4321_HW6FA08 - Assume no other charge exists. 6. Find the...

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Physics 4321 Homework Set 6 October 13, 2008 Due: October 20, 2008 1. Show that the radial part Laplace’s equation in spherical coordinates may be written as r 2 R ′′ + 2 rR - l ( l +1) = 0, where R = d R /d r . This differential equation is known as a Cauchy- Euler differential equation. Note that the power of the independent variable multiplying the derivative is the same as the order of the derivative. 2. For problem 1, assume a solution of the form R ( r ) = r p , and obtain the general solution to the differential equation by finding the allowed values of p . 3. Derive P 3 (x) from the Rodrigues formula and show that P 3 (cos θ ) satisfies the angular equation (3.60) for l = 3. 4. The potential at the surface of a sphere of radius R is given by V o = k cos 3 θ . Find the potential inside and outside the sphere. Hint: Express cos 3 in terms of Legendre polynomials using trigonometric identities. Then use the normal methods. 5. For problem 4, calculate the surface charge density σ ( ) on the surface of the sphere.
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Unformatted text preview: Assume no other charge exists. 6. Find the potential outside a charged metal sphere having charge Q and radius R that is placed in an otherwise uniform electric field E = E o z ^ . Hint: Set V = 0 on the equatorial plane far from the sphere. Then use superposition. 7. Solve Laplace’s equation in cylindrical coordinates with no z-dependence. In particular, show that the most general solution is given by V ( s , φ ) = a o + b o ln s + ∑ [ s k ( a k cos k + b k sin k ) + s-k ( c k cos k + d k sin k )], where the summation is from k = 1 to ∞ . Use separation of variables for the radial equation S ( s ), and assume you may write the solution as S = s n . 8. (a) Find the potential outside an infinitely long metal pipe of radius R that is placed at right angles to an initially uniform electric field E = E o x ^ . (b) Find the surface charge density induced on the pipe. Hint: Use the results from problem 7 with V = 0 on the yz plane....
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This note was uploaded on 02/20/2012 for the course PHYS 1101 taught by Professor Lowellwood during the Fall '10 term at University of Houston.

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