A point charge q is located at the point (??, ?', z') inside a grounded cylindrical box defined by the surfaces z = 0, z = L, ? = a. Show that the potential inside the box can be expressed in the following alternative forms:

Discuss the relation of the last expansion (with its extra summation) to the other two.

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х mnр Xmnp eim( -4)Jm *..Σ ΣΣ Ф(х, х') %3D XmnL TEnl m -00 n1 Xmmm+1(xmn) sinh X mn Xmn X sinh (L - 2,) sinh 1 πρ. E E elm( - d) sin TE,L sin d(x, x') %3D L. п па Im L. m=- n=1 η πρ. L. п та ηπρ Km Im kmz' sin Xmnp kTz X mnP eiml o -6') sin 29 ΣΣ - 2. (7) |-.t. Ф(х, х") km Pm+1(Xmn) TELA? Xmn m=-00 k-1 n-1