1. Consider a conformal mapping given by . How are lines parallel to x and y axes map?

so that . For we have , , so that, which are  hyperbola with foci at . For y=b, we have  so that the equation of trajectory is , which represent ellipses with foci at .

2. A semi-circular region is bounded by the x-axis   and a circular arc of radius 1 about the centre. The x – axis is grounded while the circular arc is maintained at a constant potential . Consider a conformal transformation to convert the region into a planar region, solve the Laplace’s equation and obtain an expression for the potential in the semi-circular region.

Consider a mapping . For  the semi-circle of unit radius, , so that,

with . The semi-circle is mapped on to the real  axis since v=0. For and for , . Thus the semicircular part is mapped along the positive real axis from 0 to infinity. 

For the line  is purely imaginary and has a value 0 for x=1 and is for . Thus the real line is mapped along the imaginary axis from the origin to infinity.

For the region bounded by the lines the transformation is given by

which gives,

Since satisfies two dimensional Laplace’s equation, ,subject to boundary condition the solution is given by .The potential as a function of is obtained by substituting for u and v the expressions above.

3. Two semi-infinite conductors make right angles with each other. The horizontally placed conductor is grounded while the vertical one is at a potential , there being a thin insulation.  Find an expression for the potential in the region between the two planes.

The solution of this problem proceeds identically with that for the tutorial problem 3 with the change that the first conformal transformation is . The expression for potential is .