There is a fixed semicircular ring of radius`R` lying in `yz` plane ,with centre at origin and it is uniformly charged with charge `+Q` A pipe is fixed along x-axis from the origin ,The inner surface is pipe is smooth and is made of insultaing material .A small ball of charge `+q` And mass `m` is projected in the pipe with negligible velocity ,ball can move in the pipe whole arrangement is in gravity free space.

The kinetic energy of the particle when the acceleration is maximum is

Options

A. `(1)/(4piepsilon_(0))(Qq)/(r )(1-sqrt((2)/(3)))`
B. `(1)/(4piepsilon_(0))(Qq)/(R )(1)/(2)`
C. `(1)/(4piepsilon_(0))(Qq)/(r )(1-(1)/(sqrt(3)))`
D. `(1)/(4piepsilon_(0))(Qq)/(r )(1+(1)/(sqrt(3)))`

Answer

A

Explanation

`dF_(x)=q(dE_(x))=q(1)/(4piepsilon_(0))(dQ)/((R^(2)+x^(2)))costheta`
`rArr F_(x)=(1)/(4piepsilon_(0))(Qqx)/((R^(2)+x^(2))^(3//2))`
`a=(1)/(4piepsilon_(0)m)(Qqx)/((R^(2)+x^(2))^(3//2))`
`(d)/(dx)[(x)/((R^(2)+x^(2))^(3//2))]=0`
`rArr (1)/((R^(2)+x^(2))^(3//2))(1)=x(-3)/(2)(R^(2)+x^(2))^(-5//2)(2x)=0`
`rArr (1)/(R^(2)+x^(2))^(3//2)=(3x^(2))/((R^(2)+x^(2))^(5//2)) rArr R^(2)+x^(2)=3x^(2)`
`rArr a_("max")=(1)/(4piepsilon_(0))(QqR//sqrt(2))/((R^(2)+(R^(2))/(2))^(3//2))`
`=(1)/(4piepsilon_(0)mR^(2))(Qq(2sqrt(2)))/(sqrt(2)3sqrt(3))=(Qq)/(6sqrt(3)piepsilon_(0)mR^(2))`
Applying mecharnical energy cosnervation
`(1)/(4piepsilon_(0))(Qq)/(R)=(1)/(4piepsilon_(0))(Qq)/(sqrt(R^(2)+x^(2)))+(1)/(2)mv^(2)`
`KE=(1)/(2)mv^(2)=(1)/(4pepsilon_(0))(Qq)/(R)(1-sqrt((2)/(3)))`