Find the potential in the region R in the first quadrant of the z plane bounded by the axes (having potential U 1 ) and the hyperbola y 1 x (having potential U 2 ) by mapping R onto a suitable infinite strip Show that is harmonic What are its boundary values

Question: Find the potential in the region R in the first quadrant of the z-plane bounded by the axes (having potential U 1 ) and

Find the potential Φ in the region R in the first quadrant of the z-plane bounded by the axes (having potential U₁) and the hyperbola y = 1/x (having potential U₂) by mapping R onto a suitable infinite strip. Show that Φ is harmonic. What are its boundary values?

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The hyperbola y 1x has the property that if a and b are two points on the hyperbola then ab is less than or equal to 1 This implies that and therefore by substitution this can be written as This is essentially a linear equation in terms of only three independent variables t x and y Furthermore it can be shown that if 1 exists then 2 is also necessarily true That is if is harmonic then xy 0 With this in hand it is possible to see that xy 0 yields two solutions one positive and one negative for t Because the region R is bounded by the axes and the hyperbola the ... View full answer

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