11exam2 - ∇ 2 V r θ = 0(30 points b What are the Frst...

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PHY 3323 October 26, 2011 Exam #2 . . . such names as Laplace, Maxwell and Einstein — do they mean anything to you? . . . Physicists, according to our not-so-reliable historians. Responsible for the rapid rise of the European-American culture . . . A Canticle for Leibowitz (1) A point charge q of mass m is released from rest at a distance d from an inFnite grounded conducting plane. How long Δ t will it take for the charge to hit the plane? a) Dimensional analysis gives you how the time Δ t depends upon the parameters q , m , d and ǫ 0 , up to an overall constant. What is this form? (30 points) b) Suppose the charge q is at height z . Use the Method of Images to Fnd the force exerted upon it. (30 points) c) What is the exact formula for Δ t ? (20 points) (2) A charge + Q is distributed uniformly along the z axis from z = a to z = + a . a) ±ind the general expansion of the potential V ( r, θ ) for r > a , up to dimensionless constants, using the symmetry of the charge distribution, dimensional analysis and
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Unformatted text preview: ∇ 2 V ( r, θ ) = 0. (30 points) b) What are the Frst three multipole moments? (30 points) c) What is the leading term in the expansion of the potential for large r ? (20 points) (3) A point dipole v p = p h z is embedded at the center of a sphere of linear dielectric material with radius R and dielectric constant ǫ r . a) The general form of the potential V ( r, θ ), both inside and outside the sphere can be guessed, up to dimensionless constants, using dimensional analysis, the presence of the dipole source and the fact that ∇ 2 V ( r, θ ) = ǫ r ǫ vp · v ∇ δ 3 ( v r ). What is this form? (30 points) b) What are the boundary conditions which apply at the edge of the sphere? (30 points) c) What is V ( r, θ ) both inside and outside the sphere? (20 points)...
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