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Download Midterm Exam No. 02 (Spring 2015) PHYS 520B: Electromagnetic Theory
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Midterm Exam No. 02 (Spring 2015) PHYS 520B: Electromagnetic Theory Date: 2014 Mar 18 1. (20 points.) The vector potential for a point magnetic moment m is given by A= µ0 m × r . 4π r 3 (1) Verify that the magnetic field due to the point dipole given by B= ∇×A (2) ∇ · B = 0. (3) satisfies 2. (30 points.) Consider a straight wire of radius a carrying current I described using the current density C J(r) = ẑ e−λρ θ(a − ρ), (4) ρ where θ(x) = 1 for x > 1 and zero otherwise. (a) Find C in terms of the current I. (b) Find the magnetic field inside and outside the wire. (c) Plot the magnetic field as a function of ρ. 3. (30 points.) The current density for a wire forming a helix and carrying a steady current I is given by ∞ X 1 z J(r) = n I δ(ρ − a)δ φ − 2π + 2π(m − 1) , (5) ρ L m=−∞ where the direction of the flow of current is described by the vector n = ẑ + 2π a φ̂. L (6) Here a is the radius of the helix and L is the pitch. The coordinates (ρ, φ, z) are the usual cylindrical coordinates. The coordinate φ generates one period of the helix for 0 < φ < 2π, and the sum periodically repeats it. 1 (a) Calculate the flux of current density Z da · J(r) (7) S passing through the surface S of the z = 0 plane. Hint: The area element for the z = 0 plane is da = ẑ dx dy = ẑ ρdρ dφ. (b) Using µ0 B(r) = 4π Z r − r′ d r J(r ) × |r − r′ |3 3 ′ ′ (8) it is possible to determine the magnetic field on the symmetry axis of the helix in terms of the modified Bessel functions, µ0 I 2πa 2πa 2πa B(0, 0, z) = + K1 . (9) K0 ẑ − φ̂ L L L L How is the ẑ-component of the magnetic field related to the magnetic field of a solenoid? (c) Using the fact that the modified Bessel functions for large arguments tends to zero determine the magnetic field on the z-axis in this limit. How well does a helix with pitch L small compared to radius a compare with a solenoid? 2