Electromagnetic field Homework Help

1. Two infinitely large conducting plates are located at x = 1 and x = 4. The space between them is free space with charge distribution- - nC/m³. Find V at x = 2 if V(1) = -50 V and V(4) = 50 V. бп

3. The cylindrical capacitor whose cross section is in Figure 6.27 has inner and outer radii of 5 mm and 15 mm, respectively. If V(p = 5 mm) = 100 V and V(p = 15 mm) = 0 V, calculate V, E, and D at p = 10 mm and ps on each plate. Take , = 2.0. 100 V 5 mm. 15 mm Figure 6.27 Cylindrical capacitor

4. A conducting sphere of radius 2 cm is surrounded by a concentric conducting sphere of radius 5 cm. If the space between the spheres is filled with sodium chloride (e, = 5.9), calculate the capacitance of the system.

5. In free space, infinite planes y 4 and y = 8 carry charges 20 nC/m² and 30 nC/m², re- spectively. If plane y = 2 is grounded, calculate E at P(0, 0, 0) and Q(-4, 6, 2).

1. A phonograph record of radius R, carrying a uniform surface charge s, is rotating at constant angular velocity w. Find its magnetic dipole moment.

• Measurement of Voltage Standing Wave Ratio (VSWR) In addition to the questions asked in the handout, do the following: 1. Using a Smith Chart, determine the magnitude (absolute value) of the load reflection coefficient and draw its locus on the Smith Chart. 2. Is it possible to determine the load impedance from your Smith Chart? Justify your answer. 3. Suppose that the phase of the load reflection coefficient is -75°, what is the load impedance? Is the load impedance inductive or capacitive? Justify your answer.

Problem 4 A segment of conductor on the z-axis extends from z = 0 to z = h. If this segment conducts current / in the +â, direction, find (Robs) where Robs = (0, y,0).

Problem 1 (30 points) - For the following current distributions, sketch the distribution and find Rai and di(R). (Remember di has a weighting, spatial differential, and direction.) a) (1 point) Loop carrying current I on the xy plane with radius a centered at the origin. Give the answers in cylindrical coordinates. b) (2 points) Loop carrying current I parallel to the xy plane at a height of z=h with radius a centered at the origin. Give the answers in cylindrical coordinates. c) (6 points) Loop carrying current I on the xy plane with radius a centered at the point x = 2a, y = 0. Give the answers in Cartesian coordinates. (Hint: use a parametric circle with 0 ≤t<2n.) d) (6 points) A plate conductor on the xy plane. Current is being injected at a rate of I into the center of the plate. The current spreads out evenly radially from the origin. (Hint: remember that the current crossing each concentric ring is also I. Calculate the surface current density first.) Give the answers in cylindrical coordinates. e) (6 points) A spherical shell of radius a [m] is centered at the origin. The spherical shell is evenly charged with ps. []. The shell is rotating around the z-axis with an angular velocity of wo . Give the answers in spherical coordinates. f) (3 points) A conducting ring defined in cylindrical coordinates by a ≤p ≤ b,0 ≤ 0 < 2ñ, – ≤ z ≤ with a total current I spread evenly throughout the conductor traveling counter-clockwise around the ring. Give the answers in cylindrical coordinates. g) (6 points) The spherical shell from part e) is now a solid sphere with radius a centered at the origin. The sphere is evenly charged with pvo [] and spinning about the z-axis with an angualar velocity of wo[ra]. Give the answers in spherical coordinates.

Problem 6 (10 points) - A 4.0 [cm] wide ribbon of current is centered about the y-axis on the xy plane and has a surface current density J₁ = 2n ây []. Determine the magnetic field intensity at the points: a) Robs = (0,0,2)[cm] b) Robs = (2,2,2)[cm]

Problem 5 (15 points) (based on Problem 3.5 from Wentworth textbook) - An infinite length line with 2.0 [A] current in the +â, direction exists at y = -3.0 [m], z = 4.0 [m]. A second infinite length line with 3.0 [A] current in the +â, direction exists at x = 0 [m], y = 3.0 [m] a) (2 points) Sketch the lines of current in this problem. b) (2 points) Define Rd, which can point to any point on the first wire. c) (2 points) Define di for the first wire. (weight, differential, direction) d) (2 points) Define Raz, which can point to any point on the second wire. e) (2 points) Define di for the second wire (weight, differential, direction) f) (2 points) Define Robs if we only want to find the magnetic field at the origin. g) (3 points) Set up Biot-Savart's law in superposition to find the magnetic field at the origin. Solve the integrals.