1 Consider the annulus S in the plane z = 8.3m, defined by a <+ <, a = 3.3 m, b 4.4m. The region surrounding S is filled with a conducting medium, where the current density is given as follows in spherical coordinates: ao A/m? r sin(0) Find the net upward current through S. ANSWER: I 4pi(4.4^2-3.3^2)
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- An iron wire with cross-section 2.6 x 10-6 m² carries current 115 A. Find the drift velocity Va, in units of milimeters per second, by assuming the existence of two carriers per iron atom. Hint: First, calculate number density n of the charge carriers. Answer:longitudinally-uniform, and axially-symmetric 2. The current distribution of an infinite, wire can be described in cylindrical coordinates by J = J(p)ź. (a) Show that · ƒ = 0. (b) Considerations of longitudinal and axial symmetry require that the mag- netic field can only depend upon p, i.e. that = Ẻ(p) = B₂(p)ô+ Bø(p)❖ + B₂(p)ź. Use Ampère's Law to determine Bo(p) in terms of I(p) = 2π ff J(p') p'dp'. (c) Use the Biot and Savart Law to show that Bp(p) = B₂ (p) = 0. Side note: it is pretty easy to show that Bp(p) must be zero using Gauss's Law for B with a cylindrical volume. I am not aware of an "easy" way to see B(p) 0 not that it is very difficult using the Biot and Savart Law... (d) Use the integral form for the vector potential [Jackson, Eq. (5.32)] to determine A for this current distribution. Hint: in order to deal with divergent integrals, you may want to limit the current distribution to -L≤ zThe figure below represents a section of a circular conductor of nonuniform diameter carrying a current of I = 5.60 A. The radius of cross-section A1 is r1 = 0.330 cm. (a) What is the magnitude of the current density across A1? A/m2 The radius r2 at A2 is larger than the radius r1 at A1. (b) Is the current at A2 larger, smaller, or the same? The current is larger.The current is smaller. The current is the same. (c) Is the current density at A2 larger, smaller, or the same? The current density is larger.The current density is smaller. The current density is the same. Assume A2 = 2A1. (d) Specify the radius at A2. mm(e) Specify the current at A2. A(f) Specify the current density at A2. A/m2A wire of circular cross-section carries current density that is not uniform but varies with distance from the center as j(r)=j0(1-(r/R)2), for radius r in the range 0 < r < R. Here, j0 is a constant with units amperes per square meter, and the radius of the wire is R = 0.49 mm. A) Find an expression for the current enclosed in a cylinder with a radius of r < R. B) If the total current in the wire is I, find an expression for the constant j0, in terms of the other variables in the problem. C) If the total current is 1.5 A, what is the constant j0, in amperes per square meter? D) Find an expression for the magnetic field inside the wire, r < R, in terms of the current I. E) Find an expression for the magnetic field outside of the wire, for r > R. F) For what r, in meters, is the current enclosed maximum? G) What is the maximum value of the enclosed current, in amperes? H) For what r, in meters, is the magnetic field maximized? I) What is the maximum value of…5. Consider two (1 and 2) wires both having constant circular cross sectional areas: wire 1 has diameter = d, ; wire 2 has diameter = d. Potential differences are applied across each wire so that they both carry current. The current in wire 1 is I, and in wire 2 it is I, ; the the magnitude of the current density in wire 1 is J, and in wire 2 it is J,. If I,= I, and J, = (2) J, then wire diameters must be such that d, = (?) d. a. 2/2 b. V212 c. 2 d. V2 e. V214Consider a hollow spherical conductor of inner radius a and outer radius 3a. The conductivity (sigma) is a function of r, measured from the center of the sphere: sigma = c / r, where a < r < 3a and c is a constant. a) If we apply a potential difference V between it's inner and outer surfaces, with the inner surface at a higher potential, find the resulting current density j and the electric field E as a function of r. b) Find the resistance R of the spherical resistor