A ring of radius 0.1m is made out of thin metallic wire of area of cross section `10^(-6)m^(2)`. The ring has a uniform charge of `pi` coulombs. Find the change in the radius of the rig when a charge of `10^(-8)C` is placed at the center of the ring. Young's modulus of hte metal is `2xx10^(11)Nm^(-2)`.

Answer

2.25 mm

Explanation

`q=10^(-8)C, lambda=(Q)/(2 pi r)=(pi)/(2 pi(0.1))=5Cm^(-1)`
Take an are of ring subtending small angle `theta` at the center. Charge on this arc is `dq=lambda r theta`

For net force to be zero on this arc,
`2Tsin((theta)/(2))=F`
or, `2Tsin((theta)/(2))=(kq(lambda r theta))/(r^2)`
or, `T=(kq lambda)/(r)`
`=(9xx10^(9)xx10^(-8)xx5)/(0.1)=4500N`
`Delta r=(Tr)/(AY)=(4500xx0.1)/(10^(-6)xx2xx10^(11))`
`=225xx10^(-5)m=2.25mm`.