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An infinite dielectric sheet having char...

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.

Text Solution

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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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Knowledge Check

  • There is a uniformly charged ring having radius R. An infinite line charge (charge per unit length lambda) is placed along a diameter of the ring (in gravity free space). Total charge on the ring Q=4sqrt(2lambda)R . An electron of mass m is released from rest on the axis of the ring at a distance x=sqrt(3) R from the centre. The distance from centre of ring on the axis where the net force on the electron is zero.

    A
    2R
    B
    `sqrt(2)`R
    C
    R
    D
    none of these

  • There is a uniformly charged ring having radius R. An infinite line charge (charge per unit length lambda) is placed along a diameter of the ring (in gravity free space). Total charge on the ring Q=4sqrt(2lambda)R . An electron of mass m is released from rest on the axis of the ring at a distance x=sqrt(3) R from the centre. Magnitude of initial acceleration of the electron.

    A
    `(elambda)/(pivarepsilon_(0)mR)((3-2sqrt(2))/(4sqrt(6)))`
    B
    `(elambda)/(pivarepsilon_(0)mR) ((3+2sqrt(2))/(4sqrt(6)))`
    C
    `(elambda)/(pivarepsilon_(0)mR)((3+2sqrt(2))/(4sqrt(3)))`
    D
    none

  • A sphere of radius R has a charge density sigma . The electric intensity at a point at a distance r from its centre is

    A
    `E=(sigmaR^(2))/(in_(0)kr^(2))`
    B
    `E=(sigma)/(in_(0)k)`
    C
    `E=R^(2)`
    D
    `E=in_(0)kr^(2)sigma^(2)`