1.

An infinite dielectric sheet having charge density `sigma` has a hole of radius `R` in it. An electron is released on the axis of the hole at a distance `sqrt(3)R` from the center. Find the speed with which it crosses the center of the hole.

Answer» Potential function is not defined for infinite conducting sheet and hence to solve this either calculate potential difference or use force equations
Electric field due to infinite dielectric sheet, `E_(1)=(sigma)/(2epsilon_(0))`
Electric field at the axis of a disc of radius `R`.
`E_(2)=(sigma)/(2epsilon_(0))[1-(x)/(sqrt(x^(2)-R^(2)))]`
Resultant electric field `E=E_(1)-E_(2)=(sigma)/(2epsilon_(0))-(x)/(sqrt(x^(2)+R^(2)))`
Force on the direction
`F=-(sigmaex)/(2epsilon_(0)sqrt(x^(2)+R^(2)))`
`mv(dv)/(dx)=-(sigmaex)/(2epsilon_(0)sqrt(x^(2)+R^(2)))`
`m int_(0)^(v)vdv=-(sigmae)/(2epsilon_(0))int_(sqrt(3R))^(0)=(x)/(sqrt(x^(2)+R^(2)))dx`
`m(v^(2))/(2)=-(sigmae)/(2epsilon_(0))[sqrt(x^(2)+R^(2))]_(sqrt(3R))^(0)`
`v=sqrt((sigmaeR)/(mepsilon_(0)))`


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