 Define energy and energy density in a capacitor.8 Derive the boundary conditions of the normal and

tangential components of electric, field at the interface of two media with different delectrics. [November 2004]

9 A parallel plate capacitor with a separation d = 1 cm has 29 Kv applied . Assume air has a dielectric strength of 30 Kv / cm. Show why the air breaks down when a thin piece of glass with a dielectric. Strength of 290 Kv / cm and thickness d2 = 0.2 cm is inserted as shown in fig (b) [November 2002]

10 Two concentric perfectly conducting spheres of radii PREPARED BY : Mr.M.SIVAPRAKASAN , LECT/ECE 35

35


‘a’ and ‘b’ contains charges +Q and – Q respectively. The region between the spheres is filled with a dielectric of permittivity . Find the total energy contained in the system.10 MARKS QUESTIONS

11 Permittivity of the dielectric of a parallel plate capacitor increases uniformly from at one plate to at the other. If A is the surface area of the plate and d is thickness of dielectric, derive an expression for capacitance.

12 A parallel plate capacitor has two dielectrics with relative permittivities and . If c1 is the capacitance when two dielectrics take same thickness and C2 Is the capacitance when one dielectric takes 3

times of other one find

13 Determine the resistance of the insulation in a length l of coaxial cable.

14 Derive an expression for serial and parallel combination of capacitors [April 2004]

15 Find capacitance of a parallel plate capacitor containing two dielectrics and each comprising one half the volume:

16 Derive an expression for inductance of a solenoid [April 2004]

17 Derive self inductance of a Toroid:18 Determine the expression for Inductance / unit length

for a air coaxial transmission line or coaxial cable.19 Derive the inductance / unit length for a Two wire

Transmission line.20 Derive an expression for energy stored in the

magnetic field of a coil possessing an inductance of L Henry when the current in the coil is I amps. [April 2004]

21 Derive an expression for energy density in a Magnetic field [April 2005]

22 Assuming static conditions derive the boundary relations for Magnetic fields [April 2003]

Calculate the inductance of solenoid of 200 turns wound on a cylindrical tube 6 cms diameter. Te


36


length of tube is 60 cm and Medium is air. [November 2002]

23 Calculate the inductance of solenoid of 200 turns wound on a cylindrical tube 6 cms diameter. Te length of tube is 60 cm and Medium is air. [November 2002]

24 Define a Magnetic circuit with a sketch and hence obtain the expression for its reluctance.[November 2004]

25 A Magnetic circuit employs an air core toroid with 500 turns, across cross sectional area 6 cm2 mean radius of 15 cm and coil current 4A. Determine reluctance of eh circuit, flux density and Magnetic field intensity. [November 2004]

26 Describe Hysteresis loop?

UNIT-IV

CIRCULAR WAVE GUIDE

1. Show graphically attenuation Vs frequency characteristics in a circular frequency wave guide for TE11, TM11 and TE01 modes. (April / May - 2004)


37


2. What are wave impedances in a circular wave guide excited in TE and TM modes in terms of frequency and medium constants? (April /May - 2004)

3. Mention the applications of circular wave-guide and write expression for cut off frequency.

Circular wave-guides are used as attenuators and phase – shiftersCut –off frequency

Where

4. Calculate cut- off frequency of copper tube with 3 cm diameter inside with air filled, in TM11 mode?

PREPARED BY : Mr.M.SIVAPRAKASAN , LECT/ECE

TM11

TE11

f

ATtenuator

TE01

38

38


= 2.566 x 102

= 12.2 GHZ

5. Why is TM01 mode preferred to TE0, mode in a circular wave guide? Mention the dominant modes in rectangular and circular wave guides?

TM01 mode is preferred to TE0, mode, since it requires a smaller diameter for the same cut – off wavelength. For Rectangular wave guide

The dominant mode TE01

For circular wave guide the dominant mode TE11

6. Define Q of a wave-guide? Give the relation between quality factor and attenuation factor of a wave guide.

Quality factor Q is given by

Q= relation between quality

factor and attenuation factor of a wave guide

7. Given a circular wave guide used for a signal at a frequency of 11 GHZ propagated in the TE 01, mode and internal diameter is 4.5 cm. Calculate i ) C ii) g

i )

or 0.0768m


39


ii).

8. Explain what is the suitable mode used in circular waveguide for making a wave meter

TM01 is the most preferred type, since for a given cur off frequency, the diameter required for the circular wave guide will be smaller, because the root of the Based function for TM01 is 2.4 where as it is 3.84 for the TE01 mode.

Therefore a circular waveguide operating in the TM01 mode is generally Preferred in applications where rotating couplings have to be used.

9. Using neat diagram show excitation method of TM01 and TE11 modes in a circular wave guide

10. Give the reason why circular wave guides are avoided, or list the reasons why it is avoided.

The frequency difference between the lowest frequency on dominant mode and next mode is smaller than is a rectangular wave guide, with b/a = 0.5.

The circular symmetry of the wave guide may reflect on the possibility of the wave not maintaining its polarization throughout the length of the guide.

For the same operating frequency, circular wave guide is bigger in size then a rectangular wave guide.


40

Antenna probe

Short circuited end

Coaxial line


11. A circular wave guide has fc = 9 GHZ in dominant mode a. Find inside diameter of guide if it is air filled

For TE11,

a.)

12. An air filled circular waveguide of 2 cm inside radius is operated in TE01 mode. Compute its cut off frequency

For TE01 mode

Cut – of frequency

= 0.92 x 1010

= 9.2 GHZ

13. Given a circular waveguide used for a signal at a frequency of 11 GHZ propagated in TE11 mode and internal diameter 4.5 cm. Calculate Cut off wavelength Given: diameter 4.5 cm

frequency of 11 GHZ

= 0.027 m For TE11

Cut off wave length


41


= 7.68 cm

14. A circular wave guide has internal diameter of 5 cm.Calculate fc for TE12 and TE02.

For TE12

= 2.95 cm

= 10.18 GHZ For TE02

= 2.21 cm

fc = 13.57 GHZ15. Write down the roots of TE and TM waves for an circular wave guide.

For TE For TM


42


16.What are disadvantages in using a circular wave-guide for transmission of signal. i) For a given signal frequency f which is greater than the cut off frequency fc for TE10 mode, in a rectangular wave guide and with a circular wave guide in the TE 11 mode and with identical cut off wavelength the area of cross section of a circular ii) Wave guide will be higher the circular waveguide will occupy a large space. In circular wave guide in TE11 mode the polarization of the field at input may be varied due to presence of wave bends and other irregularities, before reaching receiving end. This result in reflection of signal. Above problem can be overcome by using either TE01, or TM01 modes.

17. Write down the general field components or electric & magnetic field components of circular wave guide

18. Draw the circular cylindrical co-ordinate system


43


19. Write the field components of TE waves in circular wave guide.

20. What is the attenuation factor for TE10 & TM circular wave guide For TM

For the dominant TE10 mode, the attenuation factor is given by

21. What is the attenuation factor for TE on mode & Q of circular Wave guide

For TE on mode

Where

Q =

22. Given a circular waveguide used for a signal at a frequency of 11


44


GHZ propagated in TE01 mode and internal diameter 4.5 cm. Calculate Guide wave length Guide wavelength

g = 0.029 m.

23.In a material for which σ = 5.0 S/m and ρr and E = 250 sin 10^10 t(V/m). Find the conduction and displacement current densities. The conduction current density is given by, Jc = σ E = 5(250 sin 10^10 t) = 1250 sin 10^10 t A/m^2 The displacement current density is given by,

JD = ðD/ðt = ð/ðt (єE)

= ð/ðt [єo єr E ] = ð/ðt [8.854 x 10^-12 x 1 x 250 sin 10^10 t] = (8.854 x 10^-12 x 250) (10^10) (cos 10^10 t) = 22.135 cos 10^10 t A/m^2 For the two densities, the condition for magnitudes to be equal is,

│Jc│/│JD│ = σ/єω = 5/(8.854 x 10^-12 x 1) = 5.6471 x 10^11 But ω = 2 π f f = ω / 2 π = 5.6471 x 10^11 / 2 π = 89.87 GHz

24.What is motional e.m.f. ? secondly magnetic field is stationary, constant not varying with time while the

closed circuit is revolved to get the relative motion between them. This action is similar to generator action, hence the induced e.m.f. is called motional or generator e.m.f.

25.Write down the differential form of Maxwell’s equation.


45


▼xE = -∂B/∂t ▼xH = j + ∂D/∂t

▼∙D = ρv ▼∙B = 0

S.NO

8 MARKS QUESTIONS

1 Write short notes on solution of wave equation for circular wave guide (or) Derive expression for wave equation of circular wave guide?(or)derive the general field components of TMmn waves in circular wave guides(Nov/Dec-2004).

2 Write briefly a note on transverse magnetic waves in circular wave guide (or) Derive general field components of TMmn waves in Circular wave guide (November/December-2004).

3 Write shorts notes on transverse electric (TE) waves (or) Derive expression for transverse electric waves in terms of circular waveguide.

4 Derive an expression for attenuation constant of TE01, mode of an air filled circular wave guide. Show the variation of with frequency and diameter for TE11 and TE01 mode (April / May-2005).Derive expression for power transmission & attenuation constant.

5 Write short notes on Q of wave guides. 6 Write short notes on wave impedance for circular wave guide. 7 Show schematically methods of Excitation of dominant &

higher order modes in circular wave guides. (April/may-2004)

8 An air filled circular waveguide having an inner radius 1cm is excited in dominant mode at 10 GHZ. Find the guide wavelength, wave impedance and the frequency bandwidth for operation in dominant mode only. (April/May -2004)

9 An air filled circular waveguide having an inner radius of 1cm is excited in dominant mode at 10 GHZ. Find

i). Cut of frequencyii). Guide wave length iii). Wave impedance iv). Attenuation constant for wall surface

resistance of 20 mill ohm. (April/May -2005) 10 Determine the cut-off frequencies of the first two propagating

modes of a circular waveguide with a = 0.5 cm and if guide is 50 cms in length operating at f = 13 GHz, determinetheattenuation. 10 MARK QUESTIONS

11 A TE11 mode is propagating through a circular waveguide. PREPARED BY : Mr.M.SIVAPRAKASAN , LECT/ECE 46

46


The radius of the guide is 5 cm and the guide contains on air dielectric a).Determine the cut-off frequency b).Determine the wavelength in the guide for an frequency of 3 GHZ c). Determine the wave impedance in the guide.

12 Calculate the cut-off wavelength, the guide wavelength and the characteristic wave impedance of a circular wave guide whose internal diameter is 4 cm for a 9GHZ signal propagated in it in the TE11 mode.

13 A 6 GHZ signal is to be propagated in the dominant mode in a rectangular wave guide. If its group velocity is to be 90% of velocity of light, what must be the breadth of the waveguide? What impedance will it offer to this signal, if it is correctly matched?

14 Write down Maxwell’s equation in differential as well as in integral forms. Explain their significance.

15 Derive Maxwell’s curl equation from Ampere’s law and faraday’s law. Express the equation in phasor form for time harmonic fields

16 State and explain Faraday’s and Lenz’s laws of induction. Demonstrate the law using a simple network.

17 Derive general field relations for time-varying electric and magnetic fields using Maxwell’s equation

18 In free space E(z, t) = 100 cos (ωt – βz ) ax V/m. Calculate H and plot E and H waveform at time t = 0.

UNIT-V


47


TWO MARKS 1. Define wave. It’s a physical phenomenon that occurs at one place at a given time and is reproduced at other place at other place at later time. Or If a physical phenomenon that occurs at one place at a given time is reproduced at other places at later times the time delay being proportional to space separation from the first location, then the group of phenomena constitutes a wave.

2. What is a homogeneous medium (University-Apr/May-2005). Or Write short notes on homogeneous medium. A homogeneous medium is one for which the quantities ε, μ and σ are Constant throughout the medium Where ε- permittivity μ-Permeability

σ- Conductivity

3. When a medium is said to be isotropic. (Apr/May-2005) The medium is isotropic if ‘ε’ is a scalar constant, so that D and E have Everywhere the same direction.

4. Define skin depth or depth of polarization (Apr/May-2005) Skin depth (δ) is defined as the depth in which the wave has been attenuated to 1/e or approximately 37% of its original value

δ= (1/ α) = √ (2 / (μω σ)) ; for a good conductor

5. Find skin depth value for cu (σ = 5.8x107mho/m) at 10MHZ (University- Apr /May-2005)

For good conductor δ= √ (2/ (μ ω σ)) Given

σ = 5.8x107mho/m μr = 1 ω = 2x100x106

δ = 2

2Пx100x106x5.8x107


48


δ = 5.488x10-17

δ = 7.4081x10-9

6. Define polarization. It refers to the time varying nature of the electric field vector at some fixed point in space.

7. Define propagation constant. The propagation constant (γ) is a complex number, and it is given by γ = α + j β Where α is attenuation constant β is phase constant

γ = j ω μ (σ + j ω ε)

8. Define intrinsic impedance or characteristic impedance. It is the ratio of electric field to magnetic field or it is the ratio of square root of permeability to permittivity of the medium Η = E/H = √ (μ / ε) ohms

9. Calculate the characteristic impedance of free space.

Η = E/H = √( μ0 /ε0) = √ (4π x 107 x36 П x109)

= 20π = 377ohms

10.Find the velocity of a plane wave in a lossless medium having a relative permittivity of 5 and relative permeability of unity.

γ = 1/ √ (εμ) = 1 /√ (μ0 μr ε0 εr) . = 1.34 x 108 m/sec

11. At what frequencies may the earth considered to be a perfect , dielectric if σ = 6 x 10-3 s/m, μr = 1 and εr = 10

σ = 1


49


ωε This is the boundary line between dielectric and conductor

σ = 108 x 106 /f ωε σ = 1 , 108 x 106 =1 ωε f f = 108 x 106 Hz

12 hat is expression for the propagation constant of plane waves in good Conductor in terms of medium parameters? What is the ratio of magnetic and electric energy density inside such conductor? (Nov-2004)

For good conductor σ / ωε >>1 γ =√ (jωμ (σ+jωε))

=√ (jωμσ)

Magnetic & electric field intensity

η = E/H = √ (j ω μ) / σ

13 how that in conductors the wave is attenuated greatly as it progresses through the conductor.(Nov-Dec-2004)

α = β = √ (ω μ σ)/2 The velocity of the wave in a conductor

γ= 2ω/ √ (ω μ σ) intrinsic impedance of the conductor

η = √ (j ω μ)/ σ

it is found that in good conductors α & β are large since σ is large,i.e. the wave is attenuated greatly as it progresses through the conductor.

14 Give the characteristics of uniform plane traveling waves with mathematical representation.(University-Nov-Dec-2004)


50


a. At every point is space electric field (E) and magnetic field (H) are perpendicular to each other and to the direction of travel.

b. The fields vary harmonically with time and at the same frequency everywhere in space.

∂2Ex / ∂x2 = με(∂2Ex / ∂t2)

∂2Ey / ∂x2 = με(∂2Ey / ∂t2)

∂2Ez / ∂x2 = με(∂2Ez / ∂t2)

15. Write Helmholtz’s equation

2E –γ2 E = 0

Where γ = √ (jωμ (σ +jωε))

16. State Snells law When a wave is traveling from one medium to another medium, the angle of incidence is related to the angle of reflection as follows

(Sinθi / Sinθt) = √ (n1 /n2) = √ (ε 2 /ε1)

θi= angle of incidence θi =angle of reflection ε2 =dielectric constant of medium 2 ε1 =dielectric constant of medium 1

17. What is Brewster angle? Brewster angle is an incident angle at which there is no reflected wave for parallely polarized wave. θ = tan -1 √( ε2 / ε1) ε2 =dielectric constant of medium 2 ε1 =dielectric constant of medium 1

18. A wave propagates from a dielectric medium to the interface with free space if the angle of incidence is the critical angle 200.Find the relative permittivity

( Sinθi / Sinθt) = √ (ε 2 /ε1 )

√ ( Sinθi / Sinθt) = √ (ε 0 /ε0 εr ) √εr = (1 / 0.342) εr = 8.5519. Define linear polarization. If x and y component of a electric field E x & Ey are present and are in phase, the resultant electric field has a direction at an angle of tan-1 (Ey/Ex)


51


And if this angle is constant with time, the wave is said to be linearly polarized.

EY E

tan θ = (EY /EX)

E = √(EX2 + EY

2)

θ

EX

20. Define circular and elliptical polarization. If x and y component of a electric field E x & Ey have equal amplitude and 900 phase difference, the locus of the resultant electric field E is a circle & the wave is said to be circularly polarized .

If x and y component of a electric field Ex & Ey have different amplitude and if the phase shift is other than 900 or 900, the locus of the resultant electric field E is a ellipse & the wave is said to be elliptically polarized .

21. The velocity of uniform plane wave in a lossless dielectric is 1 x 108m/sec.Find dielectric constant.

For lossless dielectric μr = 1

γ = 1/ √ (μ 0μr ε 0εr)

√ εr = 3

εr = 9

22. A wave propagates from a dielectric medium to the interface with free space if angle of incidence is the critical angle 200 . Find the relative permittivity.

(Sinθ1/Sinθ2) = √ (ε2/ε1)

√ (sin20/sin90) = √ ε0/(ε 0ε r)

εr = 8.5523. A lossy dielectric is characterized by εr = 2.5 μ=4 σ= 10-3 at 10 MHz. Find α, β σ / ωε = (10- 3 x36П x109 )/ (2П x 10 x106x 2.5) = 0.72


52


i) α = σ /2 √ (μ/ε) =10-3/2(√(4/2.5)x120П = 238.43x10-3

ii) β = ω√με(1+(1/2)(σ /2ωε )2 )

= 0.6623x1.0648 =0.7052

24) Write down the wave equation for E &H in a conducting medium.

2E –με (∂ 2E / ∂ t2) –μ σ (∂E / ∂t) = 0 2H –με (∂ 2H / ∂ t2) –μ σ (∂H / ∂t) = 0

25) Derive poisson’s equation the electrostatic field.

From the Gauss’s law in the point form, poisson’s equation can be derived. Consider the Gauss’s law in the point form as,

▼∙D = ρv

Where D = Flux density and ρv = Volume charge density It is known that for a homogeneous, isotropic and linear medium, flux density and electric field intensity are directly proportional. Thus, D = єE ▼∙єE = ρv From the gradient relationship, E = -▼V Substituting both equation ▼∙є(-▼V) = ρv -є[▼∙▼V] = ρv

▼∙▼V = -ρv/є ▼^2V = -ρv/є This equation is called poisson’s equation.

S.NO 8 MARKS QUESTIONS 1. Derive the electromagnetic wave equation in a

conducting medium.

2 Give an account on wave equation for free space (or) write the expression for wave equation interms of electric & magnetic fields (or) write a


53


note on wave equation for free space or derive the expression for wave equation for free space condition

3 Give the characteristics of uniform plane traveling waves with mathematical representation (nov-2004 ) (or) Derive the expression for uniform plane

4 Derive the expression for attenuation constant& phase constant (or)Write the solution for wave in a conducting medium

5 Write short notes on how wave propagates in dielectrics

6 Write a note on wave solution for conductors (or) Wave propagation in a good conductor

7 Write short notes on polarization (or) Describe the types of polarization

8 Write short notes on reflection by a perfect conductor.(or) Write both cases for the the above i)normal ii)oblique case

9 Write short notes on reflection by a perfect dielectric with both cases

10 Establish the laws of reflection & refraction for a linearly polarized plane wave of parallel polarization incidents on a plane interface b /w 2 lossless dielectric regions 1 & 2 at angle θi with respect to normal. Obtain the expression for reflection coefficient, Brewster angle as well as the angle of incidence for which total internal reflection occurs (nov / dec -2004). 10 MARK QUESTIONS

11 A uniform plane wave propagating in a good conducting medium (εr =1, μr =20, σ= 3 mho /m ) has E = Y2 e- α z sin ( 108t – βz )v / m.Find α,β,& H. (APR / MAY -2005)

12 A plane traveling electromagnetic wave has


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1010


H=0.008 A/m in free space. Compute the energy density &also the velocity of this wave in glass, whose relative permittivity is 3.

13 A sinusoidal plane wave is transmitted through a medium whose electric field strength is 10 kv/m an relative permittivity of medium is 4.Determine mean rms power flow/unit area.

14 Find the magnetic field intensity at a distance x above an infinite straight wire carrying a steady current I.

15 Find the magnetic flux density at a distance d.16 Find H at the centre of an equilateral triangular

loop of side 4m carrying current of 5A17 Calculate B and H due to a long solenoid18 Derive expression for inductance of a toroid19 Derive the expression for coefficient of coupling

interms of mutual and self inductances20 An air co-axial transmission line has a solid inner

conductor of radius ‘a’ and a very thin outer conductor of inner radius ‘b’. Determine the inductance per unit length of the line

GNANAMANI COLLEGE OF ENGINEERING, NAMAKKALDEPT.OF.ECE

ASSIGNMENT NO-1

SUBJECT: ELECTROMAGNETIC FIELDS YEAR/BRANCH: II/ECE

1. Explain the three coordinate system.


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2. Determine the electric field intensity of an finite&infinite long, straight, line \ charge of uniform density fL in air.3. A Circular disc of radius ‘a’ is uniformly charged with a charge density of coulombs / M2. Find the electric field intensity at a point ‘h’ from the disc along its central axis.


ASSIGNMENT NO-2


1. Derive the equation for force between two parallel conductors.2. Explain Magnetic Force on a moving charge .3. Derive Torque on a Rectangular loop carrying current I


ASSIGNMENT NO-3


1. Obtain poisons & Laplace’s from Gauss’s Law2. Derive capacitance of a parallel plate capacitor having tow dielectric media.3. Derive an expression for capacitance of an isolated sphere.


MOTIVATION NO-1



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1. Give an account on wave equation for free space (or) write the expression for wave equation interms of electric & magnetic fields 2. Give the characteristics of uniform plane traveling waves with mathematical representation


MOTIVATION NO-2


1. A plane traveling electromagnetic wave has H=0.008 A/m in free space. Compute the energy density &also the velocity of this wave in glass, whose relative permittivity is 32. A uniform plane wave propagating in a good conducting medium (εr =1, μr =20, σ= 3 mho /m ) has E = Y2 e- α z sin ( 108t – βz )v / m.Find α,β,& H.


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