xˆ z ˆ. A second vector is given by B 2xˆ yˆ 2z ˆ.

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1 Directions for all homework submissions Submit your work on plain-white or engineering paper (not lined notebook paper). Write each problem statement above each solution. Report answers using decimals (3 significant figures, unless exact) and appropriate units. Do not leave fractions, radicals, transcendental numbers (such as, e), or place-holders for physical constants (such as 0, 0) in any answer. Use vector notation that is consistent with your textbook ( xˆ, yˆ, z ˆ ). Draw a box around each answer. Chapter 3 1. A first location is given by P(1, 1, 2). A second location is given by Q(2, 1, 0). Determine the unit vector that points from P to Q. 2. A first vector is given by P 2xˆ yˆ 2z ˆ. A second vector is given by Q 4 xˆ 3 yˆ 2 z ˆ. A third vector is given by R xˆ yˆ 2 z ˆ. Determine P Q R. 3. A first vector is given by A 2 xˆ z ˆ. A second vector is given by B 2xˆ yˆ 2z ˆ. A third vector is given by 2 ˆ 3ˆ ˆ C x y z. Determine A B C. 4. A first vector is given by F 3xˆ 4yˆ z ˆ. A second vector is given by G 2yˆ 5z ˆ. Determine the angle between F and G. Do not use cross product. 5. A first vector is given by J 2xˆ yˆ 3z ˆ. A second vector is given by K 3xˆ 2z ˆ. Determine a vector perpendicular to both J and K which has a magnitude of A first vector is given by A 3yˆ 4z ˆ. A second vector is given by B 4 xˆ 10 yˆ 5 z ˆ. Determine the vector component of A in the direction of B. 7. A vector field is 3 Rˆ 2 θˆ 6 ˆ. Determine the vector component of this field in the +z direction at the point (1, /3, 5 /4). Express the answer in the spherical system. 8. Point E is located at (3, /2, 1). Point F is located at (5, 3 /2, 5). Determine the straight-line distance from E to F.

2 9. A vector field is r cos rˆ sin ˆ. Determine the closed line integral of this field around the path shown in the figure. Do not use Stokes Theorem. 10. A vector field is r cos rˆ zsin. Determine the flux of this field outward from 0 z 1, r 4. Do not use the Divergence Theorem. 2 2 ˆ A scalar field is 10Rsin cos. Determine the magnitude of the gradient of this field at the point (5, 30, 45 ). 12. A vector field is given by r sin rˆ r z z cos zˆ. Are there any surfaces along which this field is solenoidal? If so, which one(s)? 2 ˆ 13. A vector field is given by RRˆ Rcos 2 ˆ. Point P is located at (4, 60, 30 ). Does this vector field circulate around point P? In which direction(s)? Chapter 4(a) 14. A point charge of 15.7 nc exists at (4 m, 0, 0). A line charge of nc/m exists along the y axis. In free space, determine the electric field intensity at (4 m, 0, 3 m). 15. A point charge q is located at point P(0, 4, 0), while a 10 nc charge is uniformly distributed along a semicircular ring as shown in the figure. Determine the value of q such that the electric field at the origin is zero. 16. A volume charge density is zero within R 1m. In the range 1 R 4 m, the charge 2 3 density is 10 R mc m. Outside this range, R 4m, the charge density is zero. Determine the net electric flux crossing through the surface R = 2 m. 17. A volume charge density in free space is 2R nc/m 3 for 0 R 10 m and zero otherwise. Determine the electric field intensity at R = 2 m.

3 18. An electric field intensity is 2xxˆ 6yy ˆ V m. Determine the potential difference from (0, 1 m, 0) to (1 m, 0, 0). 19. An electric field intensity is 2rsin rˆ rcos ˆ V m. Determine the amount of work required to move a charge of +20 C from 1m, 6, 5 m to 2 m, 2, 5 m. Chapter 4(b) A current density is 10zsin r ˆ ma m. Determine the current passing through a cylinder with a radius of 2 m, and with its top at z = 5 m and its bottom at z = 1 m. 21. It is necessary to construct a 1-M resistor from a cylinder that is 2 cm long with a radius of 4 mm. Determine the conductivity of the material throughout the cylinder. 22. A cylindrical wedge of carbon ( = 2.67 x 10 3 S/m) is depicted. Calculate the resistance between the curved faces at r = a and r = b if a = 40 mm, b = 80 mm, h = 62 mm, and 0 = The potential field V 2x yz y z (V) exists in a dielectric medium with a relative permittivity of 2. Calculate the total charge within the unit cube 0 x 3 m, 0 y 3 m, 0 z 3 m. 24. The cylindrical surface r = 10 m separates two homogeneous dielectric regions: region 1, r 10 m, r 1 2 and region 2, r 10 m, r 2 5. If the electric field intensity in region 1 is 10 rˆ 20 ˆ 40 zˆ kv m, determine the electric field intensity in region The boundary between two regions of space is defined by 8x 6z 48 m. The region including the origin is air, where the electric field intensity is 125 xˆ 75 yˆ 50 z ˆ V m. Determine the electric field intensity in the second region, where the dielectric constant is 2. The boundary is charge-free. 26. The upper plate of a parallel-plate capacitor (at z = 2 mm) is maintained at 10 V, and the lower plate (at z = 0 mm) is maintained at 0 V. Determine the potential everywhere between the plates as a function of distance from the lower plate to the upper plate (i.e. as a function of z) using Laplace s equation.

4 27. One semi-infinite conducting plane is located at 0 ; it is held at +50 V. Another conducting plane is located at 4 ; it is held at 50 V. There is a tiny gap between the planes along the z axis. The volume between the planes is occupied by free space. Determine the electric field intensity at 6 and a radius of 1 cm. 28. One infinite conducting plane is located at y = 2.00 cm. Another infinite conducting plane is located at y = cm. Between the planes, the charge density is y μc m. Both planes are charged to 30.0 kv. The material between the planes has a dielectric constant of 3. Determine the potential exactly halfway between the planes An electrostatic potential in free space is 2x 6y V. Determine the electrostatic energy stored in the region 1m x 1m, 1m y 1m, 1m z 1m. 30. Flat metal plates are attached at = 0 and = /2 to the curved bar in the figure. The conductivity of the material within the bar is 2.06 x 10 7 S/m. The dimensions of the bar are as follows: a = 5.0 cm, b = 7.0 cm, t = 1.5 cm. Determine the resistance of the bar, from one metal plate to the other. 31. An infinite line charge with a density of nc/m traces out x 16 m, y 3 m. It is located in free space. A grounded conductor occupies y 2m. Determine the electric field intensity at ( 8 m, 3 m, 7 m). Chapter A loop carries current as shown in the figure. Determine the magnetic field at point P for a = 4 cm, b = 5 cm, = /6, and I = 120 ma. Use the Biot-Savart Law. 33. A thin wire carries 3 A in the +x direction. The wire is 5 mm long. Its center is located at (0, 4 m, 3 m). Determine the magnetic field intensity at the origin. 34. An infinitely-long cylindrical conductor of radius a is placed along the z axis. The current density in the conductor is Jrz ˆ 0 (where J0 is a constant in A/m 3 ). Determine the magnetic field intensity everywhere.

5 35. An infinitely-long coaxial line centered on the z axis carries a current I in the +z direction, uniformly spread across its inner conductor of radius a. It carries a return current I uniformly spread across its outer conductor between radius b and radius b + t (where t is the thickness of the outer conductor). Let I = 9.05 A, a = 2 cm, b = 5 cm, and t = 2 cm. The coaxial line is non-magnetic. Determine the magnetic field intensity at (6 cm, 20, 8 cm). 36. A conducting loop bent into a triangle carries a current of 2 A and is located close to an infinitelylong, straight filament which carries a current of 5 A, as shown in the figure. The loop and filament are in free space. Determine the magnitude of the force on side 3 of the triangular loop due to the filament (only). 37. Three conductors comprise a three-phase transmission line, as shown in the figure. Triangle A-B-C is equilateral. At one point in time, A and B carry 75 A in the +z direction while conductor C carries a return current of 150 A in the z direction. Between the conductors is free space. Determine the force on a 1-m section of conductor C at that time. 38. A semicircular wire loop (in the x-y plane) carries a current of 4 A. The radius of the semicircle is a = 8 cm. The closed loop is 2 exposed to a flux density of 50 y ˆ mwb m. Determine the magnetic torque experienced by the loop. 39. A square loop of current, 2 m on each side, lies in the x-y plane and is centered on the origin. The loop carries 10 A of current, counter-clockwise around the z axis. Describe the motion of this loop if it is inside the magnetic field intensity 378 xˆ 557 z ˆ A m and it is free to move. Assume = The interface between two materials is y = 0. The magnetic field intensity in medium 1, y < 0, with permeability = 3 0, is 40 xˆ 120 yˆ 80 z ˆ ma m. Medium 2, y 0, has permeability = Assuming that no surface current exists at the boundary, calculate the angle between the magnetic field intensity in region 1 and the magnetic field intensity in region 2.

6 41. The boundary between two magnetic media is 12x + 5y = 0. Medium 1 contains all points for which x < 0 and y < 0. The magnetic field intensity in medium 1 is 1521 xˆ 2028 y ˆ A m. The permeability of medium 1 is 7 0. The permeability of medium 2 is Assume that there is no surface current along the boundary. Determine the magnetic field intensity in medium In a region for which = 20 0, the magnetic field intensity is x yz xˆ 10xy z yˆ 15xyz z ˆ A m. Calculate the magnetic energy stored inside 0 x 1m, 0 y 2 m, 1 z 2 m. 43. An infinitely-long thin wire is near a rectangular conducting loop. The rectangular loop has N turns and is parallel to the thin wire. This geometry is drawn, but only one turn is shown. Assume that the wire and the loop are in free space. For a = 51 mm, b = 108 mm, l = 400 mm, N = 5, determine the mutual inductance between the thin wire and the rectangular loop. Chapter A square coil of wire, with sides of 25 cm each, contains 100 turns. A small gap exists between the beginning of the first turn and the end of the last turn. The coil is centered on the origin. Each of the sides of the coil is parallel to the x or the y axis. The coil is 10cos x cos 10 3 t z ˆ Wb m 2. Determine the exposed to the magnetic flux density RMS value of the EMF induced across the gap. 45. A conducting bar slides along two conducting rails as shown in the figure. The magnetic flux density 2 is 0.5 z ˆ Wb m. The distance between the rails is 10 cm. The rod moves with a velocity of 8 x ˆ m s. The resistance is 20. Determine the power absorbed by the resistor. 46. A single-turn loop moves away from a filamentary current along the z axis, as shown in the figure. Assume I = I0t/T. The relevant quantities are I0 = 250 ma, T = 100 ns, a = 3 cm, b = 4 cm, d = 2 cm, = 0, R = 1 k. Determine the voltage induced across the resistor (according to the orientation for V given in the figure).

7 47. A parallel-plate capacitor has an area of 2.8 cm 2 and a gap of 0.2 mm. Air occupies the space between the plates. An AC voltage with a peak of +50 V at 20 MHz is connected across the plates. Determine the maximum value of the displacement current from one plate to the other. Do not use i = C dv/dt. As part of your solution, find E between the plates. 48. A sinusoidal voltage with an amplitude of 100 V is applied across a coaxial capacitor. The length of the capacitor is 6 cm. The insulator within the capacitor has a dielectric constant of 9. The inner radius of the capacitor is 0.5 cm. The outer radius is 1 cm. The frequency of operation is 60 Hz. Determine the RMS value of the displacement current. Chapter In free space, without any charge or current nearby, the magnetic field intensity is 1 cos t 3z ˆ A m. Write a complete expression for the corresponding electric field r intensity. 50. The electric field intensity of a propagating electromagnetic wave 6 25sin 2 10 t 6x z ˆ V m. Determine (a) the period of the wave, is (b) the wavelength, and (c) the wave s velocity (as a vector). 51. The electric field intensity of a plane wave propagating in a non-magnetic medium 8 754cos 10 t 5y z ˆ mv m. Determine the dielectric constant of the medium. is 52. The electric field intensity of a 1-MHz plane wave traveling in the +x direction in air points in the y direction. The peak value of the electric field intensity is 3.77 mv/m and occurs at t = 0, x = 50 m. Write a complete expression for the magnetic field intensity as a function of time and space.

ELECTRO MAGNETIC FIELDS

ELECTRO MAGNETIC FIELDS SET - 1 1. a) State and explain Gauss law in differential form and also list the limitations of Guess law. b) A square sheet defined by -2 x 2m, -2 y 2m lies in the = -2m plane. The charge density on the