EKT 241: ELECTROMAGNETIC THEORY SEM 2 2010/2011 TUTORIAL 3 1. In a cylindrical coordinate system, a 2-m-long straight wire carrying a current of 5 A in the positive z-direction is located at r 4 cm , and - 1 m z 1 m . If 2 B rˆ 0.2 cos (T) , what is the magnetic force acting on the wire? Answer: 2. Two infinitely long, parallel wires carry 6A current in opposite directions. Calculate the magnetic flux density at point P in Figure 1. Figure 1 Answer: 3. Explain the meaning of self-inductance and mutual inductance by including diagrams and equations in your explanation. Answer: Self-inductance, L of any conducting structure is the ratio of the magnetic flux linkage, Λ to the current I flowing through the structure. L Where; I H N2 N IS (Wb) Hence, l L N2 S solenoid l Mutual inductance – produced by magnetic coupling between two different conducting structures. Magnetic field B1 generated by current I1 results in a flux Φ12 through loop 2: 12 B1 dS S2 If loop 2 consists of N2 turns all coupled by B1 in exactly the same way, the total magnetic flux linkage through loop 2 due to B1 is: 12 N 212 N 2 B1 dS S2 Hence, the mutual inductance: L12 12 N 2 B1 ds H I1 I1 s2 1.5 10 6 cos a r A/m exist in free space. Find the r magnetic flux crossing the surface defined by 0 / 3 and 0 z 3 m . 4. A radial magnetic field, H Answer: B ds since B 0 H 0 H ds 1.5 10 6 0 z 0 0 r 3 0 3 z 0 /3 cos a r rddza r 1.5 10 cos ddz /3 6 0 0 1.5 10 6 sin 0 z 30 4 10 7 1.5 10 6 0.866 3 /3 4.9 Webers 5. A long cylindrical conductor whose axis is coincident with the z-axis has a radius a and carries a current characterized by a current density J zˆ J 0 / r , where J 0 is a constant and r is the radial distance from the cylinder’s axis. Derive an expression for the magnetic field H for the regions 0 r a and r a . 6. The rectangular loop shown in Figure 2 is coplanar with the inifinitely long, straight wire carrying the current I = 20 A. Determine the magnetic flux through the loop. Figure 2 Answer: The magnetic field due to the wire: I H a 2r B 0H I 4 10 7 a 2r 20 4 10 7 a 2r 4 10 6 a r The magnetic flux through the loop: Β dS S The differential area of the loop: dS drdza Β dS S 4 10 6 a drdza z 0.05 r 0.05 r 0.2 4 10 6 0.3 ln 0.05 0.35 0.2 1.66 10 6 Wb 7. A coaxial cable has inner conductor of radius a = 5 mm and outer conductor of radius b = 25 mm. The center of the coaxial cable lies on the z axis and a dc current I flow in the center conductor in the +az direction. The region between the conductors contains a magnetic material with r 2.5 located at a r 3a and air located at 3a r 5a . Find 60 A / m at r 10 mm . B everywhere between the conductors if H Answer: First, we know that H I 2r from which we construct: I 60 2 2 (10 ) Hence, I 1.2 A Since the interface between the two media lies in the aφ direction and H is in aφ direction, H is continuous across the two media. (Note: the unit of r is in milimeter) In the region 5 r 25 1.2 a 2r 0.6 a kA / m r H In the magnetic material, 5 r 15 B H ( 2.5)( 4x10 7 )(1.2) a 2r 0 .6 a mT r In the region of air, 15 r 25 B 0 H (4x10 7 )(1.2) a 2r 0.24 a mT r (Note: the unit of r is in milimeter)