(adsbygoogle = window.adsbygoogle || []).push({}); 1. The problem statement, all variables and given/known data

Find a complete set of conditions on the constants a, b, c, n such that, for Cartesian coordinates (x, y, z), V = axn + byn + czn is a solution of Laplace’s equation ##∇^2V = 0##. A mass filter for charged particles consists of 4 electrodes extended along the z direction, whose surfaces describe hyperbolae in the xy plane: ##x^2 − y^2 = d^2## for one pair, and ##x^2 − y^2 = −d^2## for the other pair, where d is the distance from the axis to the nearest point on an electrode surface. [Each hyperbola has two branches and thus describes two (diagonally opposite) electrodes.] A positive voltage ##V_0## is applied to the pair at ##x^2 − y^2 = d^2##, and ##−V_0## is applied to the other pair. Taking the electrode length along the z axis to be effectively infinite, find the electric potential in the region between the electrodes.

2. Relevant equations

3. The attempt at a solution

So the boundary conditions i get are

##V(x,\sqrt{x^2-d^2},z)=V_0##

##V(x,\sqrt{x^2+d^2},z)=-V_0##

##\nabla^2V=0##

for solutions of the form ##V=\summation a_nx^n+b_ny^n+c_nz^n ## ##c_n=0##

When I use all of these boundary contains i find that we must only have odd powers of x and that

##\summation a_n x^n +b_n(x^2-d^2)^{\frac{n}{2}}=V_0##

##\summation a_n x^n +b_n(x^2+d^2)^{\frac{n}{2}}=-V_0##

but I can't see how to formulate a solution given these as my coefficients will depend on x .

Many thanks

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# Homework Help: Laplacian for hyperbolic plates

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