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# hw04 - you can assume that the magnetic ﬁeld generated by...

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Princeton University—Physics 501 Problem Set 4 Due: Oct. 18, 2011 1) A long right cylinder of radius a and permeability μ is oriented with its symmetry axis perpendicular to a uniform field ~ B 0 . Calculate the ~ B field everywhere in space. ( Hint: Since there are no currents, ~ ∇× ~ H = 0 so that one can define a magnetic potential Φ M . You can make an argument that Φ M satisfies LaPlace’s equation.) 2) A solid conducting sphere of radius a rotates with angular velocity ω along an axis parallel to a uniform magnetic field ~ B . Calculate the induced surface charge density, and the charge induced in the sphere. ( Hint: Consider the force experienced by a free charge in the sphere, which will be the sum of the Coulomb force and the Lorentz force. The charge in the sphere will rearrange itself until these forces cancel one another. Also,
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Unformatted text preview: you can assume that the magnetic ﬁeld generated by these moving charges is negligible.) 3) (J5.3) A right circular solenoid of ﬁnite length L and radius a has N turns per unit length and carries a current I . Show that the magnetic induction on the cylinder axis in the limit NL → ∞ is B z = μ NI 2 (cos θ 1 + cos θ 2 ) where the angles are deﬁned in the ﬁgure. 2 θ θ 1 4) (J5.13) A sphere of radius a carries a uniform surface-charge distribution σ . The sphere is rotated about a diameter with constant angular speed ω . Find the vector potential and magnetic-ﬂux density both inside and outside the sphere. Hint: Decompose the Y ‘m ’s into e.g., (ˆ y cos φ-ˆ x sin φ )sin θ ∼-i 2 [ˆ x ( Y 1 , 1 + Y 1 ,-1 ) + i ˆ y ( Y 1 , 1 + Y 1 ,-1 )]...
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