2. Determine which of the following potential field distributions satisfy Laplace's equation. (a) V₁ = x² + y² - 2z² + 10 1 (x² + y² + z²)¹/2 (b) V₂ = (c) V3 = pz sino + p²

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 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.

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]

A conductive bar (green) rests on two conductive rails (blue) and is driven with a current source with current I. A magnetic flux of B = Boâz is applied evenly to the region. a. Write an expression for di. (Assume that the conductive bar is a thin wire, not a cylinder.) b. Write an expression for the differential force on a small piece of the conductive bar. c. Write an expression for the total force on the whole conductive bar.

Problem 4 - Review of surface integral, definition of flux, Gausses' Law. 10 pts. For this problem, consider a right-hand coordinate system wherein the x-axis is directed to the right on your paper, the y-axis is directed toward the top of your paper, and the z- axis is pointing out of your paper. A winding with 10 turns is wound around the rectangle with corners (x,y,z) = (0,2,1),(0,2,3), (2,2,3), and (2,2,1). The winding is wound such that the direction for positive flux is in the same direction as the positive y-axis. Now suppose that within the rectangle B = (1+x²yz²)a, cos(100t) T where a, is the unit vector is the y direction, and t represents time. The winding (coil) is open circuited. Using the definition of flux, flux linkage, and Faraday's law determine the voltage that would be measured across the two leads of the winding at t= π/200.

Problem 2 - Review of line-integral and Ampere's law. A conductor with a rectangular cross section comes out of the page in the direction of the positive z-axis. On the plane z=0 the corners of the conductor are denoted (x,y) at (0,0), (2,0), (2,1),(0,1). On this plane z=0 the field intensity may be expressed H=(1+x²y³)a +(1+y)a, How much current is flowing in the positive as direction in the conductor?

(Cooordinate systems, unit vectors, Integration Elements) Find the distance between the points P(R=1,

Problem 1 - Review of line-integral and application to MMF drop. Consider a Cartesian co-ordinate system (x,y,z). Suppose a uniform H-field of (1+k)ax -3ay-Fa: A/m is present. Calculate the MMF drop from the point (1,1,1) to the point (5,-4,-1).

(Introduction, Units, Vectors) (Use the vectors listed in Table 1 for these problems.)