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Download Homework No. 07 (Spring 2015) PHYS 420: Electricity and Magnetism II
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Homework No. 07 (Spring 2015) PHYS 420: Electricity and Magnetism II Due date: Monday, 2015 Apr 6, 4.30pm 1. (40 points.) (Based on Problem 5.58, Griffiths 4th edition.) A circular loop of wire carries a charge q. It rotates with angular velocity ω about its axis, say z-axis. (a) Show that the current density generated by this motion is given by q J(r) = ω × r δ(ρ − a)δ(z − 0). 2πa (1) Hint: Use J(r) = ρ(r)v, and v = ω × r for circular motion. (b) Using Z 1 m= d3 r r × J(r). 2 determine the magnetic dipole moment of this loop to be m= qa2 ω. 2 (2) (3) (c) Calculate the angular momentum of the rotating loop to be L = ma2 ω, (4) where m is the mass of the loop. (d) What is the gyromagnetic ratio g of the rotating loop, which is defined by the relation m = gL. 2. (20 points.) A charged spherical shell carries a charge q. It rotates with angular velocity ω about a diameter, say z-axis. (a) Show that the current density generated by this motion is given by q ω × r δ(r − a). J(r) = 4πa2 (5) Hint: Use J(r) = ρ(r)v and v = ω × r for circular motion. (b) Using Z 1 d3 r r × J(r). m= 2 determine the magnetic dipole moment of the rotating sphere to be m= 1 qa2 ω. 3 (6) (7) 3. (30 points.) A steady current I flows down a long cylindrical wire of radius a. The current density in the wire is described by, n > 0, J(r) = ẑ ρ n I (n + 2) θ(a − ρ). 2πa2 a (8) (a) Show that, indeed, Z dS · J(r) = I. (9) S (b) Using Ampere’s law show that the magnetic field inside and outside the cylinder is given by µ0 2I ρ n+2 φ̂ ρ < a, 4π ρ a (10) B(r) = µ0 2I φ̂ ρ > a. 4π ρ (c) Plot the magnitude of the magnetic field as a function of ρ. 2