hw1_p5406_s07

# hw1_p5406_s07 - HW 1 Phys5406 S07 due 1 In class we found...

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Unformatted text preview: HW 1 Phys5406 S07 due 2/1/07 1) In class we found the B ﬁeld on the axis of a circular loop of radius R carrying current I. Now everywhere on the axis there is no current, so that the ﬁeld is derivable from a scalar potential, Φ. Calculate Φ applicable for all points on the axis from knowledge of the ﬁeld. Your answer should contain a seemingly arbitrary integration constant. However, for points very far from the loop Φ reduces to that from only a magnetic moment term. This allows evaluation of the integration constant; what is it? 2) A conducting sphere of radius R has a uniform surface charge density σ . The sphere rotates about a diameter with constant angular velocity ω . Choose the direction of the angular velocity as the z axis. What is the magnetic moment? 3) For the case of question 2), what is B and what is the magnetic scalar potential Φ far from the sphere? 4) An inﬁnitely long cylinder of radius R and axis along ez with permeability µ1 is completely surrounded by material of permeability µ2 . Both permeabilities are constant. If everywhere inside the cylinder there is a constant magnetic ﬁeld B1 = B0 ex , what is B2 and B1 · B2 at the surface of the cylinder as a function of φ, the azimuthal angle? There are no free conduction currents anywhere. 5) A small cylinder of radius R, length L, and uniform magnetization M ex has its axis along ex and center at the origin. A second identical coaxial cylinder is at r = −xex , where x >> R, x >> L. a)If the magnetizations are parallel, what force is exerted on the second cylinder? b)If the magnetizations are anti-parallel, what force is exerted on the second cylinder? c)If the second cylinder is rotated so that the magnetizations are perpendicular, what force is exerted on the second cylinder? 1 ...
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