R5210205-electromagnetic Fields

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Code No: R5210205

II B.Tech I Semester(R05) Supplementary Examinations, May/June 2009 ELECTROMAGNETIC FIELDS (Common to Electrical & Electronics Engineering and Electronics & Control Engineering) Time: 3 hours Max Marks: 80 Answer any FIVE Questions All Questions carry equal marks ????? 1. (a) Explain the superposition principle governing the forces between charges at rest. (b) For the vector function A (x, y, z) = yay + zaz , f ind

P :(0,1,1)

∫

A.d`.

0:(0,0,0)

(c)

i. A line charge of 20 nC / m lies along the entire z - axis, in free space Find the electric field at (0,4,0) m. ii. A point charge of 20 nC is now added at (3,0,0) m. What is the new field at (0,4,0) m? [6+6+4]

2. (a) For a conducting body in the electric field of static charges, explain what will be the i. net electric field inside the conductor, and ii. volume charge density at any point inside the conductor. (b) Obtain, from fundamentals, an expression for the capacitance per unit area of a parallel plate capacitor. If the plates are separated by 1 mm in air, and have a potential difference of 1000 V, what is the energy stored per unit area? [8+8] 3. (a) Establish the electrostatic boundary conditions for the tangential components of electric field and electric displacement at the boundary of two linear dielectrics. (b) z<0 is a region of a linear dielectric of relative permittivity 6.5; and z>0 is free space. Electric field in the free space region is (-3ax + 4ay -2az ) V/m. Find i. D for z > 0; ii. tangential components of D & E for z < 0, on the boundary.

[8+8]

i. A steady current element 10−3 az A-m is located at the origin in free space. What is the magnetic field B due to this element at the point (0,1,0) m (in rectangular coordinates)? ii. Where should a point be located for the magnetic field due to this element to be 0? (b) A straight length of steady current I A extends from the origin to z = ` m along the z-axis. Find the magnetic field B at a distance of y m from the origin along the y-axis. [6+10]

4. (a)

5. (a) Obtain the MFI due a long filamentary conductor of infinite length. (b) A surface current density K= 10az A/m, flows on that portion of the plane y = 4 in free space bounded by x=± ∞ and z=±30. Find H at origin. [8+8] 6. (a) Prove that the force on a closed filamentary circuit in a uniform magnetic field is zero (b) If the magnetic field is H = (0.01/µ0 ) ax A/m, what is a force on a charge of 1 pC moving with a velocity of 106 ay m/s . [8+8] 7. With the help of Vector magnetic potential obtain the expression for MFI at any point due to a current carrying sheet of density K a/m. [16] 8. Using the Maxwell’s equation in the integral form, find the relation between the tangential and normal components of time varying fields at the boundary of two media characterized by parameters ε1 , µ1 , σ1 and ε2 , µ2 , σ2 . [16] ?????

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Code No: R5210205

II B.Tech I Semester(R05) Supplementary Examinations, May/June 2009 ELECTROMAGNETIC FIELDS (Common to Electrical & Electronics Engineering and Electronics & Control Engineering) Time: 3 hours Max Marks: 80 Answer any FIVE Questions All Questions carry equal marks ????? 1. (a) E is the electric field due to a point charge Q C at the origin in free space. Find ∫ E.da S

where S is a spherical surface of radius R m and center at origin. (b) Using Gauss’s law, show that the electric field due to an infinite straight line of uniform charge density λ C/m along the z-axis in free space is (λ/2πε0 r) ar N /C. (c) An infinitely long cylinder of radius 1m in free space is filled with a uniform charge density of 1nC/m3 . If the potential on the axis of the cylinder is 100 V, find the potential variation inside the cylinder. [6+6+4] 2. (a) A conducting body is in the electric field of static charges. Explain why the net electric field at any point inside the conducting body will be zero. (b) Use the result of (a) to show that i. the net volume charge density at any point inside the conductor is zero, and [8+4+4] ii. the conductor is an equipotential body. 3. (a) Obtain the continuity equation. How does it get modified for steady currents? (b) J (r, θ, φ) is given as J = r42 cos θar + 20e−2r sin θaθ − r sin θ cos φaφ A/m2 (in spherical co ordinates). Find the total current passing through the spherical area r =3 m, 0 < θ < 200 , 0 < φ < 2π. [8+8] 4. (a) State Biot-Savart’s law for the magnetic field B due to a steady current element I dl A-m situated in free space. (b) Find B due to an infinite plane sheet of steady current on the xy - plane, with a uniform surface current density of K(-ax ) A/m in free space. [Hint: B due to an infinite straight steady line current I A at a point y m from the line is (µ0 I/2πy) T , in a direction given by the right hand rule]. [6+10] 5. Obtain the expression for the magnetic field intensity and sketch the same for a co-axial cable. [16] 6. (a) A block of iron (µ = 5000 µo ) is placed in a uniform magnetic field of 1.5 T. If the iron consists of 8.5*1028 atoms/m3 , calculate the Magnetization M and the average magnetic current. (b) Define Magnetization, how it differs from the Polarization. What is unit of Magnetization? [8+8] 7. If A = 10ρ1.5 az Wb/m in free space, find H, J.

[16]

8. (a) In a region defined by σ = 106 Siemens/ m and εr = 4, at certain frequency the ratio of conduction and displacement current density is unity. Find frequency (b) Find the value of K in the following pair of fields in free space, such that they satisfy Maxwell’s equation. D = 5xax - 2yay + Kz az mC/m2 and B = 2 ay mT. [8+8] ?????

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Code No: R5210205

II B.Tech I Semester(R05) Supplementary Examinations, May/June 2009 ELECTROMAGNETIC FIELDS (Common to Electrical & Electronics Engineering and Electronics & Control Engineering) Time: 3 hours Max Marks: 80 Answer any FIVE Questions All Questions carry equal marks ????? 1. (a)

i. Distinguish between electric potential and potential difference. ii. Express electric field of static charges in terms of potential. (b) Derive Poisson’s equation in electrostatics. (c) In cylindrical co ordinates, the potential v = 500 V at r = 50 cm and the electric field intensity ¯ (r, φ, z) = E = 500 ar V/m at r = 1 m. Determine the location of the voltage reference if E K [6+6+4] r ar V /m.

2. (a) In electrostatics, what is meant by a physical dipole? (b) For a physical dipole in the z-direction, located at the origin in free space, find the potential at a point ( r,θ, φ = π2 ) (in spherical co ordinates). (c) Point charges of 1 µC and -1 µC are located at (0,0,1) m and (0,0,-1) m respectively in free space. i. Find the potential at (0,3,4) m. ii. Recalculate the same potential, treating the dipole as a pure dipole. [6+6+4] 3. (a) Obtain the continuity equation. How does it get modified for steady currents? (b) J (r, θ, φ) is given as J = r42 cos θar + 20e−2r sin θaθ − r sin θ cos φaφ A/m2 (in spherical co ordinates). Find the total current passing through the spherical area r =3 m, 0 < θ < 200 , 0 < φ < 2π. [8+8] i. A steady current element 10−3 az A-m is located at the origin in free space. What is the magnetic field B due to this element at the point (0,1,0) m (in rectangular coordinates)? ii. Where should a point be located for the magnetic field due to this element to be 0? (b) A straight length of steady current I A extends from the origin to z = ` m along the z-axis. Find the magnetic field B at a distance of y m from the origin along the y-axis. [6+10]

4. (a)

5. If the magnetic field intensity is H = x2 ax + 2yz ay + (-x2 ) az , A/m. Find the current density at point (a) 2, 3, 4 (b) ρ = 6, ϕ = 45o , z = 3 (c) r = 3.6, θ = 60o , ϕ = 90o .

[4+6+6]

6. (a) A block of iron (µ = 5000 µo ) is placed in a uniform magnetic field of 1.5 T. If the iron consists of 8.5*1028 atoms/m3 , calculate the Magnetization M and the average magnetic current. (b) Define Magnetization, how it differs from the Polarization. What is unit of Magnetization?[8+8] 7. If A = 10ρ1.5 az Wb/m in free space, find H, J.

[16]

8. A co-axial capacitor has the parameters a=5 mm, b = 30mm, I = 20 cm, εr = 8, and σ = 10 - 6 Siemens/m. If the conduction current density in the capacitor is (2/ρ) sin 106 t aρ A/m2 , find (a) The total conduction current through the capacitor. (b) The Maximum value of the displacement current density. (c) The total displacement current. ?????

[6+6+4]

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Code No: R5210205

II B.Tech I Semester(R05) Supplementary Examinations, May/June 2009 ELECTROMAGNETIC FIELDS (Common to Electrical & Electronics Engineering and Electronics & Control Engineering) Time: 3 hours Max Marks: 80 Answer any FIVE Questions All Questions carry equal marks ????? 1. (a) State and explain Coulomb’s law for the vector force between two point charges in free space. (b) A straight line of length ` m in free space has a uniform charge density of λ C/m. P is a point on the perpendicular bisector of the line charge, at a distance y m from the line. Find the electric field at P. [6+10] 2. (a)

i. Define capacitance. Express its units in 2 different ways. ii. As per the usual definition, show that a capacitance is always positive. iii. Sometimes, capacitance of a single conductor is referred to. What does this mean? (b) Two large parallel conducting plates are separated by a distance 5 mm in air. The +ve and -ve terminals of a 100 V battery are connected, one terminal to each of the plates. The battery is then removed, and a plane conducting sheet, 2 mm thick, is introduced between the two plates parallel to them and without touching them. i. Find the original potential difference between the plates. ii. After the introduction of the conducting sheet will the potential difference between the two outer plates increase or decrease? Explain. iii. Find the new potential potential difference between the two outer plates. [8+8]

3. (a) Establish the electrostatic boundary conditions for the tangential components of electric field and electric displacement at the boundary of two linear dielectrics. (b) z<0 is a region of a linear dielectric of relative permittivity 6.5; and z>0 is free space. Electric field in the free space region is (-3ax + 4ay -2az ) V/m. Find i. D for z > 0; ii. tangential components of D & E for z < 0, on the boundary.

[8+8]

4. (a) State Biot - Savart’s law for the magnetic field B due to a steady line current in free space. (b) Find B due to a straight conductor length ` m and steady current I A at a distance of y m from the center of the line current. [6+10] 5. Obtain the expression for the magnetic field intensity and sketch the same for a co-axial cable. [16] 6. (a) What is magnetic dipole? How it differs from electric dipole? (b) In a certain region of space B = 0.1xax - 0.2yay - 0.3az T. Find the total force on a rectangular loop carrying a current of 20 A, lying in z = 0 plane and bounded by 1 < x < 3 & 2 < y < 5 m. [6+10] 7. (a) Obtain the expression for coefficient of coupling between two coils. (b) A coil of 10 mH is magnetically coupled to another coil of 600 µH. The coefficient of coupling between two coils is 0.15. Calculate the inductance if these two coils are connected in parallel addition and parallel opposition. [8+8] 8. Define displacement current density? Find the displacement current density flowing across an arbitrary intermediate surface at any radius ρ, between the two surfaces of a co-axial capacitor having inner radius ’a’ and outer radius ’b’ and the permittivity of dielectric is ε. Assume voltage varies sinusoidally V=Vm sin ωt. [16] ?????