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Find the fringing field of a semi-infinite parallel plane capacitor using a conformal map

`w=f(z)=(a)/(2\pi )(1+z+e^(z))`

where

`z=x+iy`

and

`w=u+iv`

. (a) Show that

`f`

maps the plates of the infinite plate capacitor in the

`z`

-plane into the plates of the semi-infite-plane capacitor in the

`w`

-plane. (b) What is the complex potential

`g(z)=\phi (x,y)+i\psi (x,y)`

of the infinite-plane capacitor? (c) The complex potential of the semi-infinite-plane capacitor is

`h(w)=\phi ^(')(u,v)+i\psi ^(')(u,v)=g(f^(-1)(w))`

where, unfortunately, we cannot cannot compute the inverse

`z=f^(-1)(w)`

exactly. Show that the space between the plates of the semi-infinite capacitor (away from the edge

`u=0`

) corresponds to the limit

`x->-\infty `

, find

`f^(-1)(w)`

approximately, and thus determine the potential

`\phi ^(')`

and the electric field

`E^(')`

inside the semi-infinite-plane capacitor. (d) Show that the space outside the conductor (away from the edge

`u=0`

) corresponds to the limit

`x->\infty `

, find

`f^(-1)(w)`

approximately, and thus determine the potential

`\phi ^(')`

and the electric field

`E^(')`

outside the semi-infiniteplane capacitor. (e) Using your results from parts (c) and (d) sketch the equipotentials and the lines of electric field in the

`w`

-plane. For the adventurous among you, these can be computed exactly in parametric form. We can easily find that

`u=(a)/(2\pi )(1+x+e^(x)cosy),v=(a)/(2\pi )(y+e^(x)siny)`

To find equipotentials in the

`w`

plane, fix

`y`

and vary

`x`

(this corresponds to equipotentials in the

`z`

plane); to find the field lines in the

`w`

plane, fix

`x`

and vary

`y`

(this corresponds to field lines in the

`z`

plane).